Slide 1

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 2

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 3

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 4

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 5

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 6

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 7

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 8

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 9

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 10

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 11

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 12

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 13

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 14

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 15

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 16

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 17

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 18

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 19

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 20

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 21

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 22

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 23

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 24

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 25

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 26

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 27

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 28

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 29

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 30

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 31

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 32

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 33

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 34

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 35

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 36

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 37

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 38

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross


Slide 39

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 16
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross

next

Solving
Linear
Equations
(Identities and Inconsistencies)
© 2007 Herbert I. Gross

Part 2

next

In this Lesson we will look at cases in
which an equation does not have any
equilibrium points; and at the other
extreme, we will look at cases in which
there are infinitely many equilibrium points.

These cases occur whenever we have a
linear equation in the form…

mx + b = nx + c
… in which m = n.
© 2007 Herbert I. Gross

next

To see how such a case can happen, let's
return to our “candy store” example.
Suppose the first candy store charges
$4 per box and a $3 shipping
and handling charge; while
the second store charges
$4 per box and a $5 shipping
and handling charge. Then, no
Candy
matter how many boxes of
candy you want to buy,
the second store will charge
you $2 more than the first store.
© 2007 Herbert I. Gross

next

Let's now translate this situation
into an equation. The cost of buying x
boxes of candy at the first store, in dollars,
is 4x + 3 while the cost of buying x boxes of
candy at the second store is 4x + 5.
Hence the equilibrium equation is…

4x + 3 = 4x + 5
© 2007 Herbert I. Gross

next

4x + 3 = 4x + 5
In this case there are the same number of
x's on each side of the equation. Hence,
when we subtract 4x from each side, the x
terms will cancel one another. That is, if we
subtract 4x from both sides of the equation,
we obtain the equivalent equation…

4x + 3 = 4x + 5
– 4x
– 4x
3 = 5
© 2007 Herbert I. Gross

next

3 = 5
Notice that the equation above is a
numerical statement that does not
involve a variable.

It is simply a false statement.
In other words, no

matter what
value we choose for x, 3 is never
equal to 5.

© 2007 Herbert I. Gross

next

Caution
Don’t confuse…
… with the equation

3=5
x+3=5

The equation x + 3 = 5 contains an x term,
and we can solve the equation by
subtracting 3 from each side to conclude
that x = 2.
© 2007 Herbert I. Gross

next

In the equation 3 = 5, however, if we
subtract 3 from each side, we wind up
with 0 = 2, which is still a false statement.
One way to interpret 3 = 5 or 0 = 2 is that
the right hand side of the equation will
always be 2 more than the left hand side,
no matter what value we choose for x. In
still other words, x = 2 is a solution
for the equation x + 3 = 5, but there is no
number that is a solution for the
equation 3 = 5.
next

© 2007 Herbert I. Gross

In terms of the present illustration, in the
equation 3 = 5, the right hand side is
always 2 more than the left hand side.
And since the right hand side represents
the second store's price and the left hand
side represents the first store's price, the
above equation tells us that the second
store's price is always $2 more than the
first store's price.
© 2007 Herbert I. Gross

next

At the other extreme, there are equations
for which every value of x is an equilibrium
point. As a trivial example, suppose now
that both candy stores charge $4 per box
for the candy and $3 for shipping and
handling charges. Then no matter how
many boxes of candy you buy, it costs you
the same amount regardless of which store
you use. Again, in terms of a linear
equation, the price each store charges, in
dollars, is 4x + 3.
© 2007 Herbert I. Gross

next

Hence, the equilibrium equation is…
If we subtract 4x from both sides, we get…

4x + 3 = 4x + 3

– 4x

– 4x
3 = 3

which is a true statement no matter what
value we choose for x.
© 2007 Herbert I. Gross

next

Note
Both equations 3 = 5 and 3 = 3 contain
no variables. They are called numerical
equations (or numerical statements). A
numerical equation must either be
always true, or it must be always false.

© 2007 Herbert I. Gross

next

Most of us would recognize that …

4x + 3 = 4x +3
…has every value of x as a solution
because we can see that the two sides of
the equation are identical.
In fact, the equation
4x + 3 = 4x + 3
is called an identity.
© 2007 Herbert I. Gross

next

An
equation
is
called
Definition
an identity, if every
value of the variable is a solution of
the equation.
For example, the commutative property of
arithmetic tells us that for every value of x…

x+1=1+x
The fact that equation is true for every
value of x means that we may refer to
the equation as being an identity. next
© 2007 Herbert I. Gross

However, it often happens in an identity
that the two sides do not look alike until
after we apply one or more rules of
arithmetic. Therefore, we often have to
simplify the linear expressions in the same
way as we did in Lesson 15; the only
difference being that this time we wind up
with a numerical equation rather than an
algebraic equation. Since we essentially
do nothing else new in Lesson 16, we will
limit further remarks to the solutions and
commentary of the problems in the
Lesson 16 exercise set.
next
© 2007 Herbert I. Gross

However, before we do this, let's
summarize the results we've obtained
about linear relationships up until now.

♦ A relationship is called linear if the rate of
change of the output with respect to the
input is a constant. If we denote this
constant by m, the input by x, the output by
y, and the initial value of the output by b
(that is, the value of y when x = 0), every
linear relationship can be written
in the form…
© 2007 Herbert I. Gross

y = mx + b

next

♦ It isn't always obvious to see the
“y = mx + b” form. Often we have to use the
various rules of arithmetic to paraphrase the
linear relationship into this form. If a
relationship cannot be paraphrased into a
“y = mx + b” form, the relationship is
not linear.

♦ In summary, any linear expression (in
which the variable is denoted by x) is an
expression that can be paraphrased into the
“mx + b” form.
next
© 2007 Herbert I. Gross

♦ A linear equation is one in which both
sides of the equation consist of linear
expressions.

♦ In solving a linear equation, one, and only
one, of three things must happen. Namely…
(Case 1) The linear equation is satisfied by
only one value of x.
(Case 2) The linear equation is satisfied by
no value of x.
(Case 3) The linear equation is satisfied by
every value of x.
next
© 2007 Herbert I. Gross

Case 1
Case 1 will happen whenever there are a
different number of x's on one side of the
equation than on the other.
When this happens, we simply subtract the
smaller number of x's from both sides of
the equation, and proceed as before to
solve for the specific value of x.

In other words, Case 1 occurs in the solution
of the equation mx + b = nx + c
whenever m ≠ n.
next
© 2007 Herbert I. Gross

Case 2 and 3
Cases 2 and 3 occur whenever m = n.
In this case, the equation mx + b = nx + c
becomes…
mx + b = mx +c
And if we subtract mx from both sides of
this equation, we get the numerical
statement b = c which is either always true
or always false. In other words, if b = c, the
equation is an identity, and if b ≠ c, the
equation has no solutions (in which case the
equation is called inconsistent).
next
© 2007 Herbert I. Gross

Linear Equations and
Straight lines
Geometric interpretations can often shed
light on discussions that might otherwise
seem rather abstract. This applies to our
discussion of how to solve linear
equations. As a case in point, suppose
we think in terms of the graphs of the
equations rather than in terms of the
equations themselves.
next
© 2007 Herbert I. Gross

For example, in terms of the Cartesian
plane, the linear equation
mx + b = nx + c
determines the point at which the lines
whose equations are y = mx + b and
y = nx + c intersect.
Recall that m and n are the rates of change
of y with respect to x. Therefore, if m ≠ n,
the two equations define lines that have
different directions and hence are not
parallel. In that case, the two lines will
intersect at one and only one point. next
© 2007 Herbert I. Gross

y = 4x + 1
For example,
suppose we
want to solve
the equation
4x +1 = 3x + 2
geometrically.
We could
begin by
drawing the
line y = 4x +1.

(1,5)

m = 4, b = 1

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

y = 3x + 2

We could then
continue by
drawing the
line y = 3x +2.

(1,5)

(0,2)

m = 3, b = 2

(0,0)

next
© 2007 Herbert I. Gross

y = 4x + 1

Drawn on the
same grid, we
see that the
lines intersect
at (1,5).

y = 3x + 2

(1,5)

(0,2)

(0,1)

(0,0)

© 2007 Herbert I. Gross

next

Another Connection
We can interpret the point (1,5) algebraically
in the following way…
Starting with the equation 4x + 1 = 3x + 2,
we subtract 3x from both sides to obtain
x + 1 = 2; from which it follows that x = 1.
When x = 1, both 4x + 1 and 3x + 2 equal 5.
Thus, the point (1,5) may be used as an
abbreviation for saying “x = 1 is the solution
of the equation 4x + 1 = 3x + 2, and in this
case both 4x + 1 and 3x + 2 equal 5”. next
© 2007 Herbert I. Gross

However, if m = n , the two equations define
lines that have the same directions (slope).
This lead to two possibilities…

♦ either the two equations define the same

line; which means every point that satisfies one
equation will also satisfy the other equation.
For example, if we multiply both sides of the
equation x + y = 1 by 2, we obtain the
equivalent equation 2x + 2y = 2. Thus, while
the two equations are different, they define
the same line; namely the line that is
determined by the two points (0,1) and (1,0).
next
© 2007 Herbert I. Gross

♦ the two equations define different but
parallel lines; which means that no point on
one line can also be on the other line.
For example, the lines whose equations are
y = 4x + 1 and y = 4x + 3 are different but
parallel lines. Recall that when we solved
the equation 4x + 1 = 4x + 3, we obtained
the true statement 0 = 2. Geometrically,
this means that the line y = 4x + 3 is
point- by-point 2 units above the line
y = 4x + 1.
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
More
specifically,
we draw the
lines…

…and we see
that the lines
are parallel.

(1,7)

(1,5)
(0,3)

(0,1)

(0,0)

Moreover…
next
© 2007 Herbert I. Gross

y = 4x + 1

y = 4x + 3
(1,7)

the vertical
distance
between the
two lines is
always 2
units.

2

(1,5)
(0,3)

2

(0,1)

(0,0)

next
© 2007 Herbert I. Gross

Points vs. Dots
or
Thickness Matters
In theory a point has no thickness. However,
the “dot” that we use to represent the point
does have thickness. And since we may
think of a line as being generated by a
moving point, it means that any line we draw
also has thickness.
next
© 2007 Herbert I. Gross

What this means is that when we represent
equations by lines, the thickness of the
lines often forces us to estimate the correct
answer rather than to find the exact
answer.
By way of illustration, consider the line L1
whose equation is…
y = 7x + 3
…and suppose we want to find the (exact)
point on L1 whose y-coordinate is 15.
next
© 2007 Herbert I. Gross

Algebraically, we simply replace y by 15 in
the equation y = 7x + 3 and obtain the
equation…
15 = 7x + 3
If we subtract 3 from both sides of this
equation and then divide both sides of the
resulting equation by 7, we see that the
x-coordinate of the point is given exactly
by x = 12/7.
next
© 2007 Herbert I. Gross

However, if we were to rely solely on a
geometric solution, we could begin by…
♦ first drawing the line y = 7x + 3,
♦ and then drawing the line y = 15.
♦ We would then label the point at which
these two lines intersect as P(x,15).
♦ To determine the value of x, we would
then draw the line L2 that passes through
P and is parallel to the y-axis.
♦ The desired value of x is the x-coordinate
of the point at which this line crosses the
next
x-axis.

© 2007 Herbert I. Gross

P
i
c
t
o
r
i
a
l
l
y
© 2007 Herbert I. Gross

y

y = 15

L2

P = (x,15)

y = 7x + 3

L1

x
(?,0)

next

The problem is that the best we can do is
estimate this point and conclude that the
value of x is a “little more” than 1.5.
So while the geometric model is often a
good way to help us internalize what is
happening algebraically, it is the
algebraic method that gives us the exact
answer. However, there are times when an
exact algebraic solution doesn't exist and
in such cases the graph(s) of the
equation(s) is extremely useful in helping
us obtain a reasonable approximation for
next
the exact answer to the problem.

© 2007 Herbert I. Gross

Summary of Lessons 15 and 16
Lest we lose sight of the forest because of
the trees, remember that these two lessons
were concerned with solving linear
equations of the form…
mx + b = nx + c
What we showed (basically by algebra but
illustrated in terms of coordinate geometry)
was that for each such equation, one and
only one of the following solutions
next
occurrs…

© 2007 Herbert I. Gross

♦ If m ≠ n, there is one and only one value
of x for which the equation is a true
statement.

♦ If m = n but b ≠ c, there is no value of x
for which the equation is a true statement.

♦ If m = n and it is also true that b = c, the
expressions on either side of the equal
sign are equivalent and in this case the
equation becomes a true statement for
next
every value of x.

© 2007 Herbert I. Gross