Transcript System of Equations
Slide 1
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 2
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 3
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 4
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 5
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 6
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 7
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 8
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 9
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 10
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 11
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 12
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 13
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 14
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 15
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 16
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 17
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 18
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 19
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 20
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 21
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 22
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 23
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 24
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 2
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 3
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 4
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 5
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 6
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 7
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 8
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 9
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 10
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 11
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 12
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 13
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 14
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 15
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 16
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 17
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 18
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 19
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 20
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 21
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 22
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 23
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24
Slide 24
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more variables;
any equation of the form
a1 x 1 a 2 x 2
an x n b
for real numbers a1, a2, …, an (not all of which
are 0) and b, is a linear equation or a firstdegree equation in n unknowns.
5.1 - 1
Linear Systems
A set of equations is called a system of
equations. The solutions of a system of
equations must satisfy every equation in the
system. If all the equations in a system are
linear, the system is a system of linear
equations, or a linear system.
5.1 - 2
Linear Systems
The possible graphs of a
linear system in two
unknowns are as follows.
1. The graphs intersect at
exactly one point, which
gives the (single) ordered
pair solution of the system.
The system is consistent
and the equations are
independent.
5.1 - 3
Linear Systems
2. The graphs are
parallel lines, so
there is no solution
and the solution set is
ø. The system is
inconsistent and the
equations are
independent.
5.1 - 4
Linear Systems
3. The graphs are the
same line, and there
is an infinite number
of solutions. The
system is consistent
and the equations
are dependent.
5.1 - 5
Substitution Method
In a system of two equations with two
variables, the substitution method
involves using one equation to find an
expression for one variable in terms of
the other, and then substituting into the
other equation of the system.
5.1 - 6
SOLVING A SYSTEM BY
SUBSTITUTION
Solve the system.
Example 1
3 x 2y 11
(1)
x y 3
(2)
Solution
Begin by solving one of the equations for one of
the variables. We solve equation (2) for y.
x y 3
y x3
(2)
Add x.
5.1 - 7
SOLVING A SYSTEM BY
SUBSTITUTION
Now replace y with x + 3 in equation (1), and
solve for x.
Example 1
Note the careful
use of
parentheses.
3 x 2y 11
3 x 2( x 3 ) 1 1
3 x 2 x 6 11
5 x 6 11
5x 5
(1)
Let y = x + 3 in (1).
Distributive property
Combine terms.
Subtract.
x 1
5.1 - 8
SOLVING A SYSTEM BY
SUBSTITUTION
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the
ordered pair (1, 4). Check this solution in both
equations (1) and (2).
Example 1
Check:
3 x 2y 11
(1)
x y 3
(2)
3(1) 2( 4 ) 1 1
?
1 4 3
?
11 11
True
33
True
Both check; the solution set is {(1, 4)}.
5.1 - 9
Elimination Method
Another way to solve a system of two equations,
called the elimination method, uses
multiplication and addition to eliminate a variable
from one equation. To eliminate a variable, the
coefficients of that variable in the two equations
must be additive inverses. To achieve this, we
use properties of algebra to change the system
to an equivalent system, one with the same
solution set. The three transformations that
produce an equivalent system are listed here.
5.1 - 10
TransformationS of a Linear
System
1. Interchange any two equations of the
system.
2. Multiply or divide any equation of the
system by a nonzero real number.
3. Replace any equation of the system by
the sum of that equation and a multiple of
another equation in the system.
5.1 - 11
Example 2
SOLVING A SYTEM BY
ELIMINATION
Solve the system.
Solution
3x 4y 1
(1)
2 x 3 y 12
(2)
One way to eliminate a variable is to use the
second transformation and multiply both sides of
equation (2) by –3, giving the equivalent system
3x 4y 1
6 x 9 y 36
(1)
Multiply (2) by –3
(3)
5.1 - 12
SOLVING A SYTEM BY
ELIMINATION
Now multiply both sides of equation (1) by 2, and
use the third transformation to add the result to
equation (3), eliminating x. Solve the result for y.
Example 2
6x 8y 2
6 x 9 y 36
17 y 34
y 2
Multiply (1) by 2
(3)
Add.
Solve for y.
5.1 - 13
Example 2
SOLVING A SYTEM BY
ELIMINATION
Substitute 2 for y in either of the original equations
and solve for x.
3x 4y 1
(1)
3 x 4( 2 ) 1
3x 8 1
3x 9
Let y = 2 in (1).
Multiply.
Add 8.
x 3
5.1 - 14
Example 2
SOLVING A SYTEM BY
ELIMINATION
A check shows that
(3, 2) satisfies both
equations (1) and (2);
the solution set is
{(3, 2)}.
5.1 - 15
SOLVING AN INCONSISTENT
SYSTEM
Solve the system.
Example 3
3x 2y 4
(1)
6 x 4 y 7
(2)
Solution
To eliminate the variable x, multiply both sides of
equation (1) by 2.
6x 4y 8
6 x 4 y 7
0 15
Multiply (1) by 2
(2)
False.
5.1 - 16
Example 3
SOLVING AN INCONSISTENT
SYSTEM
Since 0 = 15 is false, the
system is inconsistent
and has no solution. As
suggested here by the
graph, this means that
the graphs of the
equations of the system
never intersect. (The
lines are parallel.) The
solution set is the empty
set.
5.1 - 17
SOLVING A SYSTEM WITH
Example 4
INFINITELY MANY SOLUTIONS
Solve the system.
8 x 2y 4
(1)
4x y 2
(2)
Solution
Divide both sides of equation (1) by 2, and add
the result to equation (2).
4 x y 2
4 x y 2
00
Divide (1) by 2.
(2)
True.
5.1 - 18
Example 4
SOLVING A SYSTEM WITH
INFINITELY MANY SOLUTIONS
The result, is a true statement, which indicates that
the equations of the original system are equivalent.
Any ordered pair that satisfies either equation will
satisfy the system. From equation (2),
4 x y 2
(2)
y 2 4 x.
5.1 - 19
Solving An Applied Problem By
Writing A System of Equations
Step 1 Read the problem carefully until you understand
what is given and what is to be found.
Step 2 Assign variables to represent the unknown values,
using diagrams or tables as needed. Write down
what each variable represents.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system of equations.
Step 5 State the answer to the problem. Does it seem
reasonable?
Step 6 Check the answer in the words of the original
problem.
5.1 - 20
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 21
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 22
Solving Linear Equations with three
Unknowns (Variables)
Some possible intersections of planes representing
three equations in three variables are shown here.
5.1 - 23
Solving a System
To solve a linear system with three unknowns,
first eliminate a variable from any two of the
equations. Then eliminate the same variable
from a different pair of equations. Eliminate a
second variable using the resulting two
equations in two variables to get an equation
with just one variable whose value you can now
determine. Find the values of the remaining
variables by substitution. The solution of the
system is written as an ordered triple.
5.1 - 24