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Algebra 1 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 7-1 Graphing Systems of Equations
Lesson 7-2 Substitution
Lesson 7-3 Elimination Using Addition and Subtraction
Lesson 7-4 Elimination Using Multiplication
Lesson 7-5 Graphing Systems of Inequalities
Example 1 Number of Solutions
Example 2 Solve a System of Equations
Example 3 Write and Solve a System of Equations
Use the graph to determine whether the system has
no solution, one solution, or infinitely many solutions.
Answer: Since the graphs of
and
are parallel, there are no solutions.
Use the graph to determine whether the system has
no solution, one solution, or infinitely many solutions.
Answer: Since the graphs of
are intersecting lines, there is
one solution.
and
Use the graph to determine whether the system has
no solution, one solution, or infinitely many solutions.
Answer: Since the graphs of
coincide, there are infinitely
many solutions.
and
Use the graph to determine whether each system has
no solution, one solution, or infinitely many solutions.
a.
Answer: one
b.
Answer: no solution
c.
Answer: infinitely many
Graph the system of equations.
Then determine whether the
system has no solution, one
solution, or infinitely many
solutions. If the system has
one solution, name it.
Answer:
The graphs of the equations coincide. There are
infinitely many solutions of this system of equations.
Graph the system of equations.
Then determine whether the
system has no solution, one
solution, or infinitely many
solutions. If the system has
one solution, name it.
Answer:
The graphs of the equations are parallel lines. Since
they do not intersect, there are no solutions of this
system of equations.
Graph the system of equations. Then determine
whether the system has no solution, one solution,
or infinitely many solutions. If the system has one
solution, name it.
a.
Answer:
one; (0, 3)
Graph the system of equations. Then determine
whether the system has no solution, one solution,
or infinitely many solutions. If the system has one
solution, name it.
b.
Answer:
no solution
Bicycling Tyler and Pearl went on a 20-kilometer bike
ride that lasted 3 hours. Because there were many
steep hills on the bike ride, they had to walk for most
of the trip. Their walking speed was 4 kilometers per
hour. Their riding speed was 12 kilometers per hour.
How much time did they spend walking?
Words
You have information about the amount of time
spent riding and walking. You also know the
rates and the total distance traveled.
Variables
Let
the number of hours they rode and
the number of hours they walked. Write a
system of equations to represent the situation.
Equations
The number of
the number of
hours riding plus hours walking
r
+
w
the total number of
hours of the trip.
equals
=
The distance
the distance
traveled riding plus traveled walking equals
12r
+
4w
=
3
the total distance
of the trip.
20
Graph the equations
and
.
The graphs appear to intersect at the point with the
coordinates (1, 2). Check this estimate by replacing
r with 1 and w with 2 in each equation.
Check
Answer: Tyler and Pearl walked for 3 hours.
Alex and Amber are both saving
money for a summer vacation.
Alex has already saved $100 and
plans to save $25 per week until
the trip. Amber has $75 and plans
to save $30 per week. In how
many weeks will Alex and Amber
have the same amount of money?
Answer: 5 weeks
number of weeks
amount of money saved
Example 1 Solve Using Substitution
Example 2 Solve for One Variable, Then Substitute
Example 3 Dependent System
Example 4 Write and Solve a System of Equations
Use substitution to solve the system of equations.
Since
substitute 4y for x in the second equation.
Second equation
Simplify.
Combine like terms.
Divide each side by 15.
Simplify.
Use
to find the value of x.
First equation
Simplify.
Answer: The solution is (20, 5).
Use substitution to solve the system of equations.
Answer: (1, 2)
Use substitution to solve the system of equations.
Solve the first equation for y since the coefficient of y is 1.
First equation
Subtract 4x from each side.
Simplify.
Find the value of x by substituting
second equation.
for y in the
Second equation
Distributive Property
Combine like terms.
Add 36 to each side.
Simplify.
Divide each side by 10.
Simplify.
Substitute 5 for x in either equation to find the value of y.
First equation
Simplify.
Subtract 20
from each side.
Answer: The solution is (5, –8).
The graph verifies
the solution.
Use substitution to solve the system of equations.
Answer: (–3, 2)
Use substitution to solve the system of equations.
Solve the second equation for y.
Second equation
Subtract x from each side.
Simplify.
Substitute
for y in the first equation.
First equation
Distributive Property
Simplify.
The statement
is false. This means there are no
solutions of the system of equations. This is true because
the slope-intercept form of both equations show that the
equations have the same slope, but different y-intercepts.
That is, the graphs of the lines are parallel.
Answer: no solution
Use substitution to solve the system of equations.
Answer: infinitely many solutions
Gold Gold is alloyed with
different metals to make it hard
enough to be used in jewelry. The amount of gold
present in a gold alloy is measured in 24ths called
karats. 24-karat gold is
karat gold is
or 100% gold. Similarly, 18-
or 75% gold. How many ounces of 18-
karat gold should be added to an amount of 12-karat
gold to make 4 ounces of 14-karat gold?
Let
the number of ounces of 18-karat gold and
the number of ounces of 12-karat gold. Use the table
to organize the information.
18-karat gold 12-karat gold 14-karat gold
Total Ounces
x
y
4
Ounces of Gold
The system of equations is
and
Use substitution to solve this system.
First equation
Subtract y from each side.
Simplify.
Second equation
Distributive Property
Combine like terms.
Subtract 3 from each side.
Simplify.
Multiply each side by –4.
Simplify.
First equation
Subtract
from each side.
Simplify.
Answer:
ounces of the 18-karat gold and
of the 12-karat gold should be used.
ounces
Chemistry Mikhail needs a 10 milliliters of 25% HCl
(hydrochloric acid) solution for a chemistry
experiment. There is a bottle of 10% HCl solution and
a bottle of 40% HCl solution in the lab. How much of
each solution should he use to obtain the required
amount of 25% HCl solution?
Answer: 5mL of 10% solution, 5mL of 40% solution
Example 1 Elimination Using Addition
Example 2 Write and Solve a System of Equations
Example 3 Elimination Using Subtraction
Example 4 Elimination Using Subtraction
Use elimination to solve the system of equations.
Since the coefficients of the x terms, –3 and 3, are
additive inverses, you can eliminate the x terms by
adding the equations.
Write the equation in column
form and add.
Notice that the x value is eliminated.
Divide each side by –2.
Simplify.
Now substitute –15 for y in either equation to find the
value of x.
First equation
Replace y with –15.
Simplify.
Add 60 to each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The solution is (–24, –15).
Use elimination to solve the system of equations.
Answer: (2, 1)
Four times one number minus three times another
number is 12. Two times the first number added to
three times the second number is 6. Find the numbers.
Let x represent the first number and y represent the
second number.
Four times
one number
4x
Two times
the first number
2x
minus
three times
another number
is
12.
–
3y
=
12
added to
three times
the second number
is
6.
+
3y
=
6
Use elimination to solve the system.
Write the equation in column
form and add.
Notice that the y value is eliminated.
Divide each side by 6.
Simplify.
Now substitute 3 for x in either equation to find the
value of y.
First equation
Replace x with 3.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The numbers are 3 and 0.
Four times one number added to another number is
–10. Three times the first number minus the second
number is –11. Find the numbers.
Answer: –3, 2
Use elimination to solve the system of equations.
Since the coefficients of the x terms, 4 and 4, are the same,
you can eliminate the x terms by subtracting the equations.
Write the equation in column
form and subtract.
Notice that the x value is eliminated.
Divide each side by 5.
Simplify.
Now substitute 2 for y in either equation to find the
value of x.
Second equation
Simplify.
Add 6 to each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: The solution is (6, 2).
Use elimination to solve the system of equations.
Answer: The solution is (2, –6).
Multiple-Choice Test Item
If
A (3, –8)
and
B3
what is the value of y?
C –8
D (–8, 3)
Read the Test Item
You are given a system of equations, and you are
asked to find the value of y.
Solve the Test Item
You can eliminate the y terms by subtracting one equation
from the other.
Write the equation in column
form and subtract.
Notice that the y value is eliminated.
Divide each side by 14.
Simplify.
Now substitute 3 for x in either equation to solve for y.
First equation
Simplify.
Subtract 24 from each side.
Simplify.
Notice that B is the value of x and A is the solution of the
system of equations. However, the question asks for the
value of y.
Answer: C
Multiple-Choice Test Item
If
A4
Answer: D
and
B (4, –4)
what is the value of x?
C (–4, 4)
D –4
Example 1 Multiply One Equation to Eliminate
Example 2 Multiply Both Equations to Eliminate
Example 3 Determine the Best Method
Example 4 Write and Solve a System of Equations
Use elimination to solve the system of equations.
Multiply the first equation by –2 so the coefficients of
the y terms are additive inverses. Then add
the equations.
Multiply by –2.
Add the equations.
Divide each side by –1.
Simplify.
Now substitute 9 for x in either equation to find the
value of y.
First equation
Simplify.
Subtract 18 from each side.
Simplify.
Answer: The solution is (9, 5).
Use elimination to solve the system of equations.
Answer: (5, 1)
Use elimination to solve the system of equations.
Method 1 Eliminate x.
Multiply by 3.
Multiply by –4.
Add the equations.
Divide each side
by 29.
Simplify.
Now substitute 4 for y in either equation to find x.
First equation
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: The solution is (–1, 4).
Method 2 Eliminate y.
Multiply by 5.
Multiply by 3.
Add the equations.
Divide each side
by 29.
Simplify.
Now substitute –1 for x in either equation.
First equation
Simplify.
Add 4 to each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution is (–1, 4), which matches
the result obtained with Method 1.
Use elimination to solve the system of equations.
Answer: (4, –1)
Determine the best method to solve the system of
equations. Then solve the system.
• For an exact solution, an algebraic method is best.
• Since neither the coefficients for x nor the
coefficients for y are the same or additive
inverses, you cannot use elimination using addition
or subtraction.
• Since the coefficient of the x term in the first
equation is 1, you can use the substitution
method. You could also use the elimination
method using multiplication.
The following solution uses substitution.
First equation
Subtract 5y from each side.
Simplify.
Second equation
Distributive Property
Combine like terms.
Subtract 12 from each side.
Simplify.
Divide each side by –22.
Simplify.
First equation
Simplify.
Subtract 5 from each side.
Simplify.
Answer: The solution is (–1, 1).
Determine the best method to solve the system of
equations. Then solve the system.
Answer: The best method to use is elimination
using subtraction because the coefficient of y is
the same in both equations; (3, 5).
Transportation A fishing boat travels 10 miles
downstream in 30 minutes. The return trip takes the
boat 40 minutes. Find the rate of the boat in still water.
Let
the rate of the boat in still water. Let
the rate
of the current. Use the formula rate  time distance,
or
Since the rate is miles per hour, write 30
minutes as hour and 40 minutes as hour.
r
t
d
Downstream
10
Upstream
10
This system cannot easily be solved using substitution.
It cannot be solved by just adding or subtracting
the equations.
The best way to solve this system is to use elimination
using multiplication. Since the problem asks for b,
eliminate c.
Multiply by
.
Multiply by
.
Add the
equations.
Multiply each
side by
Simplify.
Answer: The rate of the boat is 17.5 mph.
Transportation A helicopter
travels 360 miles with the
wind in 3 hours. Te return trip
against the wind takes the
helicopter 4 hours. Find the rate
of the helicopter in still air.
Answer: 102.5 mph
Example 1 Solve by Graphing
Example 2 No Solution
Example 3 Use a System of Inequalities to Solve
a Problem
Example 4 Use a System of Inequalities
Solve the system of inequalities by graphing.
Answer:
The solution includes the
ordered pairs in the intersection
of the graphs of
and
The region is shaded
in green. The graphs
and
are boundaries of
this region. The graph
is dashed and is not included in
the graph of
. The
graph of
is included
in the graph of
Solve the system of inequalities by graphing.
Answer:
Solve the system of inequalities by graphing.
Answer:
The graphs of
and
are parallel lines.
Because the two regions have
no points in common, the system
of inequalities has no solution.
Solve the system of inequalities by graphing.
Answer: Ø
Service A college service organization requires that
its members maintain at least a 3.0 grade point
average, and volunteer at least 10 hours a week.
Graph these requirements.
Words
Variables
The grade point average is at least 3.0. The
number of volunteer hours is at least 10 hours.
If
the grade point average and
the number of volunteer hours, the following
inequalities represent the requirements of the
service organization.
Inequalities The grade point average is at least 3.0.
The number of volunteer hours is at least 10.
Answer:
The solution is the set
of all ordered pairs
whose graphs are in
the intersection of the
graphs of these
inequalities.
The senior class is sponsoring a blood drive.
Anyone who wishes to give blood must be at least
17 years old and weigh at least 110 pounds. Graph
these requirements.
Answer:
Employment Jamil mows grass after school but his
job only pays $3 an hour. He has been offered another
job as a library assistant for $6 per hour. Because of
school, his parents allow him to work 15 hours per
week. How many hours can Jamil mow grass and work
in the library and still make at least $60 per week?
Let
the number of hours spent mowing grass and
the number of hours spent working in the library. Since
g and both represent a number of days, neither can be a
negative number. The following system of inequalities can
be used to represent the conditions of this problem.
The solution is the set of all ordered pairs whose graphs
are in the intersection of the graphs of these inequalities.
Only the portion of the region in the first quadrant is used
since
and
.
Answer:
Any point in the region is
a possible solution. For
example (2, 10) is a
point in the region. Jamil
could mow grass for 2
hours and work in the
library for 10 hours
during the week.
Emily works no more
than 20 hours per week
at two jobs. Her babysitting job pays $3 an
hour and her job as a
cashier at the bookstore
pays $5 per hour. How
many hours can Emily
work at each job to earn
at least $80 per week?
Answer:
number of hours baby sitting
number of hours working as a cashier
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