More with Systems
of Equations
© 2011 Carnegie Learning
In 2008, 4.7
million Americans
went on a rafting
expedition. In Georgia,
outfitters run
whitewater expeditions
for ages 8 and up
on the Chattooga
River.
12.1 Systems of Equations
Using Linear Combinations to Solve a Linear System..........................................................647
12.2What’s for Lunch?
Solving More Systems.................................................655
12.3 Making Decisions
Using the Best Method to Solve a Linear System..........................................................663
12.4Going Green
Using a Graphing Calculator to Solve Linear Systems..................................................669
12.5Beyond the Point of Intersection
Using a Graphing Calculator to Analyze a System..................................................................... 675
645
© 2011 Carnegie Learning
646 • Chapter 12 More with Systems of Equations
Systems of
Equations
Using Linear Combinations
to Solve a Linear System
Learning Goals
Key Term
In this lesson, you will:
 linear combinations
 Write a system of equations to represent a problem
method (elimination)
context.
 Solve a system of equations algebraically using linear
combinations (elimination).
M
orse code is a communication system which allows people to “speak”
with sound. Words are transmitted using short sounds called “dits,” which
are represented in writing as dots, and long sounds, called “dahs,” which are
represented in writing as dashes. The letters of the alphabet and digits each have
© 2011 Carnegie Learning
their own unique collection of dits and dahs:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
0
When you combine these codes, you can produce sentences in Morse code.
Try it out! Communicate with your friends using Morse code.
12.1 Using Linear Combinations to Solve a Linear System • 647
Problem 1 End of Year Trip
There are a total of 324 students enrolled in Armstrong Elementary School. The girls
outnumber the boys by 34.
1. Write an equation in standard form that represents the total number of students at
Armstrong Elementary School. Use x to represent the number of girls, and use y to
represent the number of boys.
2. Write an equation in standard form to represent the number of girls in relationship to
the number of boys.
3. How are these equations the same?
4. How are the equations different?
5. Complete parts (a) through (f) to write and solve a linear system of equations for
a. Write a linear system for this problem situation.
b. Add the two equations together.
c. Solve the resulting equation.
I see how you
can add equations.
(4 + 2) = 6
(4 - 2) = 2
So, if I can add 6 and 2 and
get 8, then that means I
can add (4 + 2) and (4 - 2)
and get 8 also. So,
(4 + 2) + (4 - 2) = 8.
d. Substitute the value of x that you obtained in part (c) into one
of the original equations and solve to determine
the value of y.
648 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
this situation.
e. What is the solution of your linear system?
f. Check your solution algebraically.
6. Check your solution by creating a graph of your linear system on the coordinate plane
shown. Choose your variables, bounds, and intervals. Be sure to label your graph.
Variable Quantity
Lower Bound
Upper Bound
Interval
© 2011 Carnegie Learning
Number of Boys at Armstrong Elementary
y
x
Number of Girls at Armstrong Elementary
7. Interpret the solution of the linear system in this problem situation.
8. What effect did adding the equations together have?
9. Describe how the coefficients of y in the original system are related.
12.1 Using Linear Combinations to Solve a Linear System • 649
Problem 2 Hotel Promotion
The Splash and Stay Resort has an indoor water park, an arcade, a lounge, and
restaurants. The resort is offering two winter specials. The first special is two nights and
four meals for $270, while the second is three nights and eight meals for $435.
1. Write an equation in standard form that represents the first resort special. Let n
represent the cost for one night at the resort, and let m represent the cost for
each meal.
2. Write an equation in standard form that represents the second resort special.
3. How are these equations the same?
5. Multiply each side of the equation that represents
the first resort special by 22. Simplify the equation;
maintain standard form.
650 • Chapter 12 More with Systems of Equations
If I multiply
both sides of an
equation by the same
number, is the equation
still true?
© 2011 Carnegie Learning
4. How are these equations different?
6. Write a linear system of equations using the transformed equation that represents the
first special and the equation that represents the second special.
a. How do the coefficients of the equations in your linear system of equations compare?
b. Add the equations in your linear system together. Then, simplify the result.
What does the result represent?
c. How will you determine the m value of the linear system?
When you
divide a negative
value by -1, you
make it positive.
d. Determine the value of m for the linear system.
© 2011 Carnegie Learning
e. What is the solution of the linear system?
f. Interpret the solution of the linear system in the problem situation.
g. Check your solution algebraically.
12.1 Using Linear Combinations to Solve a Linear System • 651
Problem 3 Linear Combinations
The method you used to solve the linear systems in Problems 1 and 2 is called the linear
combinations method. The linear combinations method is a process to solve a system
of equations by adding two equations to each other, resulting in an equation with one
variable. You can then determine the value of one variable and use it to find the value of
the other variable.
In many cases, one, or both, of the equations in the system must be
multiplied by a constant so that when the equations are added together,
the result is an equation in one variable. This means that the coefficients of
either the term containing x or y must be opposites.
For example, consider this system:
 
4x 1 2y 5 3
​
  ​ ​
​     
5x 1 3y 5 4 ​
You can multiply the first equation by 3 and the second equation by 22 to
produce opposite coefficients for y that will eliminate each other.
Alternatively, you can multiply the first equation by 23 and the second
equation by 2.
1. Multiply the first equation by 3 and multiply the second equation by 22. Then, rewrite
2. Solve the new linear system. Show your work.
652 • Chapter 12 More with Systems of Equations
Did you check
your solution by
substituting the
ordered pair back into the
original equations?
© 2011 Carnegie Learning
each equation.
3. For each linear system shown, describe the first step you would take to solve the
system using the linear combination method. Identify the variable that will be solved
for when you add the equations.
a. 5x 1 2y 5 10 and 3x 1 2y 5 6
b. x 1 3y 5 15 and 5x 1 2y 5 7
© 2011 Carnegie Learning
c. 4x 1 3y 5 12 and 3x 1 2y 5 4
12.1 Using Linear Combinations to Solve a Linear System • 653
4. Solve each system using linear combinations.
 
2x 1 y 5 8
​ ​
​
 
a.​     
3x 2 y 5 7 ​
 
4x 1 3y 5 24
​
   ​ ​
b.​      
3x 1 y 5 22 ​
 
© 2011 Carnegie Learning
3x 1 5y 5 17
​
  ​ ​
c.​      
2x 1 3y 5 11 ​
Be prepared to share your solutions and methods.
654 • Chapter 12 More with Systems of Equations
What’s for Lunch?
Solving More Systems
Learning Goals
In this lesson, you will:
 Write a linear system of equations to represent a problem context.
 Choose the best method to solve a linear system of equations.
S
uppose one cell phone company charges $0.10 per minute for phone calls.
Another company charges $60 per month for unlimited call time. Can you write a
system of equations to compare the two plans? Which one would be best for you?
© 2011 Carnegie Learning
Which one would be best for your parents?
12.2 Solving More Systems • 655
Problem 1 Fundraising
One day each month the Family and Consumer Science classes offer a deli lunch for the
faculty and staff of the school to purchase. The staff has a choice of either a chef salad for
$5.75 or a hoagie for $5. Today the Family and Consumer Science classes sold 85 lunches
for a total of $464. Determine how many chef salads and hoagies were sold.
1. Write an equation in standard form that gives the total number of lunches in terms of
the number of chef salads sold and the number of hoagies sold. Let x represent the
number of chef salads sold, and let y represent the number of hoagies sold.
2. Write an equation in standard form that represents the amount of money collected.
Use the same variables as those used in Question 1.
3. Write a system of linear equations to represent this problem situation.
4. What methods can you use to solve this system of linear
Think about all
the strategies you
have used in previous
lessons.
© 2011 Carnegie Learning
equations?
656 • Chapter 12 More with Systems of Equations
5. Determine the solution of this linear system of equations by using linear combinations
and check your answer.
6. Interpret your solution to the linear system in terms of the problem situation.
Problem 2 More Fundraising
The Jewelry Club is making friendship bracelets with the school colors to sell in the school
store. The bracelets are black and orange and come in two lengths: 5 inches and 7 inches.
The club has enough beads to make a total of 84 bracelets. They have made 49 bracelets,
​ 3 ​ the number of 7-inch bracelets
which represents __
​ 1 ​ the number of 5-inch bracelets plus __
2
4
they plan to make and sell.
1. Write an equation in standard form to represent the total number of bracelets the
Jewelry Club can make out of the beads that they have. Let x represent the number of
© 2011 Carnegie Learning
5-inch bracelets, and let y represent the number of 7-inch bracelets.
2. Write an equation in standard form to represent the number of bracelets the Jewelry
Club has made so far. Use the same variables as those used in Question 1.
12.2 Solving More Systems • 657
3. Write a system of linear equations that represents the problem situation.
4. Karyn says that the first step she would take to solve this system would be to
first multiply the second equation by the least common denominator (LCD) of the
fractions. Is she correct? Explain your reasoning.
5. Rewrite the equation containing fractions as an equivalent equation without fractions.
6. Determine the solution to the system of equations by using linear combinations and
7. Interpret the solution of the linear system in terms of the problem situation.
658 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
check your answer.
8. Solve each linear system using linear combinations. Check all solutions.
__
​ 1 ​  y 5 3
​ 1 ​  x 1 __
 
3   
​2
  ​ ​
b. ​      
3x 1 5y 5 36 ​
© 2011 Carnegie Learning
 
x 1 2y 5 2
​
  
  ​ ​
a. ​      
5x 2 3y 5 229 ​
12.2 Solving More Systems • 659
 
0.6x 1 0.2y 5 2.2
c. ​       
​
  ​ ​
0.5x 2 0.2y 5 1.1 ​
 
© 2011 Carnegie Learning
__
​ 3 ​  y 5 17
​ 1 ​  x 1 __
5   
2
     
  ​ ​
d. ​ ​
3
1
__
__
 ​ y
5
17
​ 
 
 ​ x
1
​ 
 
5
4
​
660 • Chapter 12 More with Systems of Equations
Talk the Talk
You have used three different methods for solving systems of equations: graphing,
substitution, and linear combinations.
1. Describe how to use each method and the characteristics of the system that makes
this method most appropriate.
a. Graphing Method:
b. Substitution Method:
c. Linear Combinations Method:
© 2011 Carnegie Learning
Which
method do
you like best?
Be prepared to share your solutions and methods.
12.2 Solving More Systems • 661
© 2011 Carnegie Learning
662 • Chapter 12 More with Systems of Equations
Making Decisions
Using the Best Method to
Solve a Linear System
Learning Goals
In this lesson, you will:
 Write a linear system of equations to represent a problem context.
 Choose the best method to solve a linear system of equations.
Problem 1 Roller Skating, Here We Come!
The activities director of the Community Center is planning a skating event for all the
students at the local middle school. There are several skating rinks in the area, but the
director does not know which one to use. Skate Fest charges a fee of $200 plus $3 per
skater, while Roller Rama charges $5 per skater.
1. Write an equation that gives the total cost of renting Skate Fest for the skating event
in terms of the number of students attending. Define your variables.
2. Write an equation that gives the total cost of renting Roller Rama for the skating event
© 2011 Carnegie Learning
in terms of the number of students attending. Use the same variables you used in
Question 1.
3. Suppose the activities director anticipates that 50 students will attend.
a. Calculate the total cost of using Skate Fest.
b. Calculate the total cost of using Roller Rama.
12.3 Using the Best Method to Solve a Linear System • 663
7719_CL_C3_CH12_pp645-684.indd 663
14/03/14 2:16 PM
4. Suppose the activities director has $650 to spend on the skating event.
a.Determine the number of students who can attend if the event is held at
Skate Fest.
b.Determine the number of students who can attend if the event is held at
Roller Rama.
6. When is the cost of each skating rink the same? Use an algebraic method to explain
your reasoning.
664 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
5. Write a system of equations to represent this problem situation.
7. Which algebraic method did you use in Question 6? Explain your reasoning.
8. Complete the table of values to show the cost for different numbers of students
attending the event at each rink.
Quantity
Name
Number of
Students
Skate Fest
Roller Rama
Unit
Expression
x
0
25
75
150
© 2011 Carnegie Learning
200
300
9. Is your solution confirmed by the table? Explain your reasoning.
12.3 Using the Best Method to Solve a Linear System • 665
10. Check your solution by creating a graph of your system of equations. First, choose
your bounds and intervals. Be sure to label the graph.
Variable Quantity
Lower Bound
Upper Bound
Interval
A graph is
sometimes not as
exact as I'd
like it to be.
12. Which skating rink would you recommend to the activities
director? Explain your reasoning.
666 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
11. Is your solution confirmed by your graph?
Problem 2 Another Consideration
Super Skates offers the use of the rink for a flat fee of $1000 for an unlimited number
of skaters.
1. Write a linear equation to represent this situation. Then, add the graph of this equation
to the grid in Problem 1, Question 10.
2. Describe when going to Super Skates is a better option than going to Skate Fest or
Roller Rama.
3. The activities director’s final budget is $895, and she has chosen to host the event at
Skate Fest.
a. Write an inequality that represents this situation.
© 2011 Carnegie Learning
b. Solve the inequality.
c.What does this solution mean in terms of the problem situation?
Be prepared to share your solutions and methods.
12.3 Using the Best Method to Solve a Linear System • 667
© 2011 Carnegie Learning
668 • Chapter 12 More with Systems of Equations
Going Green
Using a Graphing Calculator
to Solve Linear Systems
Learning Goal
In this lesson, you will:
 Write a linear system of equations to represent a problem context.
T
he world's first graphing calculator was introduced in October of 1985.
This graphing calculator contained 422 bytes of memory and could calculate to a precision of 13 digits.
Some of the graphing calculators available in 2011 can show even three-dimensional graphs and have 2.5 megabytes of data—over 2.5 million © 2011 Carnegie Learning
bytes! That's over 6000 times more memory than the first graphing calculator!
12.4 Using a Graphing Calculator to Solve Linear Systems • 669
Problem 1 When HEVs Take Over the Market
In 2008, about 16,500 hybrid cars were sold. In 2009, about 20,000 hybrids were sold.
1. What is the rate of change in the number of hybrids sold per year from 2008 to 2009?
2. Write an equation that gives the total number of hybrid cars sold since 2008. Assume
that the rate of change in sales is constant. Define the variables.
3. In 2008, 13.2 million conventional autos were sold in the United States. In 2009, 11.4
million conventional autos were sold. What is the rate of change in the number of
conventional automobiles sold in the United States per year from 2008 to 2009?
4. Write an equation that gives the total number of conventional cars sold since 2008.
© 2011 Carnegie Learning
Assume that the rate of change in sales is constant. Define the variables.
670 • Chapter 12 More with Systems of Equations
5. Write a system of linear equations that represents the total sale of hybrid cars and the
total sale of conventional automobiles since 2008.
6. If these trends were to continue, could the hybrid sales ever equal the conventional
auto sales? Use what you know about the equations of a linear system to explain
your answer.
Problem 2 Using a Graphing Calculator
There are many ways to use a graphing calculator to solve systems. You will complete a
table of values and then graph the system of equations.
© 2011 Carnegie Learning
1. Follow the steps shown to use the table features of your graphing calculator.
Step 1:
Press Y5 and enter the system of equations.
y1 5 16,500 1 3500x
y2 5 13,200,000 2 1,800,000x
12.4 Using a Graphing Calculator to Solve Linear Systems • 671
Step 2:Set up the table. Press TBLSET (press
2nd and press WINDOW ).
TblStart is the start of your table. The
range of your independent values will be
1 to 20, so enter TblStart 5 0.
Tbl is the increment (read “delta table”).
This value tells the table how to count.
Be sure
the “"Auto"” option
is selected for
both “"Indpnt"” and
“"Depend"” in the
TABLE SETUP.
Enter Tbl 5 1. Then, the independent
values in your table will appear as 1, 2, 3, and so
on.
GRAPH ).
Step 3:
Press TABLE (press 2nd
Step 4:
Use the down arrow keys to scroll through the table.
Notice that the dependent values are in scientific
notation.
a.Complete the table of values that shows the sales of hybrid cars
and conventional cars for different numbers of years.
Quantity
Name
Years Since
2008
Hybrid
Sales
Conventional
Cars
Unit
Year
Cars
Cars
Expression
x
1
7
15
20
672 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
5
b.Determine whether the sales of hybrids will ever equal the sales of conventional
cars. Explain your reasoning by using the table of values.
2. Follow the steps shown to graph the system of equations using your graphing
calculator and to determine the point of intersection.
Step 1:
Press Y5 and enter
y1 5 16,500 1 3500x
y2 5 13,200,000 2 1,800,000x
Step 2:
Press WINDOW to set the bounds and intervals for the graph. You can use
the table of values you completed in Question 1 to determine how to set the
window.
Xmin 5 0, Xmax 5 20, Xscl 5 5, Ymin 5 0,
Ymax 5 600,000, and Yscl 5 100,000.
Step 3:
Press GRAPH . The intersection of both lines should be visible. If it is
not, go back and adjust the WINDOW settings.
Step 4:
Determine the point of intersection; you must be able to view the
intersection point. Press CALC (Press 2ND
TRACE ). Press
5 or select 5 : intersect. Move the cursor toward the intersection
point and press ENTER three times. The point of intersection will be
displayed at the bottom of your screen.
© 2011 Carnegie Learning
a. What is the point of intersection?
b.What does this point of intersection mean in terms of the problem situation?
12.4 Using a Graphing Calculator to Solve Linear Systems • 673
c.In what year would the sale of hybrids equal the sale of conventional cars?
d.What can you predict about the sales of conventional cars from your graph?
Talk the Talk
Solve each system using a graphing calculator.
 
y 5 __
​ 1 ​  x 1 9
2
1. ​     
​
​
 ​  
​ 1 ​ 
y 5 __
​ x  ​1 __
2 2 ​
 
6x 1 3y 5 7
2. ​     
​
  ​ ​
3x 1 2y 5 7 ​
What
predictions
can you make
about each solution
by analyzing the
equations first?
 
 
y 5 21.57x 2 12.4
 ​ ​
4. ​       
  
​
y 5 3.65x 1 19.6 ​
Be prepared to share your solutions and methods.
674 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
6x 2 8y 5 16
3. ​      
 ​ ​
​
  
3x 2 4y 5 8 ​
Beyond the Point
of Intersection
Using a Graphing Calculator
to Analyze a System
Learning Goals
In this lesson, you will:
 Write a linear system of equations to represent a problem context.
 Use a graphing calculator to solve a linear system of equations.
Problem 1 What’s the Count?
Cinemaplex, the local movie theater, had a blockbuster weekend! On Friday, 218 people
attended the matinee and 753 people attended the evening shows, bringing in a total of
$8620. Saturday was even more profitable. There were 847 people who attended
the matinee while 1215 people attended the evening movies, bringing in sales of $16,385.
1. Write an equation in standard form that represents each night. Define x as the price of
© 2011 Carnegie Learning
matinee movies, and define y as the price of evening movies.
2. Without solving, interpret the solution to the linear system of equations.
3. Determine the solution of the linear system using your graphing calculator.
a.Rewrite each equation by solving for y. Do not perform the division.
Friday: y1 5
Saturday: y2 5
b. Enter the equations into your graphing calculator.
12.5 Using a Graphing Calculator to Analyze a System • 675
c. What is the point of solution? Explain how you know.
d. Interpret the solution of this linear system of equations.
Problem 2 Job Offers
1. Alex is applying for positions at two different electronic stores in neighboring towns.
The first job offer is a $200 weekly salary plus 5% commission on sales. The second
job offer is a $75 weekly salary plus 10% commission.
a.Write a system of equations that represents the problem situation. Describe
the variables.
b.Without solving the system of linear equations, interpret the solution.
d.Interpret the solution of the system in terms of the problem situation.
676 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
c. Solve the system of equations using your graphing calculator.
It is important to understand the point of intersection in this situation. Let’s use the
graphing calculator to analyze beyond the point of intersection.
2. Determine values for each graph by using the TRACE function on your graphing
calculator. You may have to adjust the WINDOW .
a.Determine how much money Alex would earn at the first store if he sold $3000.
First, set Xmax 5 4000 and Ymax 5 450.
Press GRAPH .
Then, press TRACE .
Move the cursor to y1.
Enter 3000 and press ENTER .
b.Determine how much money Alex would earn at the second store if he sold $3000.
Move the cursor to y2.
Enter 3000 and press ENTER .
c.What is the difference in the weekly pay between stores if Alex sells $3000?
3. What is the difference in pay if he sold $4225 weekly?
Alex’s sales targets for each job would be between $1500 and $3000 weekly. Each
manager told Alex the same thing: “Some weeks are better than others, depending on the
time of year and the new releases of technology.”
© 2011 Carnegie Learning
4. Which job offer would you recommend Alex take? Explain your reasoning.
12.5 Using a Graphing Calculator to Analyze a System • 677
Talk the Talk
Solve each linear system using the substitution method,
linear combinations, or a graphing calculator.
 
y 5 5x 1 12
 ​ ​
​
  
1. ​     
y 5 9x 2 4 ​
Look at the
structure of each
system before you
choose your solution
method.
 
15x 1 3y 5 30
 ​ ​
​
  
2. ​      
8x 2 3y 5 16 ​
 
 
15x 1 28y 5 417
​
  ​ ​
4. ​       
31x 1 23y 5 548 ​
Be prepared to share your solutions and methods.
678 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
4y 5 11 2 3x
​
  
  ​ ​
3. ​      
3x 1 2y 5 25 ​
Chapter 12 Summary
Key Term
 linear combinations method (elimination) (12.1)
Solving a System of Equations Using the Linear
Combinations ­Method
The linear combinations method is a process used to solve a system of equations by
adding two equations to each other so that they result in an equation with one variable.
Then, you can determine the value of one variable and use it to determine the value of the
other variable. In many cases, you may need to multiply one or both of the equations in
the system by a constant so that when the equations are added together, the result is an
equation in one variable. This means that the coefficients of the term containing either x or
y must be opposites.
Example
5x 1 2y 5 16 and 2x 1 6y 5 22
23(5x 1 2y) 5 23(16)
© 2011 Carnegie Learning
2x 1 6y 5 22
215x 2 6y 5 248
5x 1 2y 5 16
2x 1 6y 5 22 5(2) 1 2y 5 16
213x 5 226
_____
5 _____
​ 226 ​
​ 213x ​ 
 
 
2y 5 6
x 5 2
y53
213
213
10 1 2y 5 16
Check:
5(2) 1 2(3)  16
2(2) 1 6(3)  22
10 1 6  16
4 1 18  22
16 5 16
22 5 22
Nice work!
Keep it up!
The solution is (2, 3).
Chapter 12 Summary • 679
Example
__
​ 1 ​ x 1 __
​ 1 ​ y 5 7
2
3
__
​ 1 ​ y 5 1
​ 1 ​ x 2 __
4
9
(
(
)
)
12​ __
​ 1 ​ x 2 __
​ 1 ​ y 5 7 ​​
2
3
​ 1 ​ x 2 __
36​ __
​ 1 ​ y 5 1 ​
4
9
→
→
6x 1 4y 5 84
9x 2 4y 5 36
6x 1 4y 5 84
6x 1 4y 5 84 6(8) 1 4y 5 84
9x 2 4y 5 36 48 1 4y 5 84
15x
5 120
4y 5 36
15x ​ 
​ ____
15
 
 
5 ____
​ 120 ​
15
​4y ​36
​ ___ ​ 5 ___
​   ​​ 
4
4
x 5 8
y59
Check:
1  ​​(8) 2 __
__
​ 1 ​ (8) 1 __
​ 1 ​ (9)  7​ __
​ ​1 ​ (9)  1
2
3
4 1 3  7
4
9
2 2 1  1
7 5 7
1 5 1
The solution is (8, 9).
Writing a Linear System of Equations to Represent
a Problem Context
When two or more linear equations define a relationship between quantities, they form a
system of linear equations. Use the data in the problem to write two related equations.
Then, evaluate the equations by substituting the given value for x or y to determine the
value of the other variable. Interpret the solution in terms of the problem context.
Ling needs to print flyers. Printer A will charge $30 plus $0.50 per flyer. Printer B will
charge $15 plus $1 per flyer. Let x represent the cost per flyer and let y represent the total
cost of the order.
Printer A: y 5 30 1 0.5x
Printer B: y 5 15 1 x
To determine how much each printer will charge to print 300 flyers, you can use
substitution and evaluate each equation.
Printer A: 30 1 0.5(300) 5 180
Printer B: 15 1 300 5 315
Printer A will charge $180 for 300 flyers, and Printer B will charge $315 for 300 flyers.
680 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
Example
Choosing the Best Method to Solve a Linear
System of Equations
Use graphing or substitution to determine the solution of a linear system of equations.
Either graph the equations to determine their point of intersection, or use substitution
to set the equations equal to one another. These methods work well when y is the same
for both equations and the equations are in slope-intercept form. Interpret the solution in
terms of the problem context.
Example
y 5 2x 2 5 and y 5 x 1 1
Substitution: 2x 2 5 5 x 1 1
x56
Graphing:
y
18
16
y5611
14
y57
12
The solution is (6, 7).
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
18
x
Writing and Solving an Inequality to Represent
a Problem Context
© 2011 Carnegie Learning
When a problem situation defines an upper or lower limit for the value of y, such as a
budget, write the linear equation as an inequality rather than an equality. Solve as usual
and interpret the solution in terms of the problem context.
Example
Kyle belongs to a DVD club. He pays $14 per month plus $2 per DVD. If he sets a budget
for himself of $30 per month, you can determine how many DVDs he can buy each month.
14 1 2 x  30
2 x  16
x8
Kyle can buy up to eight DVDs each month.
Chapter 12 Summary • 681
Writing a Linear System of Equations to Represent
a Problem Context
When two or more linear equations define a relationship between quantities, they form a
system of linear equations. Use the data in the problem to write two related equations.
Calculate the slope of each equation by determining the rate of change for each situation.
Example
Two cars are depreciating at different rates. Car A’s value went from $28,000 when it was
sold new in 2007 to $20,000 in 2009. Car B’s value went from $34,000 when it was sold
new in 2007 to $30,000 in 2009.
20,000 2 28,000
  
  
 
 ​ ​5 _______
​ ​ 28000
 ​ 
​ 5 24000; y 5 28,000 2 4000x
Car A: ​ ​ ________________
2009 2 2007
2
30,000 2 34,000
  
  
 
 5 22000; y 5 34,000 2 2000x
 ​ ​5 _______
​ ​ 24000
 ​ ​
Car B: ​ ​ ________________
2009 2 2007
2
Choosing the Best Method to Solve a Linear System
of Equations
When the coefficients in a system of equations are not convenient for using substitution or
linear combinations, you may need to use a graphing calculator to solve. Follow the steps
to graph the system of equations using your graphing calculator, and then determine the
point of intersection.
Example
 
Step 1:
Press Y5 and enter
13
 
 
y1 5 2​ ___ ​ x 1 _____
​ ​25.6
 ​ ​
5
5
​ ​17  ​x 2 ___
y2 5 ___
​ ​82 ​ .
19
19
Step 2:Press WINDOW to set the bounds and intervals for the graph.
Step 3:Press GRAPH . The intersection of both lines should be visible. If it is not, go
back and adjust the WINDOW settings.
Step 4:Determine the point of intersection; you must be able to view the intersection
point. Press CALC . Press 5 or select 5: intersect.
Move the cursor toward the intersection point and press ENTER three times.
The point of intersection will be displayed at the bottom of your screen.
The solution is (2.7, 21.9).
682 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning
13x 1 5y 5 25.6
​ ​      
   ​ ​
 17x 2 19y 5 82 ​
Writing a Linear System of Equations to Represent
a Problem Context
When two or more linear equations define a relationship between quantities, they form a
system of linear equations. Use the data in the problem to write two related equations.
Find the slope of each equation by determining the rate of change for each situation.
Example
Carlos and Carrena work as waiters. Carlos earns $6 per hour plus an average of 20% of
his total bills in tips. Carrena earns $5 per hour plus an average of 23% of her total bills in
tips. Last Saturday, they worked the same number of hours and had the same total bills.
Carlos earned $322 and Carrena earned $357.
6x 1 0.2y 5 322
5x 1 0.23y 5 357
Using a Graphing Calculator to Solve a Linear System
of Equations
When the coefficients in a system of equations are not convenient for using substitution or
linear combinations, you may need to use a graphing calculator to solve. Follow the steps
to graph the system of equations using your graphing calculator and determine the point
of intersection. Input the values greater than and less than the solution to interpret the
problem situation.
Example
A Movers: y 5 0.64x 1 250y is the total spent on a moving van, and x is the
number of miles you need to drive.
© 2011 Carnegie Learning
B Movers: y 5 0.35x 1 453
The solution is (700, 698).
Try 100 miles more or less than the solution.
0.64(600) 1 250 5 634; 0.35(600) 1 453 5 663
0.64(800) 1 250 5 762; 0.35(800) 1 453 5 733
If you are moving more than 700 miles away, B Movers is the better choice. If you are
moving less than 700 miles away, A Movers is the better choice. If you are moving exactly
700 miles, there is no difference between companies.
Chapter 12 Summary • 683
Choosing the Best Method to Solve a Linear System
of Equations
Use substitution, linear combinations, or a graphing calculator to determine the solution
of a linear system of equations. Use substitution to set the equations equal to one another
when y is the same for both equations and the equations are in slope-intercept form.
Interpret the solution in terms of the problem context. Use linear combinations when
the coefficients of like terms are opposites or can be easily made into opposites by
multiplication. Use a graphing calculator when the coefficients in a system of equations
are complex numbers or are difficult to work with using other methods.
Example
 
y 5 4x 1 5
a.​     
​
  
 
​ ​
3x 1 2y 5 43 ​
Use substitution because one equation is in slope-intercept form.
3x 1 2(4x 1 5) 5 43
3x 1 8x 1 10 5 43
11x 5 33
y 5 17
x53
y 5 4(3) 1 5
The solution is (3, 17).
 
2x 1 6y 5 14
b.​     
​
  ​ ​
4x 2 6y 5 10 ​
Use linear combinations because the y-coefficients of both equations are opposites.
2x 1 6y 5 14
2(4) 1 6y 5 14
4x 2 6y 5 10
8 1 6y 5 14
6x 5 24
x 5 4
6y 5 6
y 5 1
The solution is (4, 1).
 
13x 1 5y 5 11
c.​ ​     
  
  ​ ​
3x 2 16y 5 54 ​
Use a graphing calculator because other methods are not easily used.
The solution is (2, 23).
684 • Chapter 12 More with Systems of Equations
© 2011 Carnegie Learning