5.

1 Solving Systems of Linear

Equations by Graphing
Essential Question How can you solve a system of linear
equations?

Writing a System of Linear Equations

Work with a partner. Your family opens a bed-and-breakfast. They spend $600
preparing a bedroom to rent. The cost to your family for food and utilities is
$15 per night. They charge $75 per night to rent the bedroom.
a. Write an equation that represents the costs.

Cost, C
⋅ nights, x
$15 per Number of
= + $600
(in dollars) night

b. Write an equation that represents the revenue (income).

Revenue, R
⋅ nights, x
$75 per Number of
=
(in dollars) night
MODELING WITH c. A set of two (or more) linear equations is called a system of linear equations.
MATHEMATICS Write the system of linear equations for this problem.
To be proficient in math,
you need to identify
important quantities in Using a Table or Graph to Solve a System
real-life problems and
map their relationships Work with a partner. Use the cost and revenue equations from Exploration 1 to
using tools such as determine how many nights your family needs to rent the bedroom before recovering
diagrams, tables, the cost of preparing the bedroom. This is the break-even point.
and graphs.
a. Copy and complete the table.

x (nights) 0 1 2 3 4 5 6 7 8 9 10 11
C (dollars)
R (dollars)

b. How many nights does your family need to rent the bedroom before breaking even?

c. In the same coordinate plane, graph the cost equation and the revenue equation
from Exploration 1.

d. Find the point of intersection of the two graphs. What does this point represent?
How does this compare to the break-even point in part (b)? Explain.

Communicate Your Answer

3. How can you solve a system of linear equations? How can you check your
solution?
4. Solve each system by using a table or sketching a graph. Explain why you chose
each method. Use a graphing calculator to check each solution.
a. y = −4.3x − 1.3 b. y = x c. y = −x − 1
y = 1.7x + 4.7 y = −3x + 8 y = 3x + 5

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 5.1 Lesson What You Will Learn
Check solutions of systems of linear equations.
Solve systems of linear equations by graphing.

Core Vocabul
Vocabulary
larry Use systems of linear equations to solve real-life problems.

system of linear equations,

Systems of Linear Equations
p. 236
solution of a system of linear A system of linear equations is a set of two or more linear equations in the same
equations, p. 236 variables. An example is shown below.
Previous x+y=7 Equation 1
linear equation
2x − 3y = −11 Equation 2
ordered pair
A solution of a system of linear equations in two variables is an ordered pair that is a
solution of each equation in the system.

Checking Solutions

Tell whether the ordered pair is a solution of the system of linear equations.

a. (2, 5); x + y = 7 Equation 1 b. (−2, 0); y = −2x − 4 Equation 1

2x − 3y = −11 Equation 2 y=x+4 Equation 2

SOLUTION
a. Substitute 2 for x and 5 for y in each equation.
Equation 1 Equation 2
x+y=7 2x − 3y = −11
? ?
READING 2+5= 7 2(2) − 3(5) = −11
A system of linear
equations is also called
7=7 ✓ −11 = −11 ✓
a linear system. Because the ordered pair (2, 5) is a solution of each equation, it is a solution of
the linear system.

b. Substitute −2 for x and 0 for y in each equation.

Equation 1 Equation 2
y = −2x − 4 y=x+4
? ?
0 = −2(−2) − 4 0 = −2 + 4

0=0 ✓ 0≠2 ✗
The ordered pair (−2, 0) is a solution of the first equation, but it is not a solution
of the second equation. So, (−2, 0) is not a solution of the linear system.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Tell whether the ordered pair is a solution of the system of linear equations.
2x + y = 0 y = 3x + 1
1. (1, −2); 2. (1, 4);
−x + 2y = 5 y = −x + 5

236 Chapter 5 Solving Systems of Linear Equations

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 Solving Systems of Linear Equations by Graphing
The solution of a system of linear equations is the point of intersection of the graphs of
the equations.

Core Concept
Solving a System of Linear Equations by Graphing
Step 1 Graph each equation in the same coordinate plane.
Step 2 Estimate the point of intersection.
Step 3 Check the point from Step 2 by substituting for x and y in each equation
REMEMBER of the original system.
Note that the linear
equations are in
slope-intercept form. You Solving a System of Linear Equations by Graphing
can use the method
Solve the system of linear equations by graphing.
presented in Section 3.5
to graph the equations. y = −2x + 5 Equation 1
y = 4x − 1 Equation 2

SOLUTION
Step 1 Graph each equation. y

Step 2 Estimate the point of intersection.

The graphs appear to intersect at (1, 3). y = −2x + 5 y = 4x − 1
(1,
1 3)
Step 3 Check your point from Step 2.
2
Equation 1 Equation 2
y = −2x + 5 y = 4x − 1 −4 −2 2 4 x
−1
? ?
3 = −2(1) + 5 3 = 4(1) − 1
3=3 ✓ 3=3 ✓
The solution is (1, 3).

Check
6

y = −2x + 5 y = 4x − 1

−6 6
Intersection
X=1 Y=3
−2

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Solve the system of linear equations by graphing.

1
3. y = x − 2 4. y = —2 x + 3 5. 2x + y = 5
3
y = −x + 4 y= −—2 x −5 3x − 2y = 4

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 Solving Real-Life Problems
Modeling with Mathematics

A roofing contractor buys 30 bundles of shingles and 4 rolls of roofing paper for
$$1040. In a second purchase (at the same prices), the contractor buys 8 bundles of
sshingles for $256. Find the price per bundle of shingles and the price per roll of
rroofing paper.

SOLUTION
S
11. Understand the Problem You know the total price of each purchase and how
many of each item were purchased. You are asked to find the price of each item.
22. Make a Plan Use a verbal model to write a system of linear equations that
represents the problem. Then solve the system of linear equations.
33. Solve the Problem

Words 30 ⋅ Price
bundle
per
+4 ⋅ Price
per roll
= 1040

⋅ bundle ⋅ Price
Price per
8 +0 = 256
per roll

Variables Let x be the price (in dollars) per bundle and let y be the
price (in dollars) per roll.
System 30x + 4y = 1040 Equation 1
8x = 256 Equation 2

Step 1 Graph each equation. Note that only y

the first quadrant is shown because 320
y = −7.5x + 260
x and y must be positive.
240
Step 2 Estimate the point of intersection. The x = 32
graphs appear to intersect at (32, 20). 160

Step 3 Check your point from Step 2.

80
Equation 1 Equation 2
(32, 20)
0
30x + 4y = 1040 8x = 256 0 8 16 24 32 x
? ?
30(32) + 4(20) = 1040 8(32) = 256
1040 = 1040 ✓ 256 = 256 ✓
The solution is (32, 20). So, the price per bundle of shingles is $32, and the
price per roll of roofing paper is $20.
4. Look Back You can use estimation to check that your solution is reasonable.
A bundle of shingles costs about $30. So, 30 bundles of shingles and 4 rolls of
roofing paper (at $20 per roll) cost about 30(30) + 4(20) = $980, and 8 bundles
of shingles costs about 8(30) = $240. These prices are close to the given values,
so the solution seems reasonable.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

6. You have a total of 18 math and science exercises for homework. You have
six more math exercises than science exercises. How many exercises do you
have in each subject?

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 5.1 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check

1. VOCABULARY Do the equations 5y − 2x = 18 and 6x = −4y − 10 form a system of linear
equations? Explain.

2. DIFFERENT WORDS, SAME QUESTION Consider the system of linear equations −4x + 2y = 4
and 4x − y = −6. Which is different? Find “both” answers.

Solve the system of linear equations. Solve each equation for y.

Find the point of intersection Find an ordered pair that is a solution

of the graphs of the equations. of each equation in the system.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, tell whether the ordered pair is In Exercises 13–20, solve the system of linear equations
a solution of the system of linear equations. by graphing. (See Example 2.)
(See Example 1.)
13. y = −x + 7 14. y = −x + 4
x+y=8 x−y=6
3. (2, 6); 4. (8, 2); y=x+1 y = 2x − 8
3x − y = 0 2x − 10y = 4
1 3
y = −7x − 4 15. y = —3 x + 2 16. y = —4 x − 4
5. (−1, 3);
y = 8x + 5 1
y = —23 x + 5 y = −—2 x + 11
y = 2x + 6
6. (−4, −2);
y = −3x − 14 17. 9x + 3y = −3 18. 4x − 4y = 20
2x − y = −4 y = −5
6x + 5y = −7 6x + 3y = 12
7. (−2, 1); 8. (5, −6);
2x − 4y = −8 4x + y = 14 19. x − 4y = −4 20. 3y + 4x = 3
In Exercises 9–12, use the graph to solve the system of −3x − 4y = 12 x + 3y = −6
linear equations. Check your solution.
9. x − y = 4 10. x + y = 5 ERROR ANALYSIS In Exercises 21 and 22, describe
and correct the error in solving the system of linear
4x + y = 1 y − 2x = −4
equations.
y y

✗
21.
4 y The solution of
2 4 x 2
the linear system
−2 2
x − 3y = 6 and
2 x
x −1 2x − 3y = 3
1 4 is (3, −1).

11. 6y + 3x = 18 12. 2x − y = −2
−x + 4y = 24 2x + 4y = 8

✗
y y
22.
y The solution of
4 the linear system
4
4
y = 2x − 1 and
2
2 y=x+1
is x = 2.
−2 2 x
2 4 x
−6 −4 −2 x

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 USING TOOLS In Exercises 23–26, use a graphing 31. COMPARING METHODS Consider the equation
calculator to solve the system of linear equations. x + 2 = 3x − 4.
23. 0.2x + 0.4y = 4 24. −1.6x − 3.2y = −24 a. Solve the equation using algebra.
−0.6x + 0.6y = −3 2.6x + 2.6y = 26 b. Solve the system of linear equations y = x + 2
and y = 3x − 4 by graphing.
25. −7x + 6y = 0 26. 4x − y = 1.5
0.5x + y = 2 2x + y = 1.5 c. How is the linear system and the solution in part
(b) related to the original equation and the solution
27. MODELING WITH MATHEMATICS You have in part (a)?
40 minutes to exercise at the gym, and you want to
burn 300 calories total using both machines. How 32. HOW DO YOU SEE IT? A teacher is purchasing
much time should you spend on each machine? binders for students. The graph shows the total costs
(See Example 3.) of ordering x binders from three different companies.
Elliptical Trainer Stationary Bike
Buying Binders
y
Company B
150

Cost (dollars)
125 Company A
100
Company C
75
50
0
0 15 20 25 30 35 40 45 50 x
8 calories 6 calories
per minute per minute Number of binders

28. MODELING WITH MATHEMATICS a. For what numbers of binders are the costs the
You sell small and large candles same at two different companies? Explain.
at a craft fair. You collect $144
b. How do your answers in part (a) relate to systems
selling a total of 28 candles.
of linear equations?
How many of each type of candle $6
$4
did you sell? each
each
33. MAKING AN ARGUMENT You and a friend are going
29. MATHEMATICAL CONNECTIONS Write a linear hiking but start at different locations. You start at the
equation that represents the area and a linear equation trailhead and walk 5 miles per hour. Your friend starts
that represents the perimeter of the rectangle. Solve 3 miles from the trailhead and walks 3 miles per hour.
the system of linear equations by graphing. Interpret
your solution.
you

(3x − 3) cm

6 cm
your friend
30. THOUGHT PROVOKING Your friend’s bank account
balance (in dollars) is represented by the equation
y = 25x + 250, where x is the number of months. a. Write and graph a system of linear equations that
Graph this equation. After 6 months, you want to represents this situation.
have the same account balance as your friend. Write a
b. Your friend says that after an hour of hiking you
linear equation that represents your account balance.
will both be at the same location on the trail. Is
Interpret the slope and y-intercept of the line that
your friend correct? Use the graph from part (a) to
represents your account balance.
explain your answer.

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Solve the literal equation for y. (Section 1.5)

1
34. 10x + 5y = 5x + 20 35. 9x + 18 = 6y − 3x 36. —34 x + —4 y = 5

240 Chapter 5 Solving Systems of Linear Equations

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