3-2
Solving Systems
Algebraically
Vocabulary
Review
1. Circle the equations that are in standard form.
4x 1 3y 5 2
y 5 3x 2 5
4x 2 3 5 2y
2x 1 5y 5 0
Write each equation in standard form.
2. y 5 5x 2 3
3. y 2 4 5 6x
5x 2 y 5 3
4. 2y 1 3x 2 12 5 0
6x 2 y 5 24
3x 2 y 5 12
Vocabulary Builder
LOO
shun
Main Idea: If two numbers are substituted for x and y in a
system of equations and they make both equations true,
then the ordered pair (x, y) is a solution of the system.
Definition: A solution is any ordered pair that makes an equation in two variables true.
Use Your Vocabulary
5. Write three ordered pairs that are solutions of the equation y 5 25x 2 2. Answers may vary.
Check students’ work.
Q 0 , 22 R
Q 25 , 23 R
Q 5 , 227 R Samples are given.
Use the system at the right. Write T for true or F for false.
F
6. The system has a unique solution.
F
7. The system has infinitely many solutions.
T
8. The system has no solution.
F
9. The solution is (0, 0).
T
y
2x 1 2 5 3
e
4x 1 y 5 2
10. The solution of a system can be found by graphing the equations of the system.
Chapter 3
62
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solution (noun) suh
(2, 1) is a solution of
y 3x 5 because
1 3(2) 5.
Problem 1 Solving by Substitution
Got It? What is the solution of the system of equations? e
x 1 3y 5 5
22x 2 4y 5 25
11. Follow the steps to find the solution.
Solve the first equation for x.
3y
x
1
2
5
5
3y
x
Substitute the expression for x in the
2x
second equation. Then solve for y.
2+ 3y 5 ,
4y
5
4y
5
4y
5
2
y
5
y
2.5
6y 10
3
Substitute the value for y in either
equation. Solve for x.
x
3+
x
3y
5
2.5
,
5
7.5
x
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
x
12. The solution of the system is Q 22.5 ,
5
2.5
R.
2.5
Problem 2 Using Substitution to Solve a Problem
Got It? Music An online music company offers 15 downloads for $19.75 and
40 downloads for $43.50. Each price includes a one-time registration fee. What is
the cost of each download and the registration fee?
13. Complete the model to write a system of equations.
Relate
Define
Write
total cost
Let c
is
number of
downloads
times
฀the cost of one download and let r
cost of one
download
plus
registration
fee
the registration fee.
$19.75
15
r
c
r
$ 43.50
40
r
c
r
63
Lesson 3-2
14. Circle the equation that expresses r in terms of c in the first equation.
r 5 215c 1 19.75
r 5 15c 1 19.75
r 5 215c 2 19.75
15. Substitute the equation you chose in Exercise 14 into the second equation of the
system and solve for c.
43.5 5 40c 1 r
Write the original equation.
43.5 5 40c 1
215 c 1 19.75
Substitute for r.
43.5 5
25
c 1 19.75
Simplify.
23.75 5
25
c
Use the Addition Property of Equality.
0.95 5 c
Divide.
16. Now substitute the value of c into one of the equations of the system and solve for r.
Answers may vary. Sample:
r 5 215c 1 19.75
r 5 215(0.95) 1 19.75
r 5 214.25 1 19.75
r 5 5.5
17. The cost of each download is $
and the registration fee is $ 5.50 .
Solving by Elimination
Got It? What is the solution of the system of equations?
18. Add the equations.
22x
5x
1
2
3x
1
8y
8y
e
22x 1 8y 5 28
5x 2 8y 5 20
19. Now choose one of the original equations.
Substitute and solve.
5
5
28
20
0y
5
12
x
5
4
Answers may vary. Sample:
22(4) 1 8y 5 28
28 1 8y 5 28
8y 5 0
y50
Simplify.
20. Circle the ordered pair that is the solution of the system of equations.
(24, 22)
Problem 4
(4, 22)
(4, 0)
Solving an Equivalent System
Got It? What is the solution of the system of equations?
21. Underline the correct values to complete the sentence.
e
3x 1 7y 5 15
5x 1 2y 5 24
To get additive inverses for the x-term, multiply the first equation by 2 / 3 / 5
and the second equation by 22 / 23 / 27.
Chapter 3
64
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Problem 3
.95
22. Circle the equivalent system that shows additive inverses for the x-term.
e
e
3x 1 7y 5 15
5x 1 2y 5 24
6x 1 14y 5 3c
235x 2 14y 5 28
23. Solve the system for y.
e
15x 1 35y 5 75
215x 2 6y 5 12
24. Then substitute and solve for x.
15x 1 35y 5 75
215x 2 6y 5 12
29y 5 87
y5 3
5x 1 2y 5 24
5x 1 2(3) 5 24
5x 5 210
x 5 22
25. The solution of the system is Q 22 , 3 R .
Problem 5 Solving Systems Without Unique Solutions
Got It? What is the solution of the system of equations? Explain. e
2x 1 y 5 22
2x 2 2y 5 0
26. Circle the first step in solving the system.
Multiply 2x 1 y 5 22 by 21.
Multiply 2x 1 y 5 22 by 2.
27. Add 22x 1 2y 5 24 and 2x 2 2y 5 0.
22x
2x
2y
2y
1
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
0
5
5
24
0
5
24
28. What is the solution of the system?
Place a ✓ in the box if the response is correct.
Place an ✗ if it is incorrect.
✓ The system has no solution.
✗ The system has infinitely many solutions.
Lesson Check • Do you UNDERSTAND?
Vocabulary Give an example of two equivalent systems.
29. Cross out the system of equations that is NOT equivalent to the others.
e
e
4y 1 5x 5 13
4y 2 x 5 3
e
y 1 5x 5 12
4y 2 x 5 3
8y 1 40x 5 96
8y 2 2x 5 6
Math Success
Check off the vocabulary words that you understand.
substitution
elimination
equivalent equation
unique solutions
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Lesson 3-2