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MATH GRADE 8 UNIT 5
LINEAR EQUATIONS
EXERCISES
Copyright © 2014 Pearson Education, Inc. 2
Copyright © 2014 Pearson Education, Inc., or its affiliate(s). All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. The publisher hereby grants permission to reproduce these pages, in part or in whole, for classroom use only, the number not to exceed the number of students in each class. Notice of copyright must appear on all copies. For information regarding permissions, write to Pearson Curriculum Group Rights & Permissions, One Lake Street, Upper Saddle River, New Jersey 07458. The Pearson logo, and the Pearson Always Learning logo are trademarks, in the U.S. and/or other countries, of Pearson Education, Inc. or its affiliate(s).
Copyright © 2014 Pearson Education, Inc. 3
Grade 8 Unit 5: Linear Equations
CONTENTS EXERCISES
LESSON 1: CATCHING UP �������������������������������������������������������������������������� 4
LESSON 2: ONE-VARIABLE EQUATIONS ����������������������������������������������� 5
LESSON 3: HOW MANY SOLUTIONS ���������������������������������������������������� 9
LESSON 4: RELATE ONE-VARIABLE EQUATIONS ���������������������������� 13
LESSON 5: SAME LINE–DIFFERENT EQUATIONS ���������������������������� 17
LESSON 6: A SYSTEM OF LINEAR EQUATIONS �������������������������������� 21
LESSON 7: SOLVING EQUATIONS: SUBSTITUTION ����������������������� 26
LESSON 8: SOLVING EQUATIONS: ELIMINATION ��������������������������� 31
LESSON 9: USING A SYSTEM OF EQUATIONS ���������������������������������� 36
LESSON 10: SOLVING PROBLEMS ������������������������������������������������������������� 41
LESSON 11: PUTTING IT TOGETHER ����������������������������������������������������� 51
Copyright © 2014 Pearson Education, Inc. 4
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
• Review your end of unit assessment from the previous unit.
• Write your wonderings about linear equations.
• Write a goal stating what you plan to accomplish in this unit.
• Based on your previous work, write three things you will do differently during this unit to increase your success.
LESSON 1: CATCHING UP
Copyright © 2014 Pearson Education, Inc. 5
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
1. Solve. x + 3 = 2x – 1
Solve each equation.
2. 5n + 4 = 3(n – 2)
3. 2n + 3n – 8 = 4n + 7
4. 3(n – 4) = 5(n – 5)
5. 2n + 6 + n = 4n – 3n + 4
Solve.
6. Six times a number is equal to ten less than the sum of the number and 3. What is the number?
Challenge Problem
7. Write an equation that has a variable on both sides and a solution of 5.6. Write a word problem that matches your equation.
LESSON 2: ONE-VARIABLE EQUATIONS
Copyright © 2014 Pearson Education, Inc. 6
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. x = 4
x + 3 = 2x – 1
x + 3 – x + 1 = 2x – 1 – x + 1
4 = x
2. n = –5
5n + 4 = 3(n – 2)
5n + 4 = 3n – 6
5n + 4 – 4 – 3n = 3n – 6 – 4 – 3n
2n = –10
22
–102
=n
n = –5
3. n = 15
2n + 3n – 8 = 4n + 7
5n – 8 = 4n + 7
5n – 8 + 8 – 4n = 4n + 7 + 8 – 4n
n = 15
4. n = 612
or n = 6.5
3(n – 4) = 5(n – 5)
3n – 12 = 5n – 25
3n – 12 – 3n + 25 = 5n – 25 – 3n + 25
13 = 2n
132
22
=n
612
= n
LESSON 2: ONE-VARIABLE EQUATIONS
Copyright © 2014 Pearson Education, Inc. 7
Grade 8 Unit 5: Linear Equations
EXERCISES
5. n = –1
2n + 6 + n = 4n – 3n + 4
3n + 6 = n + 4
3n + 6 – n – 6 = n + 4 – n – 6
2n = –2
22
–22
=n
n = –1
6. n = –125
or –1.4
6n = (n + 3) – 10
6n = n – 7
6n – n = n – 7 – n
5n = –7
55
–75
=n
n = –125
Challenge Problem
7. Answers will vary. Possible answer:
Erin and Pedra are biking the same trail and in the same direction. Erin is 3 mi. from the start of the trail and bikes 3 miles per hour. Pedra is 8.6 mi. from the start of the trail and bikes 2 miles per hour. When will the two girls be the same distance from the start of the trail?
3h + 3 = 2h + 8.6
3h + 3 – 2h – 3 = 2h + 8.6 – 2h – 3
h = 5.6
The two girls will be the same distance after 5.6 hr, or 5 hr 36 min.
LESSON 2: ONE-VARIABLE EQUATIONS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 8
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 2: ONE-VARIABLE EQUATIONS
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2
3
4
5
6
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
7
Copyright © 2014 Pearson Education, Inc. 9
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
1. Solve.
x + 5 = 4(x – 1)
Write whether the equation has one solution, no solution, or all numbers as solutions.
2. 3x + 4 = 6 + 3x
3. x + 9 + 3x = 4x + 9
4. 3x + 4(x – 5) = 5x + 8
5. 3(x + 2) = 3x + 2
6. 2(x – 6) = –10 + 2x – 2
7. 5x + x + 4(x – 1) = 3x – 4 + x
Challenge Problem
8. a. Write an equation that has no solution.
b. Write an equation that has only 0 as a solution.
LESSON 3: HOW MANY SOLUTIONS?
Copyright © 2014 Pearson Education, Inc. 10
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. x = 3
x + 5 = 4(x – 1)
x + 5 = 4x – 4
x + 5 – x + 4 = 4x – 4 – x + 4
9 = 3x
93
33
=x
3 = x
2. No solution
3x + 4 = 6 + 3x
3x + 4 – 3x – 4 = 6 + 3x – 3x – 4
0 ≠ 2
3. All numbers as solutions
x + 9 + 3x = 4x + 9
4x + 9 = 4x + 9
4x + 9 – 9 = 4x + 9 – 9
4x = 4x
x = x
4. One solution
3x + 4(x – 5) = 5x + 8
3x + 4x – 20 = 5x + 8
7x – 20 = 5x + 8
7x – 20 – 5x + 20 = 5x + 8 – 5x + 20
2x = 28
x = 14
LESSON 3: HOW MANY SOLUTIONS?
Copyright © 2014 Pearson Education, Inc. 11
Grade 8 Unit 5: Linear Equations
EXERCISES
5. No solution
3(x + 2) = 3x + 2
3x + 6 – 3x – 6 = 3x + 2 – 3x – 6
0 ≠ –4
6. All numbers as solutions
2(x – 6) = –10 + 2x – 2
2x – 12 = –12 + 2x
2x – 12 + 12 = –12 + 2x + 12
2x = 2x
x = x
7. One solution
5x + x + 4(x – 1) = 3x – 4 + x
5x + x + 4x – 4 = 4x – 4 + x
10x – 4 – 4x + 4 = 4x – 4 – 4x + 4
6x = 0
x = 0
Challenge Problem
8. Answers will vary. Possible answers:
a. x + 4 = x + 5
b. 2x = 3x
LESSON 3: HOW MANY SOLUTIONS?
ANSWERS
Copyright © 2014 Pearson Education, Inc. 12
Grade 8 Unit 5: Linear Equations
EXERCISES
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2
3
4
5
6
7
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
8 a�
8 b�
LESSON 3: HOW MANY SOLUTIONS?
Copyright © 2014 Pearson Education, Inc. 13
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 4: RELATE ONE-VARIABLE EQUATIONS
EXERCISES
1. Write the expression 4(x + 5) + 4x + 8 – 2 in simplest form.
Use the Graphing tool to decide if the equation has one solution, no solution, or all numbers as solutions.
2. 4x + 6 = 4x – 6
3. 3(x + 4) = 3x + 7
4. 4x + 9 = 2 + 2(x + 1) + 2x + 5
5. 3x + 8 = 8(x + 1)
6. 3x – 4x – 1 = 7x – 6
7. 12
x + 6 = 3x – 1
Challenge Problem
8. Graph y = 2x + 5 and y = 3x + 1 on the same coordinate grid. Explain how to use the graph to find the solution to 2x + 5 = 3x + 1.
Copyright © 2014 Pearson Education, Inc. 14
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. 8x + 26 or 26 + 8x
4(x + 5) + 4x + 8 – 2
4x + 20 + 4x + 8 – 2
8x + 26
2. No solution
4x + 6 = 4x – 6
Use graphing tool to show graph of y = 4x + 6 and y = 4x – 6; label each line with its equation; these lines will be parallel.
3. No solution
3(x + 4) = 3x + 7
Use graphing tool to show graph of y = 3(x + 4) and y = 3x + 7; label each line with its equation; these lines will be parallel.
4. All numbers as solutions
4x + 9 = 2 + 2(x + 1) + 2x + 5
Use graphing tool to show graph of y = 4x + 9 and y = 2 + 2(x + 1) + 2x + 5; label each line with its equation; these lines are the same line.
5. One solution
3x + 8 = 8(x + 1)
Use graphing tool to show graph of y = 3x + 8 and y = 8(x + 1); label each line with its equation; these lines will intersect.
6. One solution
3x – 4x – 1 = 7x – 6
Use graphing tool to show graph of y = 3x – 4x – 1 and y = 7x – 6; label each line with its equation; these lines will intersect.
LESSON 4: RELATE ONE-VARIABLE EQUATIONS
Copyright © 2014 Pearson Education, Inc. 15
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 4: RELATE ONE-VARIABLE EQUATIONS
7. One solution
12
x + 6 = 3x – 1
Use graphing tool to show graph of y = 12
x + 6 and y = 3x – 1; label each line with its equation; these lines will intersect.
Challenge Problem
8. The graphs will intersect at (4, 13). So, the solution to 2x + 5 = 3x + 1 is the value of x at that point, 4.
Use graphing tool to show graph of y = 2x + 5 and y = 3x + 1 on the same coordinate grid; label each line with its equation; label point of intersection (4, 13).
ANSWERS
Copyright © 2014 Pearson Education, Inc. 16
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 4: RELATE ONE-VARIABLE EQUATIONS
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2
3
4
5
6
7
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
8
Copyright © 2014 Pearson Education, Inc. 17
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
1. Which equation is equivalent to y = 4x + 5?
A 4x + y = 5
B 4x – y = –5
C 4x + y = –5
D 4x – y = 5
2. Write each equation in slope-intercept form. (y = mx + b)
a. x – y = 3
b. 2x + 2y = 10
c. –5x + 5y = 35
d. x – y = 0
3. Write each equation in standard form. (ax + by = c)
a. y = 3x – 4
b. y = –12
x + 1
c. y = –2x + 2
d. y = x
4. What are the slope and y-intercept of each equation?
a. 2x + y = 8
b. x – y = 4
LESSON 5: SAME LINE—DIFFERENT EQUATIONS
Copyright © 2014 Pearson Education, Inc. 18
Grade 8 Unit 5: Linear Equations
EXERCISES
5. Look at the graph of this line:
123456789
10
–5–4–3–2–1 10987654321–2–3–4–5–6
x
y
Write an equation in both slope-intercept form and standard form for the line.
Challenge Problem
6. Marshall likes to use interval training. He jogs at 200 meters per minute and runs 250 meters per minute. He runs 8 km every day. Write an equation in standard form. Let x represent the number of minutes of jogging and y represent the number of minutes of running for Marshall’s interval training.
LESSON 5: SAME LINE—DIFFERENT EQUATIONS
EXERCISES
Copyright © 2014 Pearson Education, Inc. 19
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. B 4x – y = –5
2. a. y = x – 3
b. y = –x + 5
c. y = x + 7
d. y = x or y = x + 0
3. a. 3x – y = 4
b. 12
x + y = 1
c. 2x + y = 2
d. –x + y = 0 or x – y = 0
4. a. Slope: –2; y-intercept: 8
2x + y = 8 written in slope-intercept form is y = –2x + 8.
b. Slope: 1; y-intercept: –4
x – y = 4 written in slope-intercept form is y = x – 4.
5. Slope-intercept form: y = –x + 4; standard form: x + y = 4
The y-intercept is (0, 4). The slope is –1.
Challenge Problem
6. 200x + 250y = 8,000
LESSON 5: SAME LINE—DIFFERENT EQUATIONS
Copyright © 2014 Pearson Education, Inc. 20
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 5: SAME LINE—DIFFERENT EQUATIONS
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2 a.
2 b.
2 c.
2 d.
3 a.
3 b.
3 c.
3 d.
4 a.
4 b.
5
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
6
Copyright © 2014 Pearson Education, Inc. 21
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 6: A SYSTEM OF LINEAR EQUATIONS
EXERCISES
1. Write the equation 6x + 3y = 12 in slope intercept form. (y = mx + b)
Solve each system of equations. If the system has no solution, write no solution. If a system has infinitely many solutions write infinitely many solutions.
2. y = 2x + 3
y = 3x – 4
3. y = 3x – 2x
y – x = 4
4. 4 + y = 4x
y = 4(x – 1)
5. 3 = 3x + y
– 3x + y = 8
6. 9 = 5x – y
–3x + y = 9
Challenge Problem
7. Write a system of two equations that has no solution. Describe what the graph of the system would look like.
Copyright © 2014 Pearson Education, Inc. 22
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. y = –2x + 4
2. (7, 17)
y = 2x + 3
y = 3x – 4
2x + 3 = 3x – 4
2x + 3 – 2x + 4 = 3x – 4 – 2x + 4
7 = x
y = 2x + 3
y = 2(7) + 3
y = 14 + 3
y = 17
3. No solution
y = 3x – 2x
y – x = 4 or y = x + 4
3x – 2x = x + 4
x = x + 4
x – x = x – x + 4
0 ≠ 4
4. Infinitely many solutions
4 + y = 4x or y = 4x – 4
y = 4(x – 1) or y = 4x – 4
4x – 4 = 4x – 4
LESSON 6: A SYSTEM OF LINEAR EQUATIONS
Copyright © 2014 Pearson Education, Inc. 23
Grade 8 Unit 5: Linear Equations
EXERCISES
5. (–56
, 512
)
3 = 3x + y or y = –3x + 3
– 3x + y = 8 or y = 3x + 8
–3x + 3 = 3x + 8
–3x + 3 + 3x – 8 = 3x + 8 + 3x – 8
–5 = 6x
x = –56
y = 3x + 8
y = 3(–56
) + 8
y = –52
+ 8
y = 112
or 512
6. (9, 36)
9 = 5x – y or y = 5x – 9
–3x + y = 9 or y = 3x + 9
5x – 9 = 3x + 9
5x – 9 –3x + 9 = 3x + 9 – 3x + 9
2x = 18
x = 9
y = 3x + 9
y = 3(9) + 9
y = 27 + 9
y = 36
LESSON 6: A SYSTEM OF LINEAR EQUATIONS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 24
Grade 8 Unit 5: Linear Equations
EXERCISES
Challenge Problem
7. RUBRIC
4 pointsResponse shows a sound approach, a correct answer, and good communication of the process by which the answer was obtained.
3 pointsResponse shows a sound approach, but errors along the way may result in an incorrect response. Work is clearly shown, but some of the detail of the steps taken may be incomplete.
2 pointsResponse has an approach that is flawed but carried through appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not follow through well, with work shown being potentially sketchy. Or the correct answer is shown, but no communication is given about how the solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 points Systems will vary. Possible system:
y = 4x + 5
y = 4x – 1
4x + 5 = 4x – 1
4x + 5 – 4x – 5 = 4x – 1 – 4x – 5
0 ≠ –6
The system of the two equations has no solution.
The lines will be parallel because both lines have the same slope, 4. The line y = 4x + 5 has a y-intercept of 5. The line y = 4x – 1 has a y-intercept of – 1.
LESSON 6: A SYSTEM OF LINEAR EQUATIONS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 25
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 6: A SYSTEM OF LINEAR EQUATIONS
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2
3
4
5
6
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
7
Copyright © 2014 Pearson Education, Inc. 26
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 7: SOLVING EQUATIONS: SUBSTITUTION
EXERCISES
1. Solve this equation.
3(2y – 1) + 5y = 8
Solve each system of equations. If the system has no solution, write no solution. If a system has infinitely many solutions write infinitely many solutions.
2. x = 4y + 2
2x – 3y = 9
3. y = 2x
3x + y = 12
4. 3x + y = 5
4x + 2y = 20
5. 4x + 5 = y
4x – y = 8
6. y = 3x – 7 + x
7 = 4x – y
Challenge Problem
7. Write a system of equations that has (4, 5) as the solution.
Copyright © 2014 Pearson Education, Inc. 27
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 7: SOLVING EQUATIONS: SUBSTITUTION
ANSWERS
1. x = 1
3(2y – 1) + 5y = 8
6y – 3 + 5y = 8
11y – 3 = 8
11y – 3 + 3 = 8 + 3
11y = 11
y = 1
2. (6, 1)
x = 4y + 2
2x – 3y = 9
2(4y + 2) – 3y = 9
8y + 4 – 3y = 9
5y = 5
y = 1
x = 4(1) + 2
y = 6
3. (2.4, 4.8) or 225
, 445
y = 2x
3x + y = 12
3x + 2x = 12
5x = 12
x =
2
125
or 225
y =
2
125
y = 245
or 445
Copyright © 2014 Pearson Education, Inc. 28
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 7: SOLVING EQUATIONS: SUBSTITUTION
4. (–5, 20)
3x + y = 5 or y = –3x + 5
4x + 2y = 20
4x + 2(–3x + 5) = 20
4x – 6x + 10 = 20
– 2x + 10 = 20
– 2x + 10 + 2x – 20 = 20 + 2x – 20
– 10 = 2x
x = –5
y = –3(–5) + 5
y = 15 + 5
y = 20
5. No solution
4x + 5 = y
4x – y = 8
4x – (4x + 5) = 8
4x – 4x – 5 = 8
– 5 ≠ 8
6. Infinitely many solutions
y = 3x – 7 + x
7 = 4x – y
7 = 4x – (3x – 7 + x)
7 = 4x – 3x + 7 – x
7 = 7
ANSWERS
Copyright © 2014 Pearson Education, Inc. 29
Grade 8 Unit 5: Linear Equations
EXERCISES
Challenge Problem
7. Answers will vary. Ask a classmate to check your system of equations.
Possible system:
y = x + 1
y = –2x + 13
x + 1 = –2x + 13
x + 1 + 2x – 1 = –2x + 13 + 2x – 1
3x = 12
x = 4
y = 4 + 1
y = 5
LESSON 7: SOLVING EQUATIONS: SUBSTITUTION
ANSWERS
Copyright © 2014 Pearson Education, Inc. 30
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 7: SOLVING EQUATIONS: SUBSTITUTION
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2
3
4
5
6
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
7
Copyright © 2014 Pearson Education, Inc. 31
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
1. Add the two equations.
3x – y = 7
2x + y = –6
Solve each system of equations.
2. 4x – y = 9
3x + y = 5
3. x + 3y = 8
–x + 5y = 4
4. 5x + 3y = 8
4x + 3y = 5
5. 2x + 8y = 6
3x – 4y = –1
Challenge Problem
6. a. Explain why the substitution method, rather than the elimination method, might be the better choice for solving this system of equations.
x = 4y + 1
3y + 5x = 10
b. Solve the system of equations.
LESSON 8: SOLVING EQUATIONS: ELIMINATION
Copyright © 2014 Pearson Education, Inc. 32
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 8: SOLVING EQUATIONS: ELIMINATION
ANSWERS
1. 5x = 1
3x – y = 7
+ 2x + y = –6
5x + 0 = 1
2. (2, –1)
4x – y = 9
+ 3x + y = 5
7x + 0 = 14
x = 2
3(2) + y = 5
6 + y = 5
y = –1
3. 312
, 112
or (3.5, 1.5)
x + 3y = 8
+ –x + 5y = 4
0 + 8y = 12
y = 128
or 32
or 112
x + 3
32
= 8
x + 92
= 8
x = 312
Copyright © 2014 Pearson Education, Inc. 33
Grade 8 Unit 5: Linear Equations
EXERCISES
4. 3, – 213
4x + 3y = 5 or –1(4x + 3y) = –1(5) or –4x –3y = –5
5x + 3y = 8
+ –4x – 3y = –5
x + 0 = 3
x = 3
5(3) + 3y = 8
15 + 3y = 8
3y = –7
y = –73
or – 213
5. 12
,58
3x – 4y = –1 or 2(3x – 4y) = 2(–1) or 6x – 8y = –2
2x + 8y = 6
+ 6x – 8y = –2
8x + 0 = 4
x = 12
2(12
) + 8y = 6
1 + 8y = 6
8y = 5
y = 58
LESSON 8: SOLVING EQUATIONS: ELIMINATION
ANSWERS
Copyright © 2014 Pearson Education, Inc. 34
Grade 8 Unit 5: Linear Equations
EXERCISES
Challenge Problem
6. a. Since the first equation is solved for x, it is easy to substitute the value for x into the second equation.
b. 12023
,523
x = 4y + 1
3y + 5x = 10
3y + 5(4y + 1) = 10
3y + 20y + 5 = 10
23y = 5
y = 523
4523
1=
+x
x = 2023
+ 1
x = 12023
LESSON 8: SOLVING EQUATIONS: ELIMINATION
ANSWERS
Copyright © 2014 Pearson Education, Inc. 35
Grade 8 Unit 5: Linear Equations
EXERCISES
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2
3
4
5
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
6 a�
6 b�
LESSON 8: SOLVING EQUATIONS: ELIMINATION
Copyright © 2014 Pearson Education, Inc. 36
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
1. The sum of two numbers is 25. Their difference is 9. What are the two numbers?
2. A fire lookout is a person who looks for fire from the top of a structure known as a fire lookout tower. These towers are used in remote areas, often on mountain tops. From these towers, the fire lookouts have a good view of the surrounding terrain and are able to spot wildfires.
The grid shows the location of two lookout towers.
123456789
10
–5–4–3–2
10987654321
x
y
Tower 1
Tower 2
A fire breaks out. The fire lookout at Tower 1 sees the fire with a line-of-sight having
slope of –12
. The fire lookout at Tower 2 sees the fire with a line-of-sight having
slope of 1.
At the coordination center, exact coordinates of the fire are needed in order to find the optimal position for a water-bombing plane.
a. What is the y-intercept and slope of each line?
b. Find the linear equations of the two lines.
c. Find the coordinates of the fire.
LESSON 9: USING A SYSTEM OF EQUATIONS
Copyright © 2014 Pearson Education, Inc. 37
Grade 8 Unit 5: Linear Equations
EXERCISES
Challenge Problem
3. A third tower is located at (5, 0). The lookout sees a fire with a line-of-sight with a
slope of –32
. Can this be the same fire as the one Tower 1 and Tower 2 see?
Explain.
LESSON 9: USING A SYSTEM OF EQUATIONS
EXERCISES
Copyright © 2014 Pearson Education, Inc. 38
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. The two numbers are 17 and 8.
x + y = 25
x – y = 9
2x + 0 = 34
x = 17
17 + y = 25
y = 18
2. RUBRIC
4 pointsResponse shows a sound approach, a correct answer, and good communication of the process by which the answer was obtained.
3 pointsResponse shows a sound approach, but errors along the way may result in an incorrect response. Work is clearly shown, but some of the detail of the steps taken may be incomplete.
2 pointsResponse has an approach that is flawed but carried through appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not follow through well, with work shown being potentially sketchy. Or the correct answer is shown, but no communication is given about how the solution was obtained.
EXAMPLE OF A 4-POINT RESPONSE:
4 pointsa. Tower 1: slope = –
12
; y-intercept = 5
Tower 2: slope = 1; y-intercept = –1
LESSON 9: USING A SYSTEM OF EQUATIONS
Copyright © 2014 Pearson Education, Inc. 39
Grade 8 Unit 5: Linear Equations
EXERCISES
b. Tower 1: y = –12
x + 5
Tower 2: y = x – 1
c. The fire is at coordinates (4, 3).
123456789
10
–5–4–3–2
10987654321
x
y
Tower 1
Tower 2
Challenge Problem
3. No. The equation for Tower 3 is y = –32
x + 712
. The point (4, 3) is not on that line.
On the line of sight for Tower 3, x = 4 and y = 112
.
LESSON 9: USING A SYSTEM OF EQUATIONS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 40
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 9: USING A SYSTEM OF EQUATIONS
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2 a.
2 b.
2 c.
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
3
Copyright © 2014 Pearson Education, Inc. 41
Grade 8 Unit 5: Linear Equations
EXERCISES
EXERCISES
1. Write the equation in slope-intercept form for a line with slope 2 that passes through the point (2, 5).
2. Each table represents a linear relationship.
Table A
x –2 –1 0 2
y –4 –2 0 4
Table B
x –4 –2 0 4
y –3 0 3 9
a. Find the slope of each line.
b. Find the equation for each line.
c. Find the coordinates of the point of intersection.
LESSON 10: SOLVING PROBLEMS
Copyright © 2014 Pearson Education, Inc. 42
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 10: SOLVING PROBLEMS
3. Jones and Janes are two competing landscaping companies. Jones charges $40 to come to the location and then $40 per hour. Janes has made a graph representing their costs.
34032030028026024022020018016014012010080604020
543 82 71 6
x
y
Hours
Dollars
Two points on the graph are (0, 60) and (3, 150).
The horizontal axis shows the number of hours. The vertical axis shows the total cost.
a. Which company is less expensive for a 3-hour job?
b. Draw the graph of Jones in the same coordinate system.
c. What are the coordinates of the point of intersection of the two lines?
d. What is the meaning of this point?
EXERCISES
Copyright © 2014 Pearson Education, Inc. 43
Grade 8 Unit 5: Linear Equations
EXERCISES
Challenge Problem
4. Points A, B, C, and D represent the locations of four airports.
1
2
3
4
5
–2
–154321–1–2–3–4
x
y
D
A
C
B
a. Find the slope of each line segment shown.
b. Find the y-intercepts of each segment.
c. Write the system of linear equations that will give you the coordinates of the point of intersection of the two segments.
d. Solve the system of equations.
LESSON 10: SOLVING PROBLEMS
EXERCISES
Copyright © 2014 Pearson Education, Inc. 44
Grade 8 Unit 5: Linear Equations
EXERCISES
ANSWERS
1. y = 2x + 1
y = 2x + b
5 = 2(2) + b
5 = 4 + b
1 = b
y = 2x + 1
2. RUBRIC
4 pointsResponse shows a sound approach, a correct answer, and good communication of the process by which the answer was obtained.
3 pointsResponse shows a sound approach, but errors along the way may result in an incorrect response. Work is clearly shown, but some of the detail of the steps taken may be incomplete.
2 pointsResponse has an approach that is flawed but carried through appropriately, with work shown to document what was done.
1 point
Response begins to take an inappropriate approach and does not follow through well, with work shown being potentially sketchy. Or the correct answer is shown, but no communication is given about how the solution was obtained.
EXAMPLE:
4 points a. Table A: slope = 2
4 02 0
( )( )
−−
= 42
= 2
Table B: slope = 32
9 34 0
( )( )
−−
= 64
= 23
LESSON 10: SOLVING PROBLEMS
Copyright © 2014 Pearson Education, Inc. 45
Grade 8 Unit 5: Linear Equations
EXERCISES
b. Use slope-intercept to write the equation. (y = mx + b)
Table A: y = 2x
When x = 0, y = 0, so y-intercept is 0. The slope is 2.
Table B: y = 32
x + 3
When x = 0, y = 3, so y-intercept is 3. The slope is 32
.
c. (6, 12)
y = 2x
– y = –32
x – 3
0 = 12
x – 3
3 = 12
x
6 = x
y = 2x
y = 2(6)
y = 12
LESSON 10: SOLVING PROBLEMS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 46
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 10: SOLVING PROBLEMS
3. a. Janes is less expensive for a 3-hour job.
Jones: y = $40x + $40
y = $40(3) + $40
y = $120 + $40
y = $160
Janes: y = $30x + $60
y = $30(3) + $60
y = $90 + $60
y = $150
b.
34032030028026024022020018016014012010080604020
543 82 71 6
x
y
Hours
Dollars
Jones
James
ANSWERS
Copyright © 2014 Pearson Education, Inc. 47
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 10: SOLVING PROBLEMS
c. (2, 120)
y = 40x + 40
+ –y = –30x – 60
0 = 10x – 20
20 = 10x
2 = x
y = 40x + 40
y = 40(2) + 40
y = 80 + 40
y = 120
d. The cost will be the same for either company, $120, for a 2-hour visit.
Challenge Problem
4. a. Segment AC: slope = 23
; Segment DB: slope = –12
Segment AC:
8 – –14 – –2
96
23
( )( ) = =
Segment DB:
4 –12
–112
– 512
3.5–7
–12
= =
ANSWERS
Copyright © 2014 Pearson Education, Inc. 48
Grade 8 Unit 5: Linear Equations
EXERCISES
b. Segment AC: y-intercept = 13
; Segment DB: y-intercept = 314
Segment AC:
–1 = 23
(–2) + b
–1 = –43
+ b
13
= b
Segment DB:
12
–12
512
=
+ b
12
= –114
+ b
134
= b
b = 314
c. y = 23
x + 13
y = –12
x + 314
LESSON 10: SOLVING PROBLEMS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 49
Grade 8 Unit 5: Linear Equations
EXERCISES
d.
2
12
, 2
y = 23
x + 13
–y = 12
x – 314
0 = 76
x – 3512
3512
= 76
x
52
= x or x = 212
y = 23
x + 13
–23
52
13
=
+y
y = 53
+ 13
y = 63
or 2
LESSON 10: SOLVING PROBLEMS
ANSWERS
Copyright © 2014 Pearson Education, Inc. 50
Grade 8 Unit 5: Linear Equations
EXERCISESLESSON 10: SOLVING PROBLEMS
Self Assessment
After you use the answer key to check your answers, use the chart below to self-assess your work. For each exercise, place a check mark in the column that best describes how you did on that exercise.
Exercise Number Yes! I got it� I was confused, but
now I get it� I need help!
1
2 a.
2 b.
2 c.
3 a.
3 b.
3 c.
3 d.
Challenge Problem
Exercise I gave it a try, but I’m not sure I did it right�
I did it, and my answer makes sense�
4 a�
4 b�
4 c�
4 d�
Copyright © 2014 Pearson Education, Inc. 51
Grade 8 Unit 5: Linear Equations
EXERCISES
• Read through your work on the Self Check task and think about your other work in this lesson.
• Write what you have learned.
• What would you do differently if you were starting the Self Check task now?
• Record your ideas. Keep track of any strategies you have learned.
• Complete any exercises that you have not finished from this unit.
LESSON 11: PUTTING IT TOGETHER
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