LINEAR EQUATIONSrusdmath.weebly.com/uploads/1/1/1/5/11156667/g8_u5...The slope is –1. 6. 10x + 35y...

ANSWERS FOR EXERCISES

MATH GRADE 8 UNIT 5

LINEAR EQUATIONS

Copyright © 2014 Pearson Education, Inc. 22

Grade 8 Unit 5: Linear Equations

ANSWERS

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

= n612

LESSON 2: ONE VARIABLE ON BOTH SIDES



ANSWERSLESSON 2: ONE VARIABLE ON BOTH SIDES

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 miles from the start of the trail and bikes 3 miles per hour. Pedra is 8.6 miles 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.



ANSWERS

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

5. No solution

3(x + 2) = 3x + 2 3x + 6 – 3x – 6 = 3x + 2 – 3x – 6 0 ≠ –4

LESSON 3: HOW MANY SOLUTIONS?



ANSWERSLESSON 3: HOW MANY 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 10x – 4 – 4x + 4 = 4x – 4 – 4x + 4 6x = 0 x = 0

Challenge Problem

8. Answers will vary. Possible answers:

a. x + 4 = x + 5b. 2x = 3x



ANSWERS

ANSWERS

1. 8x + 26 or 26 + 8x

4(x + 5) + 4x + 8 – 2 4x + 20 + 4x + 8 – 2 8x + 26

2. No solution

12345678

–5–6–7

–4–3–2

65432–1 1–2–3–4–5 x

y

g(x) = 4x – 6

f(x) = 4x + 6

3. No solution

123456789

10111213

–265432–1 1–2–3–4–5 x

y

f(x) = 3(x + 4)

g(x) = 3x + 7

LESSON 4: PARALLEL OR INTERSECTING?



ANSWERSLESSON 4: PARALLEL OR INTERSECTING?


123456789

1011

65432–1–2–3–4

1–2–3–4–5 x

y

f(x) = 4x + 9

g(x) = 2 + 2(x + 1) + 2x + 5

5. One solution

123456789

10111213

–265432–1 1–2–3–4–5 x

y

f(x) = 8(x + 1)

g(x) = 3x + 8



EXERCISESLESSON 4: PARALLEL OR INTERSECTING?

6. One solution

12345678

65432–1–2–3–4–5–6–7

1–2–3–4–5 x

y

f(x) = 3x – 4x – 1

g(x) = 7x – 6

7. One solution

123456789

–5–6

–4–3–2

65432–1 1–2–3–4–5 x

y

g(x) = 3x + 1

f(x) = x + 612



EXERCISESLESSON 4: PARALLEL OR INTERSECTING?

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.

123456789

1011121314

–265432–1 1–2–3–4–5 x

y

f(x) = 2x + 5

g(x) = 3x + 1



ANSWERS

ANSWERS

1. B 4x – y = –5

2. a. y = x – 3

b. y = –x + 5

c. y = x + 7

d. y = x + 0

3. a. 3x – y = 4

b. 12

x + y = 1

c. 2x + y = 2

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.

6. 10x + 35y = 70 10x – 10x + 35y = 70 – 10x Addition property of equality 35y = –10x + 70 Combine like terms 3535

1035

7035

yx

= − + Multiplication property of equality

y = − 27

x + 2 Simplify

Challenge Problem

7. 200x + 250y = 8,000

LESSON 5: FORMS OF LINEAR EQUATIONS



ANSWERS

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 EQUATIONS



ANSWERSLESSON 6: A SYSTEM OF EQUATIONS

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

Challenge Problem

7. 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.



ANSWERS

ANSWERS

1. y = 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

LESSON 7: SOLVING—SUBSTITUTION



ANSWERSLESSON 7: SOLVING—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

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



ANSWERS

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

LESSON 8: SOLVING—ELIMINATION



ANSWERSLESSON 8: SOLVING—ELIMINATION

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 = 5

8



ANSWERSLESSON 8: SOLVING—ELIMINATION

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



ANSWERS

ANSWERS

1. The two numbers are 17 and 8.

x + y = 25 x – y = 9 2x + 0 = 34 x = 17

17 + y = 25 y = 8

2. a. Tower 1: slope = –12

; y-intercept = 5

Tower 2: slope = 1; y-intercept = –1

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

ANSWERS

1. y = 2x + 1

y = 2x + b 5 = 2(2) + b 5 = 4 + b 1 = b y = 2x + 1

2. a. Table A: slope = 2

4 02 0

( )( )

−− =

42

= 2

Table B: slope = 32

9 34 0

( )( )

−− =

64

= 32

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. T(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



ANSWERSLESSON 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. 36034032030028026024022020018016014012010080604020

543 82 71 6 x

y

Hours

Dollars

Jones

Janes

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 for a 2-hour visit will be the same for either company, $120.




Challenge Problem

4. a. Segment AC: slope = 23

; Segment DB: slope = – 12

Segment AC:

3 14 2

46

23

– –– –

( )( )

= =

Segment DB:

4 –12

–112

– 512

3.5–7

–12

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

= =

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

(continues)




(continued)

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

y = ⎛⎝⎜

⎞⎠⎟

+23

52

13

y = 53

+ 13

y = 63

or 2

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