4.1 Graphs in the Coordinate Plane

Chapter 4 – Graphs of Linear Equations and Functions Answer Key

CK-12 Basic Algebra Concepts 1

4.1 Graphs in the Coordinate Plane

Answers

1. (-2, -2)

2. (5, 6)

3. (2, -6)

4. (3, -4)

5. (-5, 5)

6. (-2, 3)

7. – 14.

15. – 22.



23.

24.

25. (2, 0) is not in a quadrant, it is on the line between QI and QIV.

26. All coordinate pairs describe points by distance from (0, 0).

27. a) 𝑦 = 3𝑥

b) 𝑦 = {0, 3, 6, 9, 12,15

c)



28.

29.

The percentage of employed men trended downward between 1973 and 2009.

58

60

62

64

66

68

70

72

74

76

78

1973 1980 1986 1992 1997 2002 2005 2007 2009

Pe

rce

nta

ge

% of Men Employed in the U.S.

𝑥 0 2 4 6 8

𝑦 8 8.5 9 9.5 10



4.2 Graphs of Linear Equations

Answers

1. Solutions to an equation in two variables graph as a line. Solutions to an equation in one

variable are a single point.

2. 𝑦 = 3𝑥 − 7 :

3. 𝑦 = 2𝑥 + 7 :

4. 𝑦 = 0.7𝑥 − 4 :

𝑥 -1 0 1 2 3

𝑦 -10 -7 -4 -1 2

𝑥 -3 -2 -1 0 1

𝑦 1 3 5 7 9

𝑥 1 2 3 4 5 𝑦 -3.3 -2.6 -1.9 -1.2 -.5



5. 𝑦 = 6 − 1.25𝑥 :

𝑥 -1 0 1 2 3 𝑦 7.25 6 4.75 3.5 2.25



4.3 Horizontal and Vertical Line Graphs

Answers

1. 𝑦 = 0

2. 𝑥 = 0

3. 𝑥 = 6

4. 𝑦 = −2

5. 𝑦 = −7

6. 𝑦 = 5

7. 𝑥 = −4

8. – 10.



4.4 Applications of Linear Graphs

Answers

1. $9.00

2.

a) 32°

b) ≈ −17

c) 100°

3. (𝑑 − 5)0.7 = 𝑒 : (50 − 5)0.7 = 𝑒 ∶ 31.50 = 𝑒

4. ≈ 9𝑙𝑏𝑠

5. ≈ 20𝑙𝑏𝑠

6. ≈ 5.5𝑘𝑔

7. ≈ 7.5𝑘𝑔



4.5 Intercepts by Substitution

Answers

1. An intercept is a coordinate describing a value on the 𝑥 or 𝑦-axis.

2. Any point on the 𝑥-axis is an 𝑥-intercept, so the point will have the form: (𝑥, 0)

3. (0, −6)(2, 0)

4. (0, 4)(2, 0)

5. (0, −21) (1.5, 0)

6. (0, 7) (21

3, 0)

7. A vertical line (𝑥 ≠ 0) will have only an 𝑥-intercept.

8. 𝑦 = 5 has only a 𝑦-intercept as it runs above and parallel to the 𝑥-axis.

9. 𝑥 = −4

10. An infinite number.



4.6 Intercepts and the Cover-Up Method

Answers

1. The “Cover-Up” method involves solving the equation for the constant, then covering up each

variable and associated coefficient in turn and solving for the visible variable.

2. Answers will vary

3. (0, −2.5) (3, 0)

4. (0, 1.25) (−12

3, 0)

5. (0, −11

7) (−

11

2, 0)

6. (0, 2.5) (5, 0)

7. (0, 3) (−1.5, 0)

8. (2, 0) (0, −6)



9. (0, −5) (5, 0)

10. (0, 8) (8, 0)

11. (0, 0)



12. (24, 0) (0, 3)

13. (0 − 2) (4, 0)

14. (10

7, 0) (0, −2)



15. (0, 3) (−3

4, 0)

16. (0, 0)

17. (0, 5) (1, 0)



18. (0, 3) (−6

7, 0)

19. Distribute the 3 and the 2

20. 3𝑠 + 𝑏 = 10 : Where 𝑠 is the horizontal axis



21. 7.5𝑥 + 4.5𝑦 = 900 :

22. 6𝑡 + 3𝑓 = 36 : Where 𝑡 is the horizontal axis



4.7 Slope

Answers

1. Slope is the relationship between change in horizontal and vertical location between different

points on the graph of a linear equation.

2. Answers will vary

3. A vertical line has an undefined slope, since it represents division by 0.

4. A horizontal line has a slope of 0, since it represents zero divided by a constant.

5. a) (−1, −6) (3, 6) =12

4=

3

1= 3 b) (−6, −2) (0, 1) =

3

6=

1

2

6. c) (−1, 6) (5, −6) = −12

6= −2 d) (−2, −4) (4, 2) =

6

6= 1

7. d/e) (4, −6) (4,2) =8

0= undefined f) (−6, −2) (3, 1) =

3

7

8. −7

5

9. 16

6= 2

2

3

10. 14

−5= −2

4

5

11. 4

0 = undefined

12. −18

−18= 1

13. 2

−5= −

2

5

14. 5

1

4

−21

2

= −21

10= −2

1

10

15. 5

6

16. 0

21= 0



17. −21

0 = undefined

18. 2

3

19. −5

5= −1

20. 1

42

3

=3

8

21. Slope = 0

22. Slope = undefined



4.8 Rates of Change

Answers

1. Slope and Rate of Change are the same.

2. Section ‘B’ represents the stop light, and Section ‘E’ represents the tire change. All other

sections are motion, with steeper slopes representing shallower-angled hillsides.

3. 155

3= 51.66̅

4. 605

2

=24

1

5. Answers will vary but should describe a situation wherein Geoffrey is associated with an

increase in altitude of 10ft/sec



4.9 Slope-intercept Form

Answers

1. 𝑚 = 2, 𝑏 = 5

2. 𝑚 = −0.2, 𝑏 = 7

3. 𝑚 = 1, 𝑏 = 0

4. 𝑚 = 0, 𝑏 = 3.75

5. 𝑚 =2

3, 𝑏 = −9

6. 𝑚 = −0.01, 𝑏 = 10,000

7. 𝑚 =3

5, 𝑏 = 7

8. Not a line, equation describes the point 𝑥 = −8

5

9. 𝑚 = −2

4= −

1

2

10. 𝑚 = 0

11. 𝑚 = −2

1= −2

12. 𝑚 =4

1= 4

13. 𝑚 = −4

3



14. 𝑚 =2

5

15. 𝑚 = −2

8= −

1

4

16. 𝑚 = −1, 𝑏 = 0

17. 𝑚 = −2

3, 𝑏 = 1

1

3

18. 𝑚 = −1

5, 𝑏 = −1

19. 𝑚 = 3, 𝑏 = 1

20. 𝑚 = 0, 𝑏 = 3

21. 𝑚 =1

2, 𝑏 = −2



4.10 Graphs Using Slope-Intercept Form

Answers

1.

2.

3.



4.

5.

6.



7.

8.

9. 𝑚 = 2

10. 𝑚 = −0.2

11. 𝑚 = −1

12. 𝑚 = 0

13. 𝑚 = −1

5

14. 𝑚 = −5

15. 𝑚 = −3

16. 𝑚 = 3



4.11 Direct Variation

Answers

1. Direct variation indicates that both variables in a situation rise and fall at a constant relative

rate and when one variable is 0, the other is also.

2. 𝑦 = (𝑘)𝑥, 𝑘 is the constant of proportionality

3. 𝑚 = 𝑘ℎ

4. 𝑀 = 𝑘𝐸 𝑜𝑟 𝑊𝑀 = 𝑘𝑊𝐸

5. 𝑉 = 𝑘𝐾 𝑜𝑟 𝑉 = 𝑘𝑇 𝑜𝑟 𝑉𝑔 = 𝑘𝑇𝐾

6. 𝑝 = 𝑘𝑚

7. 𝑐 = 𝑘𝑝 𝑜𝑟 𝐴 = 𝑘𝑝

8. This is an inverse variation, 𝑦 decreases as 𝑥 increases.

9. There is only one variable (𝑦).

10. There is only one variable (𝑥).

11. The point (0, 0) is not a solution.

12. The point (0, 0) is not a solution.



13.

14.

15.

16.

17. No, it is not direct since (0, 0) is not a solution.



4.12 Applications Using Direct Variation

Answers

1. Answers will vary: Cross Products and Isolating the Variable

2. False

3. 𝑘 = 12

4. 𝑘 = 47

5. 𝑘 = −7

6. 𝑘 = 5.15

7. 𝑘 = 8

8. $12.50

5=

$𝑥

2 ∶ 𝑥 = $5

9. 57.14 minutes

10. 12 minutes

11. a) R = 9Ω :

b) 585 V

12. ≈ 4.78 𝑖𝑛

13. Noon (14 hrs after start)

14. $51,853.45

15. a) 𝑘 = 1.2

b) 8.4 N

c) 19.167 cm

16. a) 𝑦 = 3𝑥

b) 𝑦 = −2𝑥

c) 𝑦 = −1

5𝑥

d) 𝑦 =2

9𝑥



4.13 Function Notation and Linear Functions

Answers

1. 𝑓(𝑥) is read as “f of x”

2. Answers will vary: Function notation allows the same equation to be used with different

values, particularly useful in the sciences.

3. A function is an equation with two variables where each input relates to one and only one

output.

4. Function, each 𝑥 relates to a unique 𝑦.

5. Not a function, each 𝑥 (aside from the vertex) relates to two different 𝑦’s.

6. Not a function, each 𝑥 (aside from the vertices) relates to two different 𝑦’s.

7. Function, each 𝑥 relates to a unique 𝑦.

8. 𝑓(𝑥) = 7𝑥 − 21

9. 𝑓(𝑥) = −2

3𝑥 + 4

1

2

10. 𝑓(𝑥) =1

9𝑥 −

1

3

11. 𝑓(𝑥) = 6

12. 𝑓(𝑡) = 65𝑡 + 100

13. 𝑓(𝐶) = 1.8𝐶 + 32

14. 𝑓(𝑚) = 0.10𝑚 + 25,000



15. a) 9

b) -11

c) 3

d) −2𝑧 + 3

16. a) 1.1

b) 8.1

c) 3.2

d) 0.7𝑧 + 3.2

17. a) 25

11

b) −25

11

c) 0

d) 5(2−𝑧)

11

18. a) 8.5

b) 28.5

c) 4

d) 1

2𝑧2 + 4

19. a) 4.5

b) −1

2

c) 3

d) 3 −1

2𝑧



4.14 Graphs of Linear Functions

Answers

1. a) 𝑓(𝑥) = 8𝑥 + 100 b) 180 min c) 316 min d) 21.25 lbs

2. 212 ℉ is the boiling point of water in degrees Fahrenheit.

3. Answers may vary: 𝑚(𝑑) =𝑑

0.16 : 𝑚(20) =

20

0.16= 125 𝑚𝑖𝑛

4. 𝑏(ℎ) = 330ℎ : 𝑏(. 75) = 247.50 𝑐𝑎𝑙𝑜𝑟𝑖𝑒𝑠 : The number of calories burned in 45 mins

5. a) 𝑏(𝑤) = 𝑏 − 55𝑤 :

𝑏) 𝑏(10) = 100 : She will run out of money at 12 weeks



4.15 Problem Solving with Linear Graphs

Answers

1. a) 40 ℎ𝑟𝑠 ≈ $350

b) 30 hrs

c) 𝑚 =200

30=

20

3= 6

2

3 Aatif earns $6.75 per hr

d) 𝑏 = $50 : Aatif makes a flat rate of $50 per job before his $6.75 per hr

2. 8 inches (𝑦 = 2𝑥 + 8)

3. $668

4. 52

3𝑖𝑛 (𝑙 = 5

2

3−

1

3𝑚)

5. 0.0023𝑖𝑛

6. 56 glasses (technically 55.55̅)

7. $2.53

8. 3.375 mi @ 45 mins : 4.3 mi @ 55 mins

4.1 Graphs in the Coordinate Plane

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