chapter 2 ism - topcatmathtopcatmath.com/docs/math030/chapter2.ism.pdfchapter 2 linear equations in...

127

Chapter 2 Linear Equations in Two Variables and Functions

Section 2.1 Practice Exercises

1. a. x; y-axis

b. ordered

c. origin; (0, 0)

d. quadrants

e. negative

f. III

g. Ax By C

h. x-intercept

i. y-intercept

j. vertical

k. horizontal

2. a. y-axis

b. x-axis

3. For (x, y), if x > 0, y > 0, the point is in

quadrant I. If x < 0, y > 0, the point is

in quadrant II. If x < 0, y < 0, the point

is in quadrant III. If x > 0, y < 0, the

point is in quadrant IV.

4. The order of the x- and y-coordinates is

important. For example, (2, 3) and (3, 2)

define two different points.

5.

6.

7. 0 8. 0

9. A (–4, 5), II

B (–2, 0), x-axis

C (1, 1), I

D (4, –2), IV

10. A (–1, 3), II

B (2, 1), I

C (0, –3), y-axis

D (4, –3), IV


128

E (–5, –3), III E (–2, –2), III

11. a. 2 0 3 3 9

0 9 9

9 9

0, 3 is a solution.

12. a. 5 0 2 3 6

0 6 6

6 6

0, 3 is not a solution.

b. 2 6 3 1 9

12 3 9

15 9

6,1 is not a solution.

b. 65 2 0 6

5

6 0 6

6 6

6, 0

5

is a solution.

c. 72 1 3 9

3

2 7 9

9 9

71,

3

is a solution.

c. 5 2 2 2 6

10 4 6

6 6

2, 2 is a solution.

13. a. 11 0 1

31 0 1

1 1


14. a. 34 0 4

24 0 4

4 4

0, 4 is a solution.

b. 12 3 1

32 1 1

2 2

2, 3 is a solution.

b. 37 2 4

27 3 4

7 7

2, 7 is a solution.

c. 16 1 1

31

6 134

63

6,1 is not a solution.

c. 32 4 4

22 6 4

2 2


Section 2.1 Linear Equations in Two Variables

129

15. 3 2 4x y 16. 4 3 6x y x y x y

0 –2 0 2

4 4 3 –2

–1 72

94

–1

17. 1

5y x

18. 1

3y x

x y x y

0 0 0 0

5 –1 3 1

–5 1 6 2

19.

20.


130

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.


131

31. To find an x -intercept, substitute y = 0

and solve for x. To find a y-intercept,

substitute x = 0 and solve for y.

32. No. Neither the x- nor the y-coordinate is

zero.

33. a.

2 3 18

2 3 0 18

2 18

9

x yx

xx

The x -intercept is (9, 0).

34. a.

2 5 10

2 5 0 10

2 10

5

x yx

xx


b. 2 0 3 18

3 18

6

yyy

The y-intercept is (0, 6).

b. 2 0 5 10

5 10

2

yyy

The y-intercept is (0, –2).

c. c.

35. a. 2 4

2 0 4

4

x yx

x


36. a.

8

0 8

8

x yx

x


b. 0 2 4

2 4

2

yyy


b. 0 8

8

yy


c. c.


132

37. a.

5 3

5 3 0

5 0

0

x yxxx


38. a. 3 5

3 0 5

0 5

0

y xxx

x


b. 5 0 3

0 3

0

yy

y


b. 3 5 0

3 0

0

yyy


c. c.

39. a. 2 4

0 2 4

2 4

2

y xx

xx

The x -intercept is (–2, 0).

40. a. 3 1

0 3 1

3 1

1

3

y xx

x

x

The x -intercept is 13, 0 .

b. 2 0 4

4

yy


b. 3 0 1

1

yy


c. c.


133

41. a. 42

34

0 23

42

33 3

24 2

y x

x

x

x


42. a.

21

52

0 15

21

55 5

12 2

y x

x

x

x


b. 40 2

32

y

y


b. 20 1

51

y

y


c. c.

43. a.

1

41

040

x y

x

x


44. a.

2

32

030

x y

x

x


b. 10

40

y

y


b. 20

30

y

y


c. c.


134

45. a.

15,000 0.08

15,000 0.08 500,000

15,000 40,000

55,000

y xy

The salary is $55,000.

46. a. The charge is constant for less

than 1 mile.

b. 15,000 0.08 300,000

15,000 24,000

39,000

y

The salary is $39,000.

b. The y-intercept is (0, 3.50). The

base fare is $3.50.

c. 15,000 0.08 0

15,000 0

15,000

y

The y-intercept is (0, 15,000). For $0

in sales, the salary is $15,000.

c. After the first mile, the cost

increases with increasing miles.

d. Total sales cannot be negative. d.

3.50 2.50 1

3.50 2.50 3.5 1

3.50 2.50 2.5

3.50 6.25

9.75

C mC

It would cost $9.75 to take a cab

3.5 miles.

47. a.

1500 300

1500 300 1

1500 300

1200

y x

A computer will be worth $1200

1 yr after purchase.

48. a.

3.6 59

3.6 10 59

36 59

23

y x

The air temperature at 10,000 ft

is 23 F .

b. 1500 300

300 1500 300

1200 300

4

y xx

xx

After 4 yr the computer will be worth

$300.

b. 3.6 59

5.8 3.6 59

64.8 3.6

18

y xxx

x

The air temperature is 5.8 F at

18,000 ft.


135

c.

1500 300

1500 300 0

1500

y x

(0, 1500);

The y-intercept represents the initial

value of the computer.

c.

3.6 59

3.6 0 59

59

y x

(0, 59);

The y-intercept represents the

temperature at an altitude of 0 ft

(temperature at sea level).

d. 1500 300

0 1500 300

300 1500

5

y xx

xx

(5, 0);

The x-intercept indicates that once

the computer is 5 yr old, its value is

$0.

d. 3.6 59

0 3.6 59

3.6 59

16.4

y xx

xx

(16.4, 0);

The x-intercept indicates that at

approximately 16,400 ft in

altitude, the temperature will be

0 F .

49. 1y Horizontal;

No x-intercept; y-intercept (0, –1)

50. 3y Horizontal;

No x-intercept; y-intercept (0, 3)

51. 2x Vertical;

x-intercept (2, 0); No y-intercept

52. 5x Vertical;

x-intercept (–5, 0); No y-intercept


136

53. 2 6 5

2 1

1

2

xx

x

Vertical;

x-intercept 1

, 02

; No y-intercept

54. 3 12

4

xx

Vertical;

x-intercept 4, 0 ; No y-intercept

55. 2 1 9y

2 8

4

yy

Horizontal;

No x-intercept; y-intercept (0, – 4)

56. 5 10y

2y

Horizontal;

No x-intercept; y-intercept (0, 2)

57. A horizontal line parallel to the x-axis will

not have an x-intercept. A vertical line

parallel to the y-axis will not have a y-

intercept.

58. No, a line must cross at least one axis.

59. b, c, d 60. a, b, c


137

61. 1

2 30

12 3

12

2

x y

x

x

x

01

2 3

13

3

y

y

y

The x-intercept is (2, 0) and the

y-intercept is (0, 3).

62. 1

7 40

17 4

17

7

x y

x

x

x

01

7 4

14

4

y

y

y

The x-intercept is (7, 0) and the

y-intercept is (0, 4).

63. 1

01

1

x ya bxa b

xax a

01

1

ya b

yby b

The x-intercept is (a, 0) and the

y-intercept is (0, b).

64. 0

Ax By CAx B C

Ax CCxA

0A By C

By CCyB

The x-intercept is , 0CA and the

y-intercept is 0, CB .

65. 2 7

3 3y x

66. 2 1y x

67. y = 3

68. y= –5


138

69.

70.

71.

72.

73. The lines look nearly indistinguishable.

However, the linear equations are

different so the lines are different.

74. The lines look nearly indistinguishable.

However, the linear equations are

different so the lines are different.


1. a. slope; 2 1

2 1

y yx x

b. parallel; same

c. right

d. 1

2. a. II

b. III

c. I

d. IV

Section 2.2 Slope of a Line and Rate of Change

139

3. a.

14

21

0 42

4

x y

y

y

The ordered pair is (0, 4).

4. 2 8 0

2 8

4

xxx

The x-intercept is (–4, 0).

There is no y-intercept.

b. 10 4

21

42

8

x

x

x

The ordered pair is (8, 0).

c. 14 4

22 4

6

y

yy

The ordered pair is (–4, 6).

5. 4 2 0

2 4

2

yyy

There is no x-intercept.


6.

2 2 6 0

2 2 0 6 0

2 6

3

x yx

xx

2 0 2 6 0

2 6

3

yyy

The x-intercept is (3, 0).


7. 24 24

7 7m 8. 4 2

10 5m

9. 8 1

72 9m 10. 150 3

500 10m


140

11. 4 1

100 25m 12. 1000 1

12,000 12m

The plane gains 1 ft of elevation for every

12 ft it travels horizontally.

13. 2 1

2 1

3 0

0 63 1

6 2

y ym

x x

14.

2 1

2 1

4 0

0 5

4 4

5 5

y ym

x x

15.

2 1

2 1

7 3

4 2

10 5

6 3

y ym

x x

16.

2 1

2 1

7 4

1 5

3 1

6 2

y ym

x x

17.

2 1

2 1

3 5

2 2

82

4

y ym

x x

18.

2 1

2 1

8 2

6 46

32

y ym

x x

19.

2 1

2 1

0.8 1.1

0.1 0.3

0.3

0.4

3

4

y ym

x x

20.

2 1

2 1

0.1 0.2

0.3 0.40.1

0.1

1

y ym

x x


141

21. 2 1

2 1

7 3

2 24

Undefined0

y ym

x x

22.

2 1

2 1

0 5

1 1

5 Undefined

0

y ym

x x

23.

2 1

2 1

1 1 00

3 5 8

y ym

x x

24.

2 1

2 1

4 4 00

91 8

y ym

x x

25.

2 1

2 1

6.4 4.1 2.3 1

4.6 20 4.6

y ym

x x

26. 2 1

2 1

0.3 4 4.31

3.2 1.1 4.3

y ym

x x

27. 2 1

2 1

41

37 3

2 21

1 1 134 3 2 6

2

y ym

x x

28. 3 1

2 2

1 2

6 31

5

66 6

15 5

m

29. 2 1

2 1

1 7 7 72

03 3 3 3 01 3 2 3 1

2 4 4 4 4

y ym

x x

30. 1 2

10 51 9

24 4

1 4 3

10 10 10 Undefined9 9 0

4 4

m


142

31. The slope of a line is positive if the

graph increases from left to right. The

slope of a line is negative if the graph

decreases from left to right. The slope

of a line is zero if the graph is

horizontal. The slope of a line is

undefined if the graph is vertical.

32. 4

3 63 24

8

y

yy

The change in y will be 8 units.

33. 0m 34. 1m

35. 1

10m 36. Undefined

37. 1m 38. 1

2m

39. a. 5m 40. a. 3m b. 1

5m

b. 1

3m

41. a. 4

7m 42. a. 2

11m

b. 7

4m

b. 11

2m

43. a. 0m 44. a. is undefined.m b. is undefined.m b. 0m

45. No, because the product of the slopes

of perpendicular lines must be –1. The

product of two positive numbers is not

negative.

46. 2x is the equation of a vertical line;

thus, a perpendicular line will be a

horizontal line whose slope is 0.m

47. 5y is the equation of a horizontal

line; thus a perpendicular line will be a

vertical line whose slope is undefined.

48. 3x is the equation of a vertical line;

thus, a parallel line will also be a vertical

line whose slope is undefined.


143

49. 0m 50. 0m

51. undefined 52. undefined

53. 1

9 5 42

4 2 2Lm

2

2 4

3 1

2 1

4 2

Lm

The lines are perpendicular.

54. 1

2 5 7

1 3 2Lm

2

2 4

7 02 2

7 7

Lm


55. 1

1 2

3 41

11

Lm

2

16 1

10 5

153

5

Lm

The lines are neither parallel nor

perpendicular.

56. 1

3 0

2 03

2

Lm

2

2 5

0 2

7

27

2

Lm


perpendicular.

57. 1

2

9 3

5 56

Undefined02 2

0 40

04

L

L

m

m

The lines are perpendicular. One line is

horizontal and the other is vertical.

58. 1

2

5 5

2 30

01

4 4

0 20

02

L

L

m

m

The lines are parallel.


144

59.

1

2

3 2

2 3

51

55 1

0 4

41

4

L

L

m

m


60.

1

2

0 1

0 71 1

7 7

6 8

4 10

2 1

14 7

L

L

m

m


61. a. 313 70

2010 1998243

1220.25

m

62. a. 304 282

8 022

82.75

m

b. The number of cell phone

subscriptions increased at a rate of

20.25 million per year during this

period.

b. The U.S. population has grown at a

rate of 2.75 million people per year

since the year 2000.

63. a. 74.5 44.5

10 530

56

m

64. a. 87.5 42.5

11 545

67.5

m

b. The weight of boys tends to

increase by 6 lb/yr during this

period of growth.

b. The weight of girls tends to

increase by 7.5 lb/yr during this

period of growth.

65. a. ( 1, 4) and (0, 2)

2 ( 4)

0 ( 1)

2

12

m

b. (0, 2) and (3, 4)

66. a. ( 4, 1) and (0,1)

1 ( 1)

0 ( 4)

2

41

2

m

b. (0,1) and (2, 2)


145

4 ( 2)

3 06

32

m

c. ( 1, 4) and (3, 4)

4 ( 4)

3 ( 1)

8

42

m

2 1

2 01

2

m

c. (0,1) and (4, 3)

3 1

4 02

41

2

m

67. a. ( 2, 0) and (0, 4)

4 0

0 ( 2)

4

22

m

b. (0, 4) and (2, 0)

0 ( 4)

2 04

22

m

c. (0, 4) and (3, 5)

5 ( 4)

3 09

33

m

68. a. ( 3, 8) and ( 2, 3)

3 ( 8)

2 ( 3)

5

15

m

b. ( 2, 3) and (0,1)

1 ( 3)

0 ( 2)

4

22

m

c. (0,1) and (2, 3)

3 1

2 04

22

m

69. For example: (1, 2)



146





75.

6 3

4 2 2

y

6 3

6 23

6 62

6 9

15

15

y

y

yyy

76. 2 4 6

3 7x

6 6

3 76 7 6 3

42 18 6

24 6

4

xxx

xx

77. a. 4Pitch

241

6

b. 4

121

3

m


1. a. y mx b

b. standard

c. horizontal

d. vertical

e. slope; y-intercept

f. 1 1( )y y m x x

Section 2.3 Equations of a Line

147

2. 1

2 30

12 3

12

2

x y

x

x

x

01

2 3

13

3

y

y

y

c.

a. The x-intercept is (2, 0). b. The y-intercept is (0, 3).

3. a. 0 4. Two lines with the same slope are

parallel. b. undefined

5. If the slope of one line is the opposite of

the reciprocal of the slope of the other

line, then the lines are perpendicular

6. 2 1

2 1

y ym

x x

7. 24

3y x

Slope: 23

y-intercept: (0, –4)

8. 31

7y x

Slope: 37


9. 2 3

3 2

y xy x

Slope: 3

y-intercept: (0, 2)

10. 5 7

7 5

y xy x

Slope: –7

y-intercept: (0, 5)

11. 17 0

17

x yy x

Slope: –17

y-intercept: (0, 0)

12. 0x yy x

Slope: –1

y-intercept: (0, 0)

13. 18 2

9

0 9

yy

y x

Slope: 0

y-intercept: (0, 9)

14. 17

214

0 14

y

yy x

Slope: 0


148


15. 8 12 9

12 8 9

8 9

12 122 3

3 4

x yy x

y x

x

Slope: 2

3

y-intercept: 3

0, 4

16. 9 10 4

10 9 4

9 4

10 109 2

10 5

x yy x

y x

x

Slope: 9

10

y-intercept: 2

0, 5

17. 0.625 1.2y x

Slope: 0.625

y-intercept: (0, –1.2)

18. 2.5 1.8y x

Slope: –2.5

y-intercept: (0, 1.8)

19. d 20. c

21. f 22. a

23. b 24. e

25. 2 4

4 2

y xy x

26. 3 5

3 5

3 5

x yy x

y x


149

27. 3 2 6x y

2 3 6

33

2

y x

y x

28. 2 8x y

2 8

14

2

y x

y x

29. 2 5 0

5 2

2

5

x yy x

y x

30. 3 0

3

3

x yx yy x

31. Ax By CBy Ax C

A Cy xB B

The slope is given by AmB

.

The y-intercept is 0, CB

.

32. 3 7 9

3, 7, 9

x yA B C

3

7m

y-intercept: 9

0, 7


150

33.

1

3 5 1

5 1

3 35

3

y x

y x

m

2

6 10 12

10 6 12

3 6

5 53

5

x yy x

y x

m


34.

1

6 3

6 3

1 1

6 21

6

x yy x

y x

m

2

13 0

21

32

6

6

x y

y x

y xm


35.

1

3 4 12

4 3 12

33

43

4

x yy x

y x

m

2

1 21

2 31 2

6 6 12 3

3 4 6

4 3 6

3 3

4 23

4

x y

x y

x yy x

y x

m


36.

1

4.8 1.2 3.6

4.8 3.6 1.2

4 3

4

x yx y

y xm

2

1 4

4 1

4

y xy x

m


37.

1

3 5 6

52

35

3

y x

y x

m

2

5 3 9

3 5 9

53

35

3

x yy x

y x

m


perpendicular.

38.

1

3 2

3 2

3

y xy x

m

2

6 2 6

2 6 6

3 3

3

x yy xy x

m


perpendicular.

39. 3, point: 0, 5m

3 5

y mx by x

40. 4, point: 0, 3m

4 3

y mx by x


151

41. 2, point: 4, 3m

3 2 4

3 8

11

2 11

y mx bb

bb

y x

42. 3, point: 1, 5m

5 3 1

5 3

8

3 8

y mx bb

bb

y x

43. 4, point: 10, 0

5m

40 10

50 8

8

48

5

y mx b

b

bb

y x

44. 2, point: 3,1

7m

21 3

76

17

1

72 1

7 7

y mx b

b

b

b

y x

45. 3, -intercept: 0, 2m y

3 2 y x or

3 2x y

46. 1, -intercept: 0, 5

2m y

1 15 or 5

2 2 2 10

y x x y

x y

47. 2, point: 2, 7m

1 1

7 2 2

7 2 4

2 3

or

2 3

y y m x xy xy x

y x

x y

48. 2, point: 3,1 0m

1 1

10 2 3

10 2 6

2 16

or

2 16

y y m x xy xy x

y x

x y

49. 3, point: 2, 5m 50. 4, point: 1, 6m


152

1 1

5 3 2

5 3 6

3 11

or

3 11

y y m x x

y xy x

y x

x y

1 1

6 4 1

6 4 4

4 2

or

4 2

y y m x x

y xy x

y x

x y

51. 4, point: 6, 3

5m

1 1

43 6

54 24

35 54 9

5 5

or

4 9 5 54 5 9

y y m x x

y x

y x

y x

x y

x y

52. 7, point: 7, 2

2m

1 1

72 7

27 49

22 27 53

2 2

or

7 53

2 27 2 53

y y m x x

y x

y x

y x

x y

x y

53. 40 4 4

3 0 3 3m

1 1

44 0

34

4 034

4 3

or

44

34 3 12

y y m x x

y x

y x

y x

x y

x y

54. 7 1 63

3 1 2m

1 1

1 3 1

1 3 3

3 2

or

3 2

y y m x xy xy x

y x

x y

55. 10 12 21

4 6 2m

56.

4 1 3 3

3 2 5 5m


153

1 1

12 1 6

12 6

6

or

6

y y m x xy xy x

y x

x y

1 1

31 2

53 6

15 53 11

5 5

or

3 11

5 53 5 11

y y m x x

y x

y x

y x

x y

x y

57.

2 2

1 5

0

40

m

1 1

2 0 5

2 0

2

y y m x x

y xy

y

58.

1 1

2 4

0

60

m

1 1

1 0 4

1 0

1

y y m x x

y x

yy

59. 3, point: 3,2

4m

1 1

32 3

43 9

24 43 17

4 4

y y m x x

y x

y x

y x

or

3 17

4 43 4 17

x y

x y

60. 1, point: 1, 4

2m

1 1

14 1

21 1

42 21 9

2 2

or

1 9

2 2 2 9

y y m x x

y x

y x

y x

x y

x y

61. 4, point: 3,2

3m

62. 2, point: 2, 5m


154

1 1

42 3

34

2 434

2 3

or

42

34 3 6

y y m x x

y x

y x

y x

x y

x y

1 1

5 2 2

5 2 4

2 1

or

2 1

y y m x x

y x

y xy x

x y

63.

3 4 7

4 3 7

3 7

4 43

, point: 2, 54

x yy x

y x

m

1 1

35 2

43 3

54 23 13 3 13

or 4 2 4 2

3 4 26

y y m x x

y x

y x

y x x y

x y

64.

2 3 12

3 2 12

24

32

, point: 6, 13

x yy x

y x

m

1 1

21 6

32

1 432 2

5 or 53 3

2 3 15

y y m x x

y x

y x

y x x y

x y

65.

15 3 9

3 15 9

5 3

15, , point: 8, 1

5

x yy xy x

m m

1 1

11 8

51 8

15 51 13 1 13

or 5 5 5 5

5 13

y y m x x

y x

y x

y x x y

x y

66.

4 3 6

3 4 6

42

34 3

, , point: 4, 23 4

x yy x

y x

m m

1 1

32 4

43

2 343 3

5 or 54 4

3 4 20

y y m x x

y x

y x

y x x y

x y


155

67.

3 2

3

23

, point: 4, 02

x y

y x

m

1 1

30 4

23 3

6 or 62 2

3 2 12

y y m x x

y x

y x x y

x y

68.

5 6

5

65

, point: 3, 06

x y

y x

m

1 1

50 3

65 5 5 5

or 6 2 6 2

5 6 15

y y m x x

y x

y x x y

x y

69.

3 2 21

3 2 21

27

33

, point: 2, 42

y xy x

y x

m

1 1

34 2

23

4 323 3

1 or 12 2

3 2 2

y y m x x

y x

y x

y x x y

x y

70.

7 21

7 21

13

77, point: 14, 8

y xy x

y z

m

1 1

8 7 14

8 7 98

7 90 or 7 90

y y m x x

y xy x

y x x y

71.

1

22

1, point: 3, 5

2

y x

y x

m

1 1

15 3

21 3

52 21 7 1 7

or 2 2 2 2

2 7

y y m x x

y x

y x

y x x y

x y

72.

1

44

1, point: 1, 5

4

y x

y x

m

1 1

15 1

41 1

54 41 19 1 19

or 4 4 4 4

4 19

y y m x x

y x

y x

y x x y

x y


156

73.

3 7

3 7

3, point: 0, 0

x yy x

m

1 1

0 3 0

3

or

3 0

y y m x xy x

y x

x y

74.

2 5

2 5

2, point: 0, 0

x yy x

m

1 1

0 2 0

2

or

2 0

y y m x xy x

y x

x y

75. 0, point: 2, 3m

1 1

3 0 2

3 0

3

y y m x xy x

yy

76. A line with an undefined slope is a

vertical line, which is in the form x c .

Therefore, a line containing 52

, 0 would

have the equation 5

2x .

77. A line with an undefined slope is a


Therefore, a line containing (2, –3)

would have the equation 2x .

78. 50, point: , 0

2m

1 1

50 0

2

0

y y m x x

y x

y

79. A line parallel to the x-axis has the

form y c . Therefore, a line

containing the point (4, 5) would have

the equation 5y .

80. A line perpendicular to the x-axis is a


Therefore, a line containing (4, 5) would

have the equation 4x .

81. A line parallel to the line 4x is a

vertical line and has the form x c .

Therefore, a line containing the point

(5, 1) would have the equation 5x .

82. A line parallel to the line 2y is a

horizontal line and has the form y c .

Therefore, a line containing the point (–3,

4) would have the equation 4y .


157

83. 2x is not in the slope-intercept

form. It has no y-intercept and its slope

is undefined.

84. 1x is not in the slope-intercept form. It

has no y-intercept and its slope is

undefined.

85. 3y is in the slope-intercept form,

0 3y x . Its slope is 0 and the y-

intercept is (0, 3).

86. 5y is in the slope-intercept form,

0 5y x . Its slope is 0 and the y-

intercept is (0, –5).

87. Two points on the line are (0, 3) and

(1, 1).

1 3 22

1 0 1m

1 1 1 1( ) ( , ) (0, 3)

3 2( 0)

2 3

y y m x x x yy x

y x

88. Two points on the line are (1, 1) and (2,

4).

4 1 33

2 1 1m

1 1 1 1( ) ( , ) (1,1)

1 3( 1)

1 3 3

3 2

y y m x x x yy xy x

y x

89. This is a horizontal line of the form

y k . The y-intercept is 2, so the line is

2y .

90. This is a vertical line of the form x k .

The x-intercept is 4 , so the line is

4x .

91. The lines have the same slope but

different y-intercepts; they are parallel

lines.

92. The lines have the same slope but

different y-intercepts; they are parallel

lines.

93. The lines have different slopes but the

same y-intercept.

94. The lines have different slopes but the

same y-intercept.


158

95. The lines are perpendicular.

96. The lines are perpendicular.

97.

98.

Problem Recognition Exercises 1. b, f 2. a, c, d, h

3. a 4. b, g

5. c, e 6. a

Section 2.4 Applications of Linear Equations and Modeling

159

7. c, h 8. b

9. e 10. g

11. c, h 12. a

13. g 14. e

15. h 16. f

17. e 18. g

19. d, h 20. b, f


1. model 2. 30 40 1200x y x-intercept: 30 40(0) 1200

30 1200

40

(40, 0)

xxx

y-intercept: 30(0) 40 1200

40 1200

30

(0, 30)

yyy

3. a.

2 0 2 1

3 3 6 3m

4. a. 5 1 42

3 1 2m

b. 10 3

31 1

1 or 13 3

3 3

y x

y x x y

x y

b. 1 2 1

1 2 2

2 1 or 2 1

y xy x

y x x y


160

c.

c.

5. a. b. c.

3 3

2 4

0

20

m

3 0 4

3 0

3

y xy

y

6. undefined

7. a. 120 65y x 8. a. 0.04 1000y x

b.

b.

c. The y-intercept is (0, 65). The base

cost to rent the car is $65.

c. The y-intercept is (0, 1000).

Alex’s base salary (with $0 sales) is

$1000. d. 120 2 65

240 65

305

y

A 2-day rental costs $305.

d. 40.04

100m .

Alex receives $4 for every $100 sold.


161

120 5 65

600 65

665

y


120 7 65

840 65

905

y


e. Yes; $799 is less expensive than

$905.

e. 0.04 30,000 1000

1200 1000 2200

y

Alex will make $2200 if he has sales

of $30,000.

f.

1.06 120 65

1.06 120 4 65

1.06 480 65

1.06 545

577.7

y x

A 4-day rental with 6% sales tax

costs $577.70.

g. No, the car cannot be driven for a

negative number of days.

9. a. 52 2742y x 10. a. 0.019 156y x

b.

b.

0.019 156

0.019 90,000 156

1710 156

1866

y x

The amount of property tax is

$1866.

c. 52m . The taxes increase $52 per

year.

c. 0.019 156y x

2436 0.019 156

2280 0.019

xx

120,000 x The taxable value of the home is

$120,000.

d. The y-intercept is (0, 2742). In the

initial year (x = 0) the taxes were

$2742.

e. 52 10 2742

520 2742 3262

y


162

After 10 years the taxes are $3262.

52 15 2742

780 2742 3522

y

After 15 years the taxes are $3522.

11. a. 0.2 4 0.8y

The storm is 0.8 mi away when the

time difference is 4 sec.

0.2 12 2.4y



0.2 16 3.2y



12. a. 2.5 6 15y

15 pounds of force stretch the

spring 6 in.

2.5 12 30y

30 pounds of force stretch the

spring 12 in.

b. 4.2 0.221

xx

When the storm is 4.2 miles away,

the time difference is 21 sec.

b. 45 2.5

18

xx

When 45 pounds of force are

applied, the spring will stretch 18

in.

13. a. The average amount spent per

person in Year 4 is calculated as

follows:

9.4 35.7

9.4 4 35.7

37.6 35.7

73.3

y x

The average amount spent per

person in Year 4 is $73.30.

14. a. The year 25 is represented by x =

25.

142 25 9060

3550 9060

12,610

y

The average yearly mileage for

passenger cars in the United States

in 25 is 12,610 mi.

b. The average amount spent per

person in Year 2 is calculated as

follows:

9.4 35.7

9.4 2 35.7

18.8 35.7

54.5

y x

b. The Year 5 is represented by x = 5.

142 5 9060

710 9060

9770

y

The average yearly mileage for

Year 5 is 9770 mi which is 70 mi

more than the actual value.


163

55.80 54.50 1.30 The approximate value differs from

the actual value by $1.30.

c. 9.4m ; The amount spent per

person on video games increased by

an average rate of $9.40 per year.

c. 142m . There is a 142 mi

increase per year.

d. (0, 35.7); The y-intercept means that

the average amount spent on video

games per person was $35.70 at the

start of the study (Year 0).

d. The y-intercept is (0, 9060). The

average mileage was 9060 mi at the

start of the study (Year 0).

15. a.

16. a. (4, 6) and (12, 4)

4 6 20.25

12 4 8m

1 1 1 1( ) ( , ) (4, 6)

6 0.25( 4)

6 0.25 1

0.25 7

y y m x x x yy xy x

y x

b.

24 15 9 30.3

50 20 30 1015 0.3 20

15 0.3 6

0.3 9

m

y xy x

y x

b. 0.25 15 7

1.25 7

3.25

y

3.25 ft

c. 0.3 40 9

12 9

21º F

y

c. 0.25 25 7

6.25 7

0.75

y

0.75 ft.

d. 0.3 50 9

15 9

24º F

y

d. 0.25m ; The slope means that the

water level decreases at a rate of

0.25 ft/day.

e. 0.3m . This means that

temperature decreases at a rate of

0.3ºF for every 1 mph increase in

wind speed.

e. 0.25 7

0 0.25 7

0.25 7

28

y xx

xx

(28, 0); The x-intercept represents


164

the number of days for the water

level to reach 0 ft. The linear trend

cannot continue for 28x , because

there will be no more water in the

pond.

17. a. 665 455 210

34 20 1415

m

455 15 20

455 15 300

15 155

y xy x

y x

18. a 1220 1003 217

8 4 4 =54.25

m

1220 54.25 8

1220 54.25 434

54.25 786

y xy xy x

b. The slope is 15 and means that the

number of associate degrees

awarded in the United States

increased by 15 thousand per year.

b. The number of prisoners increased at

a rate of 54.25 thousand per year

during this time period.

c. 15 45 155 675 155 830y The number of associate degrees

awarded in the United States for

Year 45 will be about 830

thousand (equivalently 830,000).

c. 54.25 25 786

1356.25 786 2142.25

y

2142.25 thousand (equivalently

2,142,250)

19. a.

20. a

b.

475 650 175175

3.50 2.50 1.00650 175 2.50

650 175 437.5

175 1087.5

m

y xy x

y x

b.

820 1020 200400

1.50 1.00 0.501020 400 1.00

1020 400 400

400 1420

m

y xy x

y x

c. 175 4.00 1087.5

700 1087.5 387.5

y

c. 400 2.00 1420

800 1420 620

y


165

Approximately 388 hotdogs would

be sold when the price of a hotdog

is $4.00.

Approximately 620 drinks would be

sold when the price of a drink is

$2.00.

21. a.

22. a.

b. Yes, there is a linear trend. b. Yes, there is a linear trend.

c.

3.1 2.2 0.90.05

28 10 182.2 0.05 10

2.2 0.05 0.5

0.05 1.7

m

y xy x

y x

c.

74 69 5 1

70 60 10 21

74 7021

74 3521

392

m

y x

y x

y x

d. 0.05 30 1.7 1.5 1.7 3.2y

A student who studies 30 hours per

week will have a GPA of

approximately 3.2.

d. 190 39

21512

102

x

x

x

Loraine needs to study for 102

minutes per day to make at least 90%

on her exam. This model is not

appropriate for other students because

it is based on Loraine’s scores.

130 39 15 39 54

2y

Loraine’s score will be 54% for a

week when she studies 30 min/day.

e. 0.05 46 1.7 2.3 1.7 4.0y

This model is not reasonable for

study times greater than 46 hours

per week, because the GPA would

exceed 4.0.


166

23.

5 4 1 1

0 3 3 32 5 3 1

9 0 9 3

2 4 2 1

9 3 6 3

m

m

m

Since the slopes are equal, the points

are collinear.

24.

41 3 1

4 4 8 2

2 1 3 1

6 22 4

2 3 1 1

2 4 2 2

m

m

m

Since the slopes are equal, the points are

collinear.

25.

12 2

2 010

=-2

= 5

6 12

1 2

-6=

1= 6

m

m

Since the slopes are not equal, the

points are not collinear.

26.

3 2

0 2

1=

21

=21 3

4 0

2=

4

1=

21 2

4 2

1=

21

=2

m

m

m

Since the slopes are equal, the points are

collinear.

27.

28.

Section 2.5 Introduction to Relations

167

29.

30.


1. a. relation 2. points 3, 4 and 1, 6

6 4 105

1 3 2m

1 1 1 1( ) ( , ) (3, 4)

4 5( 3)

4 5 15

5 11

y y m x x x yy x

y xy x

b. domain

c. range

3. a. 2 3 4

2 7

7

2

xx

x

vertical line

4. a. 2 3 4

3 2 4

2 4

3 3

x yy x

y x

slanted line

b. The slope is undefined because this

is a vertical line.

b. 2

3m

c. The x-intercept is

7, 0

2

. c. 2 3 0 4

2 4

2

xxx

The x-intercept is 2, 0 .

d. There is no y-intercept. d. The y-intercept is

40,

3

.


168

5. a. 3 2 4

2 3 4

32

2

x yy x

y x

slanted line

6. a. 2 3 4

3 2

2

3

yy

y

horizontal line

b. 3

2m

b. 0m

c. 3 2 0 4

3 4

43

xx

x

The x-intercept is 4

, 03

.

c. There is no x-intercept.

d. The y-intercept is 0, 2 . d. The y-intercept is

20,

3

.

7. a. Northeast, 54.1 , Midwest, 65.6 , South, 110.7 , West, 70.7

b. Domain: Northeast, Midwest, South, West ; Range: 54.1, 65.6, 110.7, 70.7

8. a. 1

20, 3 , 2, , 7,1 , 2, 8 , 5,1

b. Domain: 0, 2, 5, 7 ; Range: 12

3, ,1 , 8

9. a. USSR, 1961 , USA, 1962 , Poland, 1978 , Vietnam, 1980 , Cuba, 1980

b. Domain: USSR, USA, Poland, Vietnam, Cuba ; Range: 1961,1 962,1 978,1 980

10. a. Maine, 1820 , Nebraska, 1823 , Utah, 1847 , Hawaii, 1959 , Alaska, 1959

b. Domain: Maine, Nebraska, Utah, Hawaii, Alaska ; Range: 1820,1 823,1 847,1 959

11. a. ,1 , , 2 , , 2 , , 3 , , 5 , , 4A A B C D E

b. Domain: A, B, C, D, E ; Range: 1, 2, 3, 4, 5 12. a. 5, 3 , 10, 2 , 15, 2 , 20, 2

b. Domain: 5,1 0,1 5, 20 ; Range: 2, 3

Section 2.5 Introduction to Relations

169

13. a. 4, 4 , 1,1 , 2,1 , 3,1 , 4, 2

b. Domain: 4,1 , 2, 3, 4 ; Range: 2,1 , 4 14. a. 3, 3 , 3, 2 , 0, 0 , 2, 3 , 2, 0

b. Domain: 3, 0, 2 ; Range: 2, 0, 3

15. Domain: 0, 4 ; Range: 2, 2 16. Domain: 3, 3 ; Range: 4, 4

17. Domain: 5, 3 ; Range: 2.1, 2.8 18. Domain: 3.1, 3.2 ; Range: 3, 3

19. Domain: , 2

Range: ,

20. Domain: ,

Range: , 3

21. Domain: ,

Range: ,

22. Domain: ,

Range: ,

23. Domain: 3

Range: ,

24. Domain: ,

Range: 2

25. Domain: , 2

Range: 1.3,

26. Domain: 4,

Range: 2,

27. Domain: 3, 1,1 , 3

Range: 0,1 , 2, 3

28. Domain: 4, 3, 1, 2, 4

Range: 4, 2, 1, 0, 4

29. Domain: 4, 5

Range: 2,1 , 3

30. Domain: 5,1 1, 5

Range: 1,1 , 3

31. a. 2.85 32. a. {20, 30, 40, 50, 60}

b. 9.33 b. {200. 190, 180, 170, 160}

c. December c. 20


170

d. (November, 2.66) d. (50, 170)

e. (Sept., 7.63) e. (30, 190)

f. {Jan., Feb., Mar., Apr., May, June,

July, Aug., Sept., Oct., Nov.,

Dec.}

33. a.

0.146 31

0.146 6 31 0.876 31

31.876 million or 31,876,000

y xyy

34. a.

0.159 10.79

0.159 500 10.79

79.5 10.79 68.71 sec

y xyy

b. 32.752 0.146 31

1.752 0.146

12

xx

x

The year 2012.

b. 0.159 1000 10.79

159 10.79 148.21 sec

yy

The time is not exactly the same

because the equation represents

approximate times.

35. a. For example:

{(Julie, New York), (Peggy,

Florida),(Stephen, Kansas), (Pat,

New York)}

36. a. For example:

{(Michigan, Lansing), (Florida,

Tallahassee), (Texas, Austin),(New

York, Albany)}

b. Domain: {Julie, Peggy, Stephen,

Pat}

Range: {New York, Florida,

Kansas}

b. Domain: {Michigan, Florida,

Texas, New York}

Range: {Lansing, Tallahassee,

Austin, Albany}

37. 2 1y x 38. 3y x

39. 2y x 40. 1

4y x


1. a. function e. denominator; negative

b. vertical f. 2

c. 2 1x g. 3

d. domain; range h. (1, 6)

Section 2.6 Introduction to Functions

171

2. a. 00

6

c. 16 4

b. 8

0 is undefined.

d. 16 is not a real number.

3. a. {(Kevin, Kayla), (Kevin, Kira),

(Kathleen, Katie), (Kathleen, Kira)}

4. a. 2, 4 , 1, 1 , 0,0 , 1, 1 , 2, 4

b. Domain: {Kevin, Kathleen} b. Domain: 2, 1, 0,1 , 2

c. Range: {Kayla, Katie, Kira} c. Range: 4, 1, 0

5. Function 6. Not a function

7. Not a function 8. Function

9. Function 10. Function

11. Not a function 12. Function

13. Function 14. Not a function

15. Not a function 16. Not a function

17. When x is 2, the function value y is 5. 18. When x is 7 , the function value y is 8.

19. 0, 2 20. 10, 11

21. 22 2 4 2 1

4 8 1 11

g

22. 2 2 2

0 0

k

23. 20 0 4 0 1

0 0 1 1

g

24. 0 7h

25. 0 0 2

2 2

k

26. 0 6 0 2

0 2 2

f


172

27. 6 2

6 2

f t tt

28. 2

2

4 1

4 1

g a a a

a a

29. 7h u 30. 2k v v

31. 23 3 4 3 1

9 12 1

4

g

32. 5 7h

33. 2 2 2

4 4

k

34. 6 6 6 2

36 2

38

f

35. 1 6 1 2

6 6 2

6 4

f x xxx

36. 1 7h x

37.

2

2

2

2 2 4 2 1

4 8 1

4 8 1

g x x x

x x

x x

38. 3 3 2

5

k x x

x

39. 2

2

4 1

4 1

g

40.

22 2 2

4 2

4 2

4 1

4 1

4 1

g a a a

a a

a a

41. 7h a b 42. 6 2

6 6 2

f x h x hx h

43. 6 2

6 2

f a aa

44. 2

2

4 1

4 1

g b b b

b b


173

45. 2k c c 46. 7h x

47. 1 16 2

2 2

3 2

1

f

48. 2

1 1 14 1

4 4 4

11 1

161

16

g

49. 17

7h

50. 3 3

22 2

1

2

1

2

k

51. 2.8 6 2.8 2

16.8 2

18.8

f

52. 5.4 5.4 2

7.4

7.4

k

53. 2 7p 54. 1 0p

55. 3 2p 56. 16

2p

57. 2 5q 58. 3 1

4 5q

59. 6 4q 60. 0 9q

61. a. 0 3f 62. a. 1 2g

b. 3 1f b. 1 1g

c. 2 1f c. 4 1g

d. 3x d. 2x

e. 0, 2x x e. 3, 5x x


174

f. Domain: , 3 f. Domain: 3,

g. Range: , 5 g. Range: , 3

63. a. 3 3H 64. a. 0 1K

b. 4 not defined (4 not in domain)H b. 5 not defined (–5 not in domain)K

c. 3 4H c. 1 0K

d. 3 and 2x x d. 1, 1, and 3x x x

e. all in the interval 2,1 x e. 4x

f. Domain: 4, 4 f. Domain: 5, 5

g. Range: 2, 5 g. Range: 1, 4

65. a. 2 4p 66. a. 3 4q

b. 1 0p b. 1 2q

c. 1 3p c. 2 1q

d. 1x d. 3x

e. There are no such values of x. e. There are no such values of x.

f. , f. ,

g. , 3 2, g. ,1 3,

67. Domain: 32

3, 7, ,1 .2 68. Range: 5, 3, 4

69. Range: 6, 0 70. Domain: 0, 2, 6,1

71. –3 and 1.2 72. –7

73. 6 and 1 74. 0 and 2

75. 7 3f 76. 0 6g

77. The domain is the set of all real numbers for which the denominator is not zero. Set the

denominator equal to zero, and solve the resulting equation. The solution(s) to the equation


175

must be excluded from the domain. In this case setting 2 0x indicates that 2x must

be excluded from the domain, The domain is , 2 2, .

78. The domain is the set of all real numbers for which 3x is nonnegative. Set the quantity

3 0x and solve the inequality. The solution set for the inequality is the domain of the

function. Thus the domain is 3, .

79.

3

66 0

6

Domain: , 6 6,

xk xx

xx

80.

1

44 0

4

Domain: , 4 4,

xm xx

xx

81.

5

0

Domain: , 0 0,

f tt

t

82.

7

0

Domain: , 0 0,

tg tt

t

83.

2

2

4

1

1 will never equal zero.

Domain: ,

ph pp

p

84.

2

2

8

2

2 will never equal zero.

Domain: ,

pn pp

p

85.

7

7 0

7

Domain: 7,

h t tt

t

86.

5

5 0

5

Domain: 5,

k t tt

t

87.

3

3 0

3

Domain: 3,

f a aa

a

88.

2

2 0

2

Domain: 2,

g a aa

a


176

89.

12

1 2

1 2 0

2 1

1

2

Domain: ,

m x xxx

x

90.

12 6

12 6 0

6 12

2

Domain: , 2

n x xxxx

91.

22 1

There are no restrictions on the domain.

Domain: ,

p t t t

92.

3 1


Domain: ,

q t t t

93.

6


Domain: ,

f x x

94.

8


Domain: ,

g x x

95. a.

2

2

2

16 80

1 16 1 80

16 80

64

1.5 16 1.5 80

16 2.25 80

36 80

44

h t t

h

h

96. a.

2

2

2

4.9 50

1 4.9 1 50

4.9 50

45.1

1.5 4.9 1.5 50

4.9 2.25 50

11.025 50

38.975

h t t

h

h

b. After 1 sec, the height of the ball is

64 ft. After 1.5 sec, the height of the

ball is 44 ft.

b. After 1 sec, the height of the ball is

45.1 m. After 1.5 sec, the height of

the ball is 38.975 m.

97. a.

11.5

1 11.5 1

11.5

1.5 11.5 1.5

17.25

d t td

d

b. After 1 hr, the distance is 11.5 mi.

After 1.5 hr, the distance is 17.25 mi.


177

98.

2

2

2

2

2

2

100

50

100 0 00

5050 0

0%

100 5 100 25 2500 1005

50 25 75 350 5

33.3%

xP xx

P

P

2

2

100 10 100 100 10,000 20010

50 100 150 350 10

66.7%

P

2

2

100 15 100 225 22,500 90015

50 225 275 1150 15

81.8%

P

2

2

2

2

100 20 100 400 40,000 80020

50 400 450 950 20

88.9%

100 25 100 625 62,500 250025

50 625 675 2750 25

92.6%

P

P

If Brian studies for 25 hours, he will get a score of 92.6%.

99. 2 3f x x 100. 2 4f x x

101. 10f x x 102. 16f x x

103.

2

22 0

2

Domain: 2,

q xx

xx

104.

8

44 0

4

Domain: 4,

p xx

xx


178

105.

106.


1. a. linear 2. a. Yes, the relation is a function.

b. constant b. Domain: {6, 5, 4, 3}

c. quadratic c. Range: {1, 2, 3, 4}

d. parabola

e. 0

f. y

3. a. Yes, the relation is a function. b. Domain: {7, 2, –5} c. Range: {3}

4. a. 12

1h x

x

120

0 112

121

h

121

1 112

undefined0

h

122

2 112

121

h

123

3 112

62

h

b. Domain: 4, ,1 1,

Section 2.7 Graphs of Functions

179

5. 4f x x 6. 3f x x

3 3 3 9f

10 3 10 30f

3 9f means that 9 lb of force is

required to stretch the spring 3 in.

10 30f means that 30 lb of force is

required to stretch the spring 10 in.

a. 0 0 4 4 2f

3 3 4 1 1f

4 4 4 0 0f

5 5 4

1 cannot be evaluated

because 5 is

not in the domain of .

f

xf

b. Domain: 4,

7. Horizontal 8. m is the slope, and (0, b) is the y-

intercept.

9. 2f x

Domain: , ; Range: {2}

10. 2 1g x x

Domain: , ; Range: ,

11. 12

12

14

14

12

12

1

2

1 1

2

4

4

2

1 1

2

f xx

x f x

12.

2 2

1 1

0 0

1 1

2 2

g x x

x g x


180

13.

3

2 8

1 1

0 0

1 1

2 8

h x xx h x

14.

2 2

1 1

0 0

1 1

2 2

k x xx k x

15.

2

2 4

1 1

0 0

1 1

2 4

q x xx q x

16.

0 0

1 1

4 2

9 3

16 4

p x xx p x


181

17. 22 3 1f x x x Quadratic function 18. 2 4 12g x x x Quadratic function

19. 3 7k x x Linear function 20. 3h x x Linear function

21. 4

3m x Constant function

22. 0.8n x Constant function

23. 2 1

3 4p x

x None of these

24. 13

5Q x

x None of these

25. 2 1

3 4t x x Linear function

26. 13

5r x x Linear function

27. 4w x x None of these 28. 10T x x None of these

29.

5 10

5 10 0

5 10

2 -intercept: 2, 0

f x xx

xx x

0 5 0 10

0 10

10 -intercept: 0, 10

f

y

30.

3 12

3 12 0

3 12

4 -intercept: 4, 0

f x xx

xx x

0 3 0 12

0 12

12 -intercept: 0,1 2

f

y

31. 6 5g x x

6 5 0

6 5

xx

32. 2 9h x x

2 9 0

2 9

xx


182

5 5 -intercept: , 0

6 6x x

0 6 0 5

0 5

5 -intercept: 0, 5

g

y

9 9 -intercept: , 0

2 2x x

0 2 0 9

0 9

9 -intercept: 0, 9

h

y

33.

18

18 0 -intercept: none

0 18 -intercept: 0,1 8

f xx

f y

34.

7

7 0 -intercept: none

0 7 -intercept: 0, 7

g xx

g y

35.

22

32

2 03

22

32 6

3 -intercept: 3, 0

20 0 2

3 0 2

2 -intercept: 0, 2

g x x

x

x

xx x

g

y

36.

33

53

3 05

33

53 15

5 -intercept: 5, 0

30 0 3

5 0 3

3 -intercept: 0, 3

h x x

x

x

xx x

h

y


183

37. a. 0 when 1f x x 38. a. 0 when 1f x x

b. 0 1f b. 0 1f

39. a. 0 when 2 and 2f x x x 40. a. 0 There are none.f x

b. 0 2f b. 0 1f

41. a. 0 There are none.f x 42. a. 0 when 1 and 1f x x x

b. 0 2f b. 0 1f

43. 22q x x 44. 22 1p x x

a. , a. ,

b.

20 2 0

2 0

0 -intercept is 0, 0 .

q

y

b.

20 2 0 1

2 0 1

0 1

1 -intercept is 0,1 .

p

y

c. Graph: vi c. Graph: iii

45. 3 1h x x 46. 3 2k x x

a. , a. ,

b.

30 0 1

0 1


h

y

b.

30 0 2

0 2


k

y

c. Graph: viii c. Graph: v


184

47. 1r x x 48. 4s x x

a. 1, a. 4,

b.

0 0 1

1


r

y

b.

0 0 4

4


s

y

c. Graph: vii c. Graph: i

49. 1

3f x

x

50. 1

1g x

x

a. , 3 3, a. , 1 1,

b. 10

0 31 1

-intercept is 0, .3 3

f

y

b.

10

0 11


g

y

c. Graph: ii c. Graph: x

51. 2k x x 52. 1 2h x x

a. , a. ,

b.

0 0 2

2


k

y

b.

0 0 1 2

1 2

1 2


h

y

c. Graph: iv c. Graph: ix

53. a. Linear 54. a. Quadratic

b. 3 190 80 90

4 460 22.5

82.5

G

This means that if the student gets a

90% on her final exam, then her

overall course average is 82.5%.

b.

25 0.14 5 7.8 5 540

0.14 25 7.8 5 540

3.5 39 540

582.5

E


185

This means that for x = 5 (the year

2005), the median weekly earnings

for women was $582.50.

c. 3 150 80 50

4 460 12.5

72.5

G

This means that if the student gets a

50% on her final exam, then her

overall course average is 72.5%.

c.

210 0.14 10 7.8 10 540

0.14 100 7.8 10 540

14 78 540

632

E

This means that for x = 10 (the

year 2010), the median weekly

earnings for women was $632.

55. 2 2f x x 56. 22f x x

57. 3f x 58. 5f x

59. 12

21

22

f x x

x

60. 1

43

f x x

61.

62.

63.

64.


186

65.

66.

67. x-intercept: (8, 0); y-intercept: (0, 1) 68. x-intercept: (6, 0); y-intercept: (0,3)

69. x-intercept: (5, 0); y-intercept: (0, 4) 70. x-intercept: (2, 0); y-intercept: (0,7)

Problem Recognition Exercises

1. a, c, d, f, g 2. a, b, d, f, h

3. 21 3 1 2 1 1c

3 1 2 1 1

3 2 1 4

4. 4 2f

5. 12

0,1 , , 3, 2 6. 4, 3, 6,1 ,1 0

7. 2, 4 8. 0,

9. 0, 3 10. 5 9 0

5 9

9 9 , 0

5 5

xx

x

is the x-intercept.

11. 0,1 and 0, 1 12. x = 1

13. c 14. d

15. 5 9 6

5 15

3

xxx

Chapter 2 Review Exercises

187

Group Activity: Deciphering a Coded Message

1. a. Message: M A T H _ I S _ T H E _ K E Y _ T O _ T H E _ S C I E N C E S

Original: 13 1 20 8 27 9 19 27 20 8 5 27 11 5 25 27 20 15 27 20 8 5 27

19 3 9 5 14 3 5 19

Coded:

54,6,82,34,110,38,78,110,82,34,22,110,46,22,102,110,82,62,110,82,34,22,110,78,14,38,

22,58,14,22,78

b. Message: M A T H _ I S _ N O T _ A _ S P E C T A T O R _ S P O R T

Original: 13 1 20 8 27 9 19 27 14 15 20 27 1 27 19 16 5 3 20 1 20 15 18 27 19 16

15 18 20

Coded:

38,2,59,23,80,26,56,80,41,44,59,80, 2,80,56,47,14, 8,59,2,59,44,53,80,56,47,44,53,59


Section 2.1

1.

2.

2 4 16

2 2 4 5 4 20

16

16

x y

(2, 5) is not a solution of the equation.

3.

5 15

5 3 15

15

x

(–3, 4) is a solution of the equation.

4. 0, 0 ; 2,1 ; 0, 4 ; 2, 4 ;

2, 0 ; 5,1 ; 4, 3

A B C D

E F G

5. 3 2 6x y 6. 2 3 10y x y x y


188

0 3 0 132

–2 0 5 132

1 92

–4 132

7. 6 2x

x y

4 0

4 1

4 –2

8.

2 3 6

2 3 0 6

2 6

3

x yxxx

2 0 3 6

6 3

2

yy

y



A third point is (3, 4).

9.

5 2 0

5 2 0 0

5 0

0

x yx

xx

5 0 2 0

2 0

0

yyy



A second point is (2, 5).

A third point is (–2, –5).


189

10. 2 6

3

yy



A second point is (–2, 3).

A third point is (3, 3).

11. 3 6

2

xx



A second point is (–2, 5).

A third point is (–2, –1).

Section 2.2

12. a. 1

2m

13. For example: 2y x

b. 3

13

m

c. 0m

14. For example:

3

4y x

15. 0 6

1 26

23

m

16. 5 2 7 7

3 7 10 10m

17. 2 2 00

3 8 5m


190

18.

12

12

1

4 4

Undefined0

m

19.

1

2

2 6

3 44

41

0 1

7 31

4

m

m


20. The lines are perpendicular. 21. The lines are neither parallel nor

perpendicular.

22. The lines are parallel. 23. a. 3080 2020

2010 19901060

2053

m

b. The enrollment increases at a rate of

53 students per year.

24. 36

483

4

m

Section 2.3

25. a. y k 26. 1, -intercept: 0, 6

91

6 9

m y

y x

or

16

99 54

x y

x y

b. 1 1y y m x x

c. Ax By C

d. x k

e. y mx b


191

27. 2, -intercept: 3, 0

3m x

20 3

3y x

22

3y x

or

2 23

2 3 6

x y

x y

28.

9 1 10

35 8m

101 8

3y x

10 801

3 310 77

3 3

y x

y x

or

10 77

3 310 3 77

x y

x y

29.

12

33, point: 6, 2

2 3 6

2 3 18

3 20

or

3 20

y x

my x

y xy x

x y

30.

4 3 1

3 4 1

4 1

3 34

, point: 0, 33

3 3

or

43

34 3 9

x yy x

y x

m

y x

x y

x y

31. a. 2y 32. Yes; a and d are the same, b and c are the

same. b. 3x

c. 3x

d. 2y

Section 2.4

33. a. 0.75 50y x d. 0.75 450 50

337.50 50

387.50

y


192

b.

e. The cost is $387.50 when 450 ice

cream products are sold.

c. The y-intercept represents the daily

fixed cost of $50 when no ice cream

is sold

f. The slope of the line is 0.75.

The cost increases at a rate of $0.75

per ice cream product.

34. a.

b.

20 7

100 5013

0.2650

7 0.26 50

7 0.26 13

0.26 6

m

y xy x

y x

c. The margin of victory would be –6

points which means they would lose

by 6 points.

Section 2.5 35.

Domain: 1 1

, 6, , 73 4

Range: 1 2

10, , 4, 2 5

36. Domain: 3, 1, 0, 2, 3

Range: 52

2, 0,1 ,

37. Domain: 3, 9

Range: 0, 60

38. Domain: 4, 4

Range: 1, 3

Section 2.6

39. Answers will vary. For example: 40. Answers will vary. For example:


193

41. a. Not a function 42. a. A function

b. 1, 3 b. ,

c. 4, 4 c. , 0.35

43. a. A function 44. a. Not a function

b. 1, 2, 3, 4 b. 0, 4

c. 3 c. 2, 3, 4, 5

45. a. Not a function 46. a. A function

b. 4, 9 b. 6, 7, 8, 9

c. 2, 2, 3, 3 c. 9,1 0,1 1,1 2

47.

20 6 0 4

6 0 4

0 4

4

f

48.

21 6 1 4

6 1 4

6 4

2

f

49.

21 6 1 4

6 1 4

6 4 2

f

50. 2

2

6 4

6 4

f t tt

51. 2

2

6 4

6 4

f b bb

52. 2

2

6 4

6 4

f

53. 2

2

6 4

6 4

a afa

54.

22 6 2 4

6 4 4

24 4 20

f


194

55. 37 1g x x

Domain: ,

56.

10

1111 0

11

Domain: ,1 1 11,

xh xx

xx

57.

8

8 0

8

Domain: 8,

k x xx

x

58.

2

2 0

2

Domain: 2,

w x xx

x

59. 48 5p x x

a. 10 48 5 10

48 50 $98

p

b. 15 48 5 15

48 75 $123

p

c. 20 48 5 20

48 100 $148

p

Section 2.7

60. h x x

61. 2f x x

62. 3g x x 63. w x x


195

64. s x x

65. 1r xx

66. 3q x

67. 2 1k x x

68. 22s x x 69. 2 4r x x

a. 2

2

4 4 2

2 4

s

2

2

2

2

3 3 2

5 25

2 2 2

0 0

s

s

2

2

1 1 2

1 1

s

a. 2 2 2 4

2 2 not a real number

r

4 2 4 4

2 0 2 0 0

5 2 5 4

2 1 2 1 2

r

r

8 2 8 4

2 4 2 2 4

r


196

2

2

0 0 2

2 4

s

b. Domain: , b.

4 0

4

Domain: 4,

xx

70. 3

3h x

x

71. 3k x x

a. 33

3 33 1

6 2

h

30

0 33

13

32

2 33

31

3 35

5 3 23 0

3

h

h

h

xx

a. 5 5 3

2 2

k

4 4 3

1 1

3 3 3

0 0

2 2 3

5 5

k

k

k

b. Domain: , 3 3,

b. Domain: ,

72. 4 7p x x

4 7 0

4 7

xx

7 7 -intercept: , 0

4 4x x

0 4 0 7

0 7

7

-intercept: 0, 7

f

y

73. 2 9q x x

2 9 0

2 9

9 9 -intercept: , 0

2 2

xx

x x

0 2 0 9

0 9

9

-intercept: 0, 9

f

y

Chapter 2 Test

197

74. a.

0.7 4.5

0 0.7 0 4.5

0 4.5 4.5

7 0.7 7 4.5

4.9 4.5 9.4

b t tb

b

In 1985 consumption was 4.5 gal of

bottled water per capita. In 1992

consumption was 9.4 gal of bottled

water per capita.

75. 76. 77.

2 1g

4 0g

0 when 0 and 4.g x x x

b. 0.7m

Consumption increased by 0.7

gal/year.

78. There are no values of x for which

4g x on this graph. 79. Domain: 4,

80. Range: ,1

Chapter 2 Test

1.

26

32

0 63

36 9

20, 9

x y

y

y

20 6

36

6, 0

x

x

23 6

32 6

4

4, 3

x

xx

2. a. False; the product is positive in

quadrants I and III. In quadrant III

both x and y are negative, so their

product is positive.

c. True

b. False; the quotient is negative in

quadrants II and IV. Each of these

quadrants contains points in which

d. True


198

one coordinate is positive and one is

negative.

3.

13

21

1 4 32

2 3 1

y x

(–4, –1) is a solution of the equation.

4. To find the x-intercept, let y = 0 and solve

for x. To find the y-intercept, let x = 0 and

solve for y.

5.

6 8 24

6 8 0 24

6 24

4

x yx

xx

6 0 8 24

8 24

3

yyy



6. 4x



7.

3 5

3 5 0

3 0

0

x yxxx

3 0 5

0 5

0

yy

y



8. 2 6

3

yy



Chapter 2 Test

199

9. a. 8 3

1 75 5

8 8

m

10. a. 3 4 4

4 3 4

31

4

x yy x

y x

b. 6 5 1

5 6 1

6 15 5

6 15 565

x yy x

y x

y x

m

b. 3, -intercept: 0,1

4m y

c.

11. a. 7m 12. a. The slopes of parallel lines are the

same.

b. 1

7m

b. The slope of one line is the opposite

of the reciprocal of the slope of the

other line.

14. a. 4 1

3 1

y x my x m


c. 3 6 horizontal

0.5 vertical

yx


13. 1

6 4 2 2

1 4 3 3m

,

2

3 0 3

0 2 2m

The lines are neither parallel nor perpendicular


200

b. 9 3 1

3 9 1

13

33

x yy x

y x

m

15 5 10

5 15 10

3 2

3

x yy xy x

m


d. 53

35 3 9

3 5 9

5

3

y x

x yy x

m

3 5 10

5 3 10

32

53

5

x yy x

y x

m


perpendicular.

15. a. For example: 3 2y x 16. 0 3 3

4 2 2m

b. For example: 2x 33 2

2y x

c. For example: 3 0y m 33 3

23

62

y x

y x

d. For example: 2y x or

3 62

3 2 12

x y

x y

17.

6 3 1

3 6 1

12

32 point: 4, 3

3 2 4

3 2 8

2 11 or 2 11

x yy x

y x

my x

y xy x x y

18.

12

2, point: 8,

12 8

2

12 16

231

22

m

y x

y x

y x

19. 3 7

3 7

x yy x

20. a. 300 800y x

Chapter 2 Test

201

1 point: 10, 3

3m

13 10

31 10

33 31 1

3 3

y x

y x

y x

b.

c. The y-intercept represents Jack’s

base salary of $800.

d. 300 17 800

5100 800 5900

y

Jack earns $5900 when he sells 17

automobiles.

21. a. (0, 66) For a woman born in

1940, the life expectancy was

about 66 years.

c. 366

10y x

b. 75 66

30 09

303

10

m

Life expectancy increases by 3

years for every 10 years that

elapse.

d. 1994 corresponds to x = 54.

354 66

1016.2 66

82.2

y

Life expectancy is 82.2 years for a

woman born in 1994. This is 3.2

years longer than 79 years reported.

22.

3 1f x x

23.

2k x


202

24.

2p x x

25.

w x x

26. a. Not a function. x is paired with two

different values of y.

27. a. A function.

b. Domain: {–3, –1, 1, 3} b. Domain: ,

c. Range: {–2, –1, 1, 3} c. Range: , 0

28.

5

77 0

7

Domain: , 7 7,

xf xx

xx

29.

7

7 0

7

Domain: 7,

f x xx

x

30. 7 5 Domain: , h x x x

31. 2 2 1r x x x 32. 0.008 0.96s t t

a.

2

2

2

2 2 2 2 1

4 4 1 9

0 0 2 0 1

0 0 1 1

3 3 2 3 1

9 6 1 4

r

r

r

a.

0 0.008 0 0.96

0 0.96 0.96

7 0.008 20 0.96

0.16 0.96 0.8

s

s

(0) 0.96s ,(Year 0) the per

capita consumption was 0.96 c

per day. (20) 0.8s . In year 20,

the per capita consumption was

0.8 c per day.

b. Domain: , b. 0.008m . Consumption

decreased at a rate of 0.008 c per

day.

Chapters 1 – 2 Cumulative Review Exercises

203

33. 23f x x Quadratic function 34. 3g x x Linear function

35. 3h x Constant function 36. 3k xx

None of these

37. To find the x-intercept(s), solve for the

real solutions of the equation 0f x .

To find the y-intercept, find 0f .

38. 39

43

9 04

f x x

x

39

43 36

12 -intercept: 12, 0

30 0 9

4 0 9 9

-intercept: 0, 9

x

xx x

f

y

39. 1 1f 40. 4 2f

41. Domain: 1, 7 42. Range: 1, 4

43. False 44. x-intercept: 6, 0

45. 0 when 6f x x 46. All x in the interval 1, 3 and 5x .


1.

35 2 4 7 5 8 4 7

1 3 4 1 1 3 3

5 2 7

1 910

101

2. 3 25 8 9 6 3 5 8 3 6

3 5 24 6

3 5 4

4


204

3. 4 3 5 2 3 7 6

4 3 5 10 3 42 7

4 7 5 3 42 7

28 20 12 42 7

21 62 12

x y x y xx y x y x

x y y xx y y xx y

4.

2 3 12

6 42 3 1

24 24 26 4

4 2 3 6 1 48

8 12 6 6 48

2 18 48

2 30

15 15

x x

x x

x xx x

xxx

5.

3 2 5 5

3 2 5 5

2 5

2 5

2 5

z z zz z z

z zz z z z

6.

5 7 10

5 3

5 3 or 5 3

8 or 2 8, 2

x

xx x

x x

7.

14 3

28 1 6

7 7 7, 7

x

xx

8.

3 2 11 and 4 2

3 9 and 2

3 and 2 2, 3

x xx xx x

9. 3 2 11 or 4 2x x

3 9 or 2

3 or 2 ,

x x

x x

10. 2 1 3 12

2 1 9

x

x

9 2 1 9

8 2 10

4 5 4, 5

xx

x

11.

32

5

3 32 or 2

5 53 10 or 3 10

13 or 7 , 7 13,

x

x x

x x

x x

12. a. 14 4 1

22 1

1

f

b.

23 3 3 2 3

3 9 2 3

27 6

33

g


205

13. 3 5

6 4

2

101

5

m

14. 6 5

17 6 5

3

2 5

3

y x


15. a.

3 5 10

3 5 0 10

x yx

3 10

10

3

x

x

3 0 5 10

5 10

2

yyy

The x-intercept is 103

, 0 .


16. a. 2 4 10

2 6

3

yyy



b. 103

103

2 0

0

2

3 6 32

10 10 5

m

b. 0m

c.

c.

17. 7x 18. 5 2 10

2 5 10

55

25

2

x yy x

y x

m


206

19. Domain: 3, 4 Range: 1, 5, 8

This is not a function.

20.

2 6

2 6

2 point: 1, 4

4 2 1

4 2 2

2 2

x yy x

m

y xy x

y x

21.

12

4

4 point: 1, 4

4 4 1

4 4 4

4

y x

m

y xy x

y x

22. Let x = the amount spent on drinks

2x = the amount spent on popcorn

(amt on drinks) + (amt on popcorn) =

(total)

2 17.94

3 17.94

5.98

2 2 5.98 11.96

x xxxx

Laquita spent $5.98 on drinks and $11.96

on popcorn.

23. 15 0

15

xx

Domain: ,1 5 15,

24. 20% Salt 50% Salt 38% Salt

Solution Solution Solution

Amount of

Solution x 15 x + 15 _

Amount of

Salt 0.20x 0.50(15) 0.38(x+15)

(amt of 20%) + (amt of 50%) = (amt of 38%)


207

0.20 0.50 15 0.38 15

0.20 7.5 0.38 5.7

0.18 7.5 5.7

0.18 1.8

0.18 1.8

0.18 0.1810

x xx xx

xx

x

10 L of 20% solution must be used.

25. Let x = the yearly rainfall in Los Angeles

2x – 0.7 = the yearly rainfall in Seattle

2 0.7 50

3 0.7 50

3 50.7

16.9

2 0.7 2 16.9 0.7 33.1

x xx

xx

x

Los Angeles gets 16.9 in of rain per year and Seattle gets 33.1 in of rain per year.

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