Holt McDougal Algebra 1

b.

Juan’s Workout

0 10 20

40

80

120

160

Time (min)

Calo

ries

bur

ned

c. The graph forms a line.

55. No; each constant change in time (+1 s) is not accompanied by a constant change in height.

56. Yes; the equation can be written in standard form with A = 1, B = 0, and C = 9. No; all solutions are ordered pairs with x-value 9. The x-value 9 corresponds to more than one y-value.

teSt prep

57. C; C is not linear, because x appears in the denominator of a fraction.

58. G; Each second, sound will move 331 meters. So the distance covered is 331 times the number of seconds.

59. Possible answer: 3x + 2y = 7; it is a linear equation because it can be written in standard form with A = 3, B = 2, and C = 7.

A table of values also shows it is a linear function:

x y

-3 8

-2 6.5

-1 5

0 3.5

1 2

2 0.5

Each constant change in x (+1) is accompanied by a constant change in y (-1.5).

The graph shows a linear function.

2

4

x

y

0 2-2

Both the graph and the table show solutions to the equation.

challenge and extend

60. y = 0; x = 0; the first describes a linear function, but the second does not.

61. Perimeter of a square

Side Length Perimeter

1 4

2 8

3 12

4 16

62. area of a square

Side Length Area

1 1

2 4

3 9

4 16

linear not linear

63. Volume of a Cube

Side Length Volume

1 1

2 8

3 27

4 64

not linear

usIng Intercepts

CHECK IT OUT!

1a. The y-intercept is 3. The x-intercept is -2.

b. -3x + 5y = 30 -3x + 5(0) = 30 -3x + 0 = 30 -3x = 30

-3x _ -3

= 30 _ -3

x = -10 The x-intercept is -10.

-3x + 5y = 30 -3(0) + 5y = 30 0 + 5y = 30 5y = 30

5y

_ 5

= 30 _ 5

y = 6 The y-intercept is 6.

c. 4x + 2y = 16 4x + 2(0) = 16 4x + 0 = 16 4x = 16

4x _ 4 = 16 _

4

x = 4 The x-intercept is 4.

4x + 2y = 16 4(0) + 2y = 16 0 + 2y = 16 2y = 16

2y

_ 2

= 16 _ 2


2a. 2x + 3y = 60 ________ -2x ____ -2x 3y = 60 - 2x

3y

_ 3 = 60 - 2x _

3

y = 20 - 2 _ 3 x

x 0 6 15 24 30

y = 20 - 2 _ 3 x 20 16 10 4 0

School Store Purchases

0 10 20 30

10

20

Pens

Not

eboo

ks

The x-intercept is 30. The y-intercept is 20.

109 Holt McDougal Algebra 1

x-2

109 Holt McDougal Algebra 1

4-2

CS10_A1_MESK710372_C04.indd 109 3/30/11 10:57:26 PM

b. x-intercept: pens that can be purchased if no notebooks are purchased.

y-intercept: notebooks that can be purchased if no pens are purchased.

3a. x-intercept: -3x + 4y = -12 -3x + 4(0) = -12 -3x = -12

-3x _ -3

= -12 _ -3

x = 4

y-intercept: -3x + 4y = -12 -3(0) + 4y = -12 4y = -12

4y

_ 4 = -12 _

4

y = -3

-2

x y

0 2 4

b. y = 1 _ 3 x - 2

3(y) = 3 ( 1 _ 3 x - 2)

3y = x - 6 -x + 3y = -6 x-intercept: -x + 3y = -6 -x + 3(0) = -6 -x = -6 -1(-x) = -1(-6) x = 6

y-intercept: -x + 3y = -6 -(0) + 3y = -6 3y = -6

3y

_ 3 = -6 _

3

y = -2

2

-4

x

y

0 2 6

THInK and dIsCUss

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

2. 4.318; -21.5489

3.

1. Find the x-intercept by letting y equal 0 and solving for x.

Graphing Ax + By = C Using Intercepts

2. Find the y-intercept by letting x equal 0 and solving for y.

3. Graph the line by plotting the points containing the intercepts and then connecting the points with a straight line.

ExErCIsEsguided practice

1. y-intercept

2. The x-intercept is -5. The y-intercept is 1.

3. The x-intercept is 2. The y-intercept is -4.

4. The x-intercept is -3. The y-intercept is -2.

5. 2x - 4y = 4 2x - 4(0) = 4 2x - 0 = 4 2x = 4

2x _ 2 = 4 _

2


2x - 4y = 4 2(0) - 4y = 4 0 - 4y = 4 -4y = 4

-4y

_ -4

= 4 _ -4

y = -1 The y-intercept is -1.

6. -2y = 3x - 6 ____ -3x _______ -3x -3x - 2y = -6

-3x - 2y = -6 -3x - 2(0) = -6 -3x - 0 = -6 -3x = -6

-3x _ -3

= -6 _ -3


-3x - 2y = -6 -3(0) - 2y = -6 0 - 2y = -6 -2y = -6

-2y

_ -2

= -6 _ -2


7. 4y + 5x = 2y - 3x + 16 ________ -2y _____________ -2y 2y + 5x = -3x + 16 _______ + 3x ________ +3x 2y + 8x = 16 8x + 2y = 16 8x + 2y = 16 8x + 2(0) = 16 8x + 0 = 16 8x = 16

8x _ 8 = 16 _

8


8x + 2y = 16 8(0) + 2y = 16 0 + 2y = 16 2y = 16

2y

_ 2 = 16 _

2


8a. x 0 1 2 3 4 5

f(x) = -25 + 5x -25 -20 -15 -10 -5 0

Refrigeration Tank Temperature

0 2 4 6

-15

-20

-25

-10

-5

Time (h)

Tem

pera

ture

(°C)

The x-intercept is 5. The y-intercept is -25.

b. x-intercept: time when temperature is 0°C y-intercept: initial temperature

110 Holt McDougal Algebra 1 110 Holt McDougal Algebra 1

CS10_A1_MESK710372_C04.indd 110 3/30/11 10:57:29 PM

9. x-intercept: 4x - 5y = 20 4x - 5(0) = 20 4x = 20

4x _ 4 = 20 _

4

x = 5

y-intercept: 4x - 5y = 20 4(0) - 5y = 20 -5y = 20

-5y

_ -5

= 20 _ -5

y = -4

-2

-4

x

y

0 2

10. y = 2x + 4 -2x + y = 4

x-intercept: -2x + y = 4 -2x + 0 = 4 -2x = 4

-2x _ -2

= 4 _ -2

x = -2

y-intercept: -2x + y = 4 -2(0) + y = 4 y = 4

4

x

y

0 2 -4

11. x-intercept:

1 _ 3 x - 1 _

4 y = 2

1 _ 3 x - 1 _

4 (0) = 2

1 _ 3 x = 2

3 ( 1 _ 3 x) = 3(2)

x = 6

y-intercept:

1 _ 3 x - 1 _

4 y = 2

1 _ 3 (0) - 1 _

4 y = 2

- 1 _ 4 y = 2

-4 (- 1 _ 4 y) = -4(2)

y = -8

-4

-8

x y

0 4 -4

12. x-intercept: -5y + 2x = -10 -5(0) + 2x = -10 2x = -10

2x _ 2 = -10 _

2

x = -5

y-intercept: -5y + 2x = -10 -5y + 2(0) = -10 -5y = -10

-5y

_ -5

= -10 _ -5

y = 2

x

y

0 -2 -4 -2

practice and problem Solving


14. The x-intercept is -5. The y-intercept is -1.


16. 6x + 3y = 12 6x + 3(0) = 12 6x + 0 = 12 6x = 12

6x _ 6 = 12 _

6


6x + 3y = 12 6(0) + 3y = 12 0 + 3y = 12 3y = 12

3y

_ 3

= 12 _ 3


17. 4y - 8 = 2x ______ -2x -2x 4y - 2x - 8 = 0 __________ + 8 ___ +8 4y - 2x = 8 -2x + 4y = 8 -2x + 4y = 8 -2x + 4(0) = 8 -2x + 0 = 8 -2x = 8

-2x _ -2

= 8 _ -2


-2x + 4y = 8 -2(0) + 4y = 8 0 + 4y = 8

4y

_ 4

= 8 _ 4


18. -2y + x = 2y - 8 _______ -2y _______ -2y -4y + x = -8 x - 4y = -8 x - 4y = -8 x - 4(0) = -8 x - 0 = -8 x = -8 The x-intercept is -8.

x - 4y = -8 0 - 4y = -8 -4y = -8

-4y

_ -4

= -8 _ -4


19. 4x + y = 8 4x + 0 = 8 4x = 8

4x _ 4 = 8 _

4


4x + y = 8 4(0) + y = 8 0 + y = 8 y = 8 The y-intercept is 8.

20. y - 3x = -15 0 - 3x = -15 -3x = -15

-3x _ -3

= -15 _ -3


y - 3x = -15 y - 3(0) = -15 y - 0 = -15 y = -15 The y-intercept is -15.


CS10_A1_MESK710372_C04.indd 111 3/30/11 10:57:32 PM

21. 2x + y = 10x - 1 ________ -10x ________ -10x -8x + y = -1 -8x + y = -1 -8x + 0 = -1 -8x = -1

-8x _ -8

= -1 _ -8

x = 1 _ 8

The x-intercept is 1 _ 8 .

-8x + y = -1 -8(0) + y = -1 0 + y = -1 y = -1 The y-intercept is -1.

22a. x 0 3 6 9 12

f(x) = 300 - 25x 300 225 150 75 0

Bass Population

0 4 8 12

70

140

210

280

Time (yr)

Popu

lati

on


b. x-intercept: time when bass population is 0 y-intercept: number of bass originally in the lake

23a. x 0 5 15 20 25

f(x) = 5 - 1 _ 5 x 5 4 2 1 0

5K Race

0 10 20

1

2

3

4

Time (min)

Dist

ance

to fi

nish

line

(km

)


b. x-intercept: total time to run the race (when the distance to the finish line is 0)

y-intercept: total length of the race (when time is 0)

24. x-intercept: 4x - 6y = 12 4x - 6(0) = 12 4x = 12

4x _ 4

= 12 _ 4

x = 3

y-intercept: 4x - 6y = 12 4(0) - 6y = 12 -6y = 12

-6y

_ -6

= 12 _ -6

y = -2

2

4

-2

-4

x

y

0 4 -2

25. x-intercept: 2x + 3y = 18 2x + 3(0) = 18 2x = 18

2x _ 2 = 18 _

2

x = 9

y-intercept: 2x + 3y = 18 2(0) + 3y = 18 3y = 18

3y

_ 3 = 18 _

3

y = 6

2

4

6

x 0 4 8

y

26. x-intercept:

1 _ 2 x - 4y = 4

1 _ 2 x - 4(0) = 4

1 _ 2 x = 4

2 ( 1 _ 2 x) = 2(4)

x = 8

y-intercept:

1 _ 2 x - 4y = 4

1 _ 2 (0) - 4y = 4

-4y = 4

-4y

_ -4

= 4 _ -4

y = -1

4

-4

x

y

4 8

27. x-intercept: y - x = -1 0 - x = -1 -x = -1 -1(-x) = -1(-1) x = 1

y-intercept: y - x = -1 y - 0 = -1 y = -1

2

x

y

0 2 -2

28. x-intercept: 5x + 3y = 15 5x + 3(0) = 15 5x = 15

5x _ 5 = 15 _

5

x = 3

y-intercept: 5x + 3y = 15 5(0) + 3y = 15 3y = 15

3y

_ 3 = 15 _

3

y = 5

2

4

x

y

0 2 -2


CS10_A1_MESK710372_C04.indd 112 3/30/11 10:57:36 PM

29. x-intercept: x - 3y = -1 x - 3(0) = -1 x = -1

2

-2

x

y

0 2

y-intercept: x - 3y = -1 0 - 3y = -1 -3y = -1

-3y

_ -3

= -1 _ -3

y = 1 _ 3

30a. y = x + 4

b. y = x + 4 ___ -x ______ -x -x + y = 4

-x + y = 4 -(0) + y = 4 0 + y = 4 y = 4 The y-intercept is 4.

c. The original height of the bamboo plant.

31a. The y-intercept is approximately 7.5. The x-intercept is approximately 600.

b. x-intercept: the number of years after 1800 when there will be no acres of tropical forest

y-intercept: million acres of tropical forest in 1800

32a. m 0 20 40 60 80

b = 412 - 4m 412 332 252 172 92

BUA111FT-BKMADD-A732

0

300

340

380

420

642

Account Balance

Time (mo)

Bala

nce

($)

The number of months must be a whole number so the domain is (0, 1, 2, 3, ...}. The range is {$412, $408, $404, $400, ...}.

b. y = 412 - 4x ____ +4x ________ + 4x 4x + y = 412 4x + y = 412 4x + 0 = 412 4x = 412

4x _ 4 = 412 _

4


4x + y = 412 4(0) + y = 412 0 + y = 412 y = 412

The y-intercept is 412. x-intercept: number of months from that time until

the account has $0 y-intercept: balance when bank employee noticed

the account

c. After 103 months or 8 years and 7 months.

33a.

4

-4

-8

x

y

0 -4 -8 4 8

x = -6 x = 1 x = 5

x = -6: x-intercept: -6, no y-intercept

x = 1: x-intercept: 1, no y-intercept

x = 5: x-intercept: 5, no y-intercept

b.

4

-4

-8

x

y

0 -4 -8 4 8

y = -3

y = 2

y = 7 y = -3: no x-intercept,

y-intercept: -3 y = 2: no x-intercept, y-intercept: 2 y = 7: no x-intercept, y-intercept: 7

c. Horizontal: For y = c, the y-intercept is c and there is no x-intercept.

Vertical: For x = k, the x-intercept is k, and there is no y-intercept.

34. -2x - y = 4 -2x - 0 = 4 -2x = 4

-2x _ -2

= 4 _ -2


-2x - y = 4 -2(0) - y = 4 0 - y = 4 -y = 4 -1(-y) = -1(4) y = -4 The y-intercept is -4.

Graph D has an x-intercept of -2 and a y-intercept of -4.

35. y = 4 - 2x ____ +2x ______ + 2x 2x + y = 4 2x + y = 4 2x + 0 = 4 2x = 4

2x _ 2 = 4 _

2


2x + y = 4 2(0) + y = 4 0 + y = 4 y = 4 The y-intercept is 4.

Graph A has an x-intercept of 2 and a y-intercept of 4.

36. 2y + 4x = 8 2(0) + 4x = 8 0 + 4x = 8 4x = 8

4x _ 4 = 8 _

4


2y + 4x = 8 2y + 4(0) = 8 2y + 0 = 8 2y = 8

2y

_ 2

= 8 _ 2


Graph A has an x-intercept of 2 and a y-intercept of 4.

37. 4x - 2y = 8 4x - 2(0) = 8 4x - 0 = 8 4x = 8

4x _ 4 = 8 _

4


4x - 2y = 8 4(0) - 2y = 8 0 - 2y = 8 -2y = 8

-2y

_ -2

= 8 _ -2

y = -4 The y-intercept is -4.

Graph B has an x-intercept of 2 and a y-intercept of -4.

38a. The x-intercept is 20. The y-intercept is 1.75.

b. x-intercept: time remaining when Kristyn started her workout

y-intercept: total distance Kristyn covered

39. Possible answer: Jen wants to save $60. Each week she will earn $12. The function shows how much money Jen has left to save each week.


CS10_A1_MESK710372_C04.indd 113 3/30/11 10:57:39 PM

teSt prep

40. D; Notice that -2(9) = -18 = 0 - 18 = 9(0) - 18. So, (9, 0) is on -2x = 9y - 18 and therefore is the x-intercept.

41. F; The y-intercept is -200 so Jamie owed her uncle $200. The x-intercept is 40 so Jamie was paying her uncle for 40 weeks.

42. 60x + 55y = 660 60(0) + 55y = 660 0 + 55y = 660 55y = 660

55y

_ 55

= 660 _ 55


challenge and extend

43. x-intercept:

1 _ 2 x + 1 _

5 y = 1

1 _ 2 x + 1 _

5 (0) = 1

1 _ 2

x = 1

2 ( 1 _ 2

x) = 2(1)

x = 2

y-intercept:

1 _ 2 x + 1 _

5 y = 1

1 _ 2 (0) + 1 _

5 y = 1

1 _ 5 y = 1

5 ( 1 _ 5 y) = 5(1)

y = 5

2

4

x

y

0 2 -2

44. x-intercept: 0.5x - 0.2y = 0.75 0.5x - 0.2(0) = 0.75 0.5x = 0.75

0.5x _ 0.5

= 0.75 _ 0.5

x = 1.5

y-intercept: 0.5x - 0.2y = 0.75 0.5(0) - 0.2y = 0.75 -0.2y = 0.75

-0.2y

_ -0.2

= 0.75 _ -0.2

y = -3.75

-2

-4

x y

0 2 -2

45. y = 3 _ 8 x + 6

____

- 3 _ 8 x

________ - 3 _

8 x

- 3 _ 8 x + y = 6

x-intercept:

- 3 _ 8 x + y = 6

- 3 _ 8 x + 0 = 6

- 3 _ 8 x = 6

- 8 _ 3 (- 3 _

8 x) = - 8 _

3 (6)

x = -16

y-intercept:

- 3 _ 8 x + y = 6

- 3 _ 8 (0) + y = 6

y = 6

-6

x

y

0 -6 -12

46. Ax + By = C Ax + B(0) = C Ax + 0 = C Ax = C

Ax _ A

= C _ A

x = C _ A

The x-intercept is C _ A

.

Ax + By = C A(0) +By = C 0 + By = C By = C

By

_ B

= C _ B

y = C _ B

The y-intercept is C _ B

.

47. 22x - 380y = 20,900 22x - 380(0) = 20,900 22x - 0 = 20,900 22x = 20,900

22x _ 22

= 20,900

_ 22


22x - 380y = 20,900 22(0) - 380y = 20,900 0 - 380y = 20,900 -380y = 20,900

-380y

_ -380

= 20,900

_ -280

y = -55 The y-interecpt is -55.

Possible answer: scale on the x-axis should include numbers from 0 to a number a little greater than 950; scale on y-axis should include numbers from a little less than -55 to 0.


CS10_A1_MESK710372_C04.indd 114 3/30/11 10:57:42 PM

Holt McDougal Algebra 1

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