44 Algebra Connections

Chapter 7

Lesson 7.1.1 7-2. Customer A should order y = 4x ! 3 instead; Customer B should order y = !3x + 2

instead; Customer C’s order is correct; Customer D’s table is not linear, so the customer should revise his or her order; Customer E’s order is correct; Customer F’s order is not linear either, since a line must have constant growth everywhere.

7-4. a. See diagrams at right.

b. y = 5x +1 ; 51 tiles

7-5. a. 3 b. 1 c. 2

7-6. a. 250 ÷ 3 !5 " 416.7 pounds of food b. 700 ÷ 45 ! 15.6 hours

7-7. x: (12, 0) and y: (0, 6)

7-8. a. y = 8 ! 6x b. x = !1

6y + 8

6 c. x = 3

7-9. Let t = number of teddy bears and d = number of dogs. Then t + d = 356 and

d = 2t +17 , so t = 113 and d = 243 . Thus, he has 113 teddy bears.

Lesson 7.1.2

7-12. Answers can vary depending on data and the trend line that is chosen.

7-13. a. 16 b. 2 c. undefined d. 0

7-14. a. 5 b. 21 c. the positive value of x ! c d. 36

e. 38 f. the positive value of y ! f

7-15. a. –3 b. 12

7-16. a. (3, –0.25) b. infinite solutions because the lines coincide 7-17. x = 15 , y = 9

7-18. a. The baby is growing 1.5 inches per month; it is the slope (m). b. It was 23 inches long when it was born; it is the y-intercept (b). c. 27.5 inches

d. 11 months after it was born, so in December

Figure 0 Figure 4

Answer Key 45

Lesson 7.1.3

7-19. a. Both have 3 tiles in Figure 0. However, A grows by 2 tiles and B grows by 6 tiles. B is steeper.

c. Typical response. It tells how much a pattern grows each figure. The steeper the line, the greater or faster the growth.

d. A. y = 2x + 3 , B. y = 6x + 3

7-20. 27 ÷ 3 = 9

7-21. a. 2 b. 4 c. 12

or 0.5 d. divide !y by !x e. y = 1

2x + 2

7-22. a. 6 ÷ 4 = 3÷ 2 = 1.5 ; Both give the same result.

b. 4.5; The ratio must be equal. c. 1.5; This height is equal to the slope.

7-23. If !y = 0 , then the line is horizontal; if !x = 0 , then the line is vertical.

7-24. a. Essie is correct; !y = "3 means that the vertical change is 3 and that the line is

pointing downward.

b. yes; !34= !

3

4 c. y = !

3

4x + 9

7-25. Possible response. It is a parabola because it has an x2 -term. (This conclusion is acceptable at this point.)

7-26. r = 3y and r + y = 124 ; r = 93 and y = 31

7-27. any horizontal line that is not y = 0

7-28. a. 5.5 b. 42 c. ! 25

d. 2

7-29. Let x represent the number of hours she swims. Then x + 2x = 4.5 and x = 1.5 . Thus, she will swim for 1.5 hours and play volleyball for 3 hours.

7-30. a. 5x2 ! 32x ! 21 b. !24x2 +18x

46 Algebra Connections

Lesson 7.1.4

7-31. a. B is the steepest; C is the least steep. b. Downward; !y is –2.

c. A. m = 1, B. m = 3, C. m = 12

, D. m = ! 25

d. Possible responses. The value of slope indicates how steep the graph is. The larger the slope, the steeper the line. If the slope is positive, the line travels upward from left to right, while a negative slope indicates that the line travels downward from left to right. e. In this case, she is correct. The steepest line (B) has the largest value for slope (3). Note that for negative slope, the steeper the line, the more negative (smaller) the slope.

7-32. a. A is the steepest; B is steeper than C.

b. A. !x = 1, !y = 2; B. !x = 2, !y = 3; C. !x = 3, !y = 2; D. !x = 5, !y = –1

c. A. m = 2, B. m =3

2, C. m =

2

3, D. m = !

1

5

d. It would point downward from left to right because its slope is negative.

e. It would point downward from left to right because its slope is negative. It would

be steeper.

7-33. A. m = !1

2, B. m = –2, C. m =

2

3, D. m = 0

7-34. a. The line should have slope m = 6 . b. m =3

5 c. m = !

3

2 d. m = 0

7-35. a. The slope of each is 12

. b. no c. !y = 1

2

7-37. a. line a. y = 2x ! 2 , line b. y = 2x + 3

b. It would lie between lines a and b, because its y-intercept is at (0, 1).

c. It would travel downward but would have the same y-intercept as the line from part (b).

7-38. a. (–1, –7)

b. ( 12

, 2)

c. These coincide, so there are an infinite number of points of intersection. 7-39. a. 12 b. 0 c. –8 d. no solution

7-40. a. 6(13x ! 21) = 78x !126 b. (x + 3)(x ! 5) = x2 ! 2x !15

c. 4(4x2 ! 6x +1) = 16x2 ! 24x + 4 d. (3x ! 2)(x + 4) = 3x2 +10x ! 8

7-41. 55 hops

7-42. y = 4

3x ! 4

Answer Key 47

Lesson 7.1.5

7-43. a. This is not enough information. An infinite number of lines pass through (2, 5).

b. This is enough information. y = !3x .

c. This is enough information. y = 2x + 4 .

d. This is not enough information. An infinite number of lines, all parallel to each

other, have slope of 4.

e. This is enough information. y = !3

4x + 5 .

f. You need the slope and a point or two points to determine a unique line.

7-44. a. 12

; parallel lines must have the same growth factor. b. They are equal.

c. It should have a slope of –2 and pass through (0, –5). d. y = 1

2x ! 6

7-45. a. She subtracted the x- and y-coordinates. b. ! 23

c. 42= 2 d. ! 3

6= !

1

2d

e. The student did not make !x negative and forgot to divide !y by !x .

7-46. The steepest line is vertical. Its slope is undefined since its !x is 0.

7-47. No; she would be on the same page as Sam on page 310, but there are only 300 pages in the book. Sam only needs 29 minutes and 45 seconds to finish the book, while Jimmica needs 30 minutes.

7-48. a. See graph at right. b. (3, –2)

7-49. a. m = !2 b. m = 0.5 c. undefined

7-50. a. 4 b. no solution c. 0.5 d. 4

7-51. a. y = 2x – 4 b. x =y!4

!2= !

1

2y + 2 c. x = 1 d. x = –7

7-52. x-intercepts. (–2, 0) and (4, 0); y-intercept. (0, –24)

Lesson 7.2.1 7-55. A person cannot be different distances from the motion detector at the same time.

7-56. $2,982,000 7-57. a. 4 b. 16 c. It would get steeper.

7-58. a. There is no solution, so the lines do not intersect. b. Yes; both lines have the

same slope. y = 2

3x ! 10

3

7-59. y = 3x ! 2 , y = x

7-60. Answers vary. Possible solutions. 5x(3x !1) , 5(3x2 ! x) , x(15x ! 5) , etc.

7-61. a. ! 43

b. (0, –5) c. y = !4

3x ! 5

3

–2

x

y

48 Algebra Connections

Lesson 7.2.2

7-62. a. Leslie. y = 2x , Kristin. y = 2

5x + 8 , Evie. y = 5

4x + 6

b. After 8 seconds, both racers were 16 meters from the starting line.

c. Leslie won in 10 seconds, Kristin took 30 seconds and Evie took 11.2 seconds.

d. Her slope is 25

. Her slope represents her rate of travel.

e. She traveled 2 meters every 3 seconds and got a 1-meter head start.

7-63. a. The customer started walking slowly toward the motion detector from 4 feet away, stopped for about 4 seconds, and then walked away from the machine at a faster rate

b. The customer started from 3.5 feet in front of the machine and walked quickly away, stopped for about 4 seconds, and then started to walk slowly toward the machine

c. The customer started walking quickly toward the machine from about 10 feet away, after 3 seconds, stopped for 4 seconds, and then started walking slowly toward the machine.

7-64. a. The slope represents the number of feet a tree grows per year, and the y-intercept represents the height when the measuring began (perhaps when it was planted), m = 2 feet per year, b = (0, 4)

b. The slope represents the amount of money being spent from a bank account each month, while the y-intercept represents the beginning balance, m = !10 dollars per month, b = (0, 170)

c. The slope represents the distance that can be traveled on a gallon of gas, m = 22 miles per gallon, b = (0, 0).

7-65. a. Elizabeth won the race, finishing in 5 seconds.

b. Barbara: y = 3

2x + 3 , Elizabeth: y = 4x

c. 5 meters every 2 seconds, or 52

meters per second

d. 2 seconds after the start of the race, when each is 6 meters from the starting line

7-66. a. 0 pounds

b. The graph should show a line with positive slope. Units labeled on the axes should be in pounds (vertical axis) and feet (horizontal axis).

c. 43

pounds d. " 9 feet

7-67. x-intercept. (2, 0), y-intercept. (0, –10)

7-68. a. 1 b. 3 c. 2 7-69. y = 3x ! 2

7-70. Let t = number of t-shirts and s = number of shorts. Then 12t + 8s = 780 and t + s = 77 ; t = 41 and s = 36 ; they sold 41 t-shirts.

Answer Key 49

Lesson 7.2.3

7-71. Rider A: y = 3

2x ,

Rider B: y = 1

2x +10 ,

Rider C: y = x + 4 ,

Rider D: y = 5

2x +1,

Elizabeth: y = 1

4x +10 ,

Leslie: y = 2x ! 4 ;

Rider D wins the race. Solution graph is shown at right.

7-72. a. Rider D; Elizabeth

b. Rider D. 5 meters every 2 seconds; Elizabeth. 1 meter every 4 seconds

c. 8 seconds after the start of the race, Rider A, Rider C, Elizabeth, and Leslie were all 12 meters from the starting line.

7-73. (–2, –15)

7-74. a. (4, 0) and (0, –2) b. (8, 0) and (0, 4)

7-75. a. –1 b. ! 12

c. 32

d. ! 15

e. The line travels downward from left to right, so m = !1 .

7-76. a. 4 b. 3 c. 1 d. 2

7-77. a. !12x b. 8x + 7 c. x2 ! 3 d. 5x + 7 ! 3y

7-78. C

D Leslie A C

B

Dis

tan

ce

fro

m S

tart

ing

Lin

e

Time (seconds)

Elizabeth

50 Algebra Connections

Lesson 7.3.1

7-79. While the graph will only approximate the solution, a table helps show that the chick weighed 51.6 grams when it was born and will weigh 140 grams after 17 days

7-80. a. It grows 5.2 grams per day, so the slope of the line is 5.2.

b. slope. m = 5.2 ; should pass through (9, 98.4)

c. Units on the axes should be in grams (vertical axis) and days (horizontal axis).

d. While the y-intercept is (0, 51.6), answers close to 50 grams are acceptable.

e. when the chick is 17 days old

7-81. a. The chick weighed 98.4 grams after 9 days.

b. On Day 0, the chick weighed 51.6 grams.

c. Colleen’s chick weighed 51.6 grams when hatched. The chick will weigh 140 grams when it is 17 days old. For this problem, a table can produce exact results, while the graph is approximate.

7-82. a. The variable m is the slope, or the chick’s daily rate of growth, 5.2 grams per day; b is the y-intercept, or the chick’s weight when hatched, which is unknown.

b. The slope is 5.2, so y = 5.2x + b . The y-intercept is still unknown.

c. Possible response. A point on a line is a solution to the equation, so when substituting the x- and y-values into the equation, the equation is still true.

d. Substituting x = 9 and y = 98.4 into the equation, the equation becomes

98.4 = 5.2(9) + b . After solving, b = 51.6 and y = 5.2x + 51.6 .

e. The equation is exact.

f. Starting with 140 = 5.2x + 51.6 , the solution becomes x = 17 days.

7-83. a. y = !3x ! 5 b. y = 0.5x !14

7-84. Mt. Everest was 8750.05 meters tall in the year 0; y = 0.05x + 8750.05 .

7-85. a. y = 1.5x + 0.5 b. Answers vary, but solutions should lie on y = 1.5x + 0.5 .

7-86.

7-87. a. ( 7, 11) b. (–8, –10)

7-88. a. 14 b. 52

c. –3 d. 11

7-89. a. (3.5, 0) and (0, –2.33) b. y = 133

" 4.33

7-90. a. 3; (0, 5) b. ! 54

; (0, 0) c. 0; (0, 3) d. 4; (0, 7)

e. ! 3

4; (0, –1) f. ! 1

5; (0, 6)

1 2

3

13

6

13

6

–15

5 –3

2

–11

11 –1

10

–14

–7 2

–5

Answer Key 51

Lesson 7.3.2

7-91. b. ! 3

2

c. Answers vary. d. They are opposite reciprocals.

e. ! 53

.

7-92. a. y = 5

2x ! 8 b. y = 3

2x +1

7-93. line L. y = !1

6x + 6 ;

line M. y = 2

3x +1; (6, 5)

7-94. Lines A and E are perpendicular,

so the slope of line E is ! 56

.

7-96. a. 0.75 meters

b. Dean. y = 3

4x + 5 , Carlos.

y = x c. after 20 seconds

7-97. a. The slope represents the cost

per mile for a ride in a taxi, m = $3.50 per mile; b. The slope represents the

gallons per minute of water draining from a tub, m = ! 4 gallons per minute.

7-98. If d = number of Democrats, then

d + (d + 30) +1 = 435 , and d =

202 Democrats. Thus, there were 202 + 30 = 232 Republicans.

7-99. (–5, 0) and (3, 0)

7-100. a. 2x2 + 5x ! 3

b. 15x2 ! 33x

c. 5x2 ! 27x +10

d. 300x ! 50

7-101. B

Lesson 7.3.3

7-102. a. m = 3

b. Substitute the point into the equation y = 3x + b in order

to solve for b; y = 3x +10 .

c. No; both will give the same equation.

d. Substitute both points into the equation to verify that both are solutions.

7-104. b. Although there are many possible solutions, the points for 1961 and 1989 are good choices. This will create a line. y = 9.82x !19234.82 .

c. See part (b) above.

e. Answers will vary. In the case stated in part (b), the y-intercept is (0, –19234.82); Dizzyland did not exist in the year 0.

f. Answers will vary. Approximately the year 2060.

7-105. a. y = 1

5x + 4 b. y = !3x + 376

7-106. y = 9

4x + 9 7-107. ( 11

3, –3)

7-108. a. There are no x-intercepts; y-intercept. (0, 10). b. (3, 1)

7-109. a. –8 b. –56

c. 3 d. 6

7-110. a. m = !5

6 b. m = 0

c. m = !5

6 d. undefined,

e. m = – 4 f. m =1

4

g. m = 4 h. m = !6

5;

Lines (b) and (d) are perpendicular; lines (e) and (f) are perpendicular; lines (a) and (c) are parallel.

52 Algebra Connections

Lesson 7.3.4

7-111. Logo A: y = 2 , y = x + 2 , y = 1

2x + 2 , y = !

3

2x + 6 , y = !

1

2x + 6 , y = 3

2x ! 6 ,

y = !1

2x +10 ; Logo B: y = 3 , y = 2x , y = 3

4x , y = !2x + 6 , y = !

3

4x + 6 ,

y = 3

4x ! 3 , y = !

3

4x + 9

7-112. a. (2y + 4)(3+ 5x) = 6y + 20x +10xy +12

b. (5 + 2x)(2x ! 3) = 4x2 + 4x !15

7-113. If b = number of brownies sold and c = the number of cookies sold, then 3b + 2.50c = 218 and b = c + 3 ; she sold 41 brownies.

7-114. a. y = !2

3x +1 b. y = –11 c. y = 3

2x

7-115. a. The slope represents the change in height of a candle per minute, m = 0 cm per minute

b. The slope represents the gallons per month of water being removed from a well, m = –900 gallons per month.

7-116.

7-117. a. 15x2 b. 8x c. 6x2 d. 7x

– 4

4 –1

3

16

– 4 – 4

–8

–33

11 –3

8

–7

–1 7

6