Algebra Connections 34

Chapter 6

Lesson 6.1.1

6-1. a. There are 23 students in the class. b. There are three fewer boys than girls in the class.

c. There are 13 girls in the class.

6-2. a. 3p + 8 b. 3p + 8 = 176 c. 56; 3(56) + 8 = 176

6-3. Answers vary, but typical responses: a) the value of some quarters and nickels is $5, and b) The area of a rectangle is 30.

6-4. a. Mountain View has 100 more students because 100 is subtracted to get the number of students at Ferguson. b. The total number of students served by these two high schools is 5980. c. Chapter 2 is the longest, Chapter 3 is the shortest.

d. p + p +12 +p

2= 182 , p = 68 pages in Chapter 1.

6-5. a. 2m !10 b. 5c + 2p = 9.50 c. 4(a + p + b) = 84

6-7. side #1 = side #2 = 8 cm and side #3 = 2 · 8 – 1 = 15cm

6-8. a. x ! 342 b. 2w c. 3c + 400

6-9. a = !3, !!b = !12

6-10. no, when x = 12, y = 102 so it would have 102 tiles.

6-11. 21.1 minutes

6-12. a. y = !2x + 2

3 b. Yes c. growth factor: –2, y-intercept: (0, 2

3)

Answer Key 35

Lesson 6.1.2 6-13. C, A, D, B 6-14 a. Side #2 = x, Side#3 = 2x !1

b. x + x + (2x !1) = 31

c. x = 8, so Side #1 = Side #2 = 8 and Side #3 = 2 · 8 – 1 = 15; Yes. 6-15. a. If m = the number of months they have saved money, then

15,000 +1000m = 12,000 +1300m , and m = 10 months

b. If s = the number of sheep, then (21 – s) = the number of chickens. Then

4s + 2(21! s) = 56 , and s = 7 sheep. 21 – 7 = 14 chickens

c. If p represents the number of pencils Mr. Williams usually orders, then 2p +12 = 60 ; p = 24 pencils

d. If g represents the number of CDs that George buys, then 15.95g = 13.95g + 8 ; g =

4 CDs

e. If s represents the number of slices on an extra-large pizza, then 4s + 3 = 51 ; s = 12 slices

6-16. a. no solution b. x = 13 6-17 (-1, 3) 6-18. 31

6-19. Lakeisha, Samantha, Carly, Barbara, and Kendra 6-20. She combined terms from opposite sides of the equation. Instead, line 4 should read

2x = 14. Then x = 7 is the solution. 6-21. This statement is false because the Distributive Property states that

a(b + c) = ab + ac .

Algebra Connections 36

Lesson 6.1.3

6-22. a. 17 cans and 4 bottles

b. Let b = number of bottles, then 4b +1 = number of cans; 10(4b +1) +12b = 218

c. c = 17 cans; yes

d. Let b = number of bottles and c = number of cans, then c = 4b +1 and

10c +12b = 218 e. 17 = 4(4) +1 and 10(17) +12(4) = 218

6-23. a. 8t +16c = 400 , t = 5 + c

b. t = 50 ! 2c , t = 5 + c

c. t = 20, c = 15

6-24. 2y + 8x = 10 becomes y = ! 4x + 5 ; (–2, –13).

6-25. a. (6, 1) b. (–2, 8)

6-26. a. t ! 4 ; 2(t ! 4) b. 150 ! c c. 14.95c + 39.99v

6-27. If Nina has n nickels, then 5n + 9 + 5(2n) = 84 , and n = 5 nickels.

6-28. 8

x=3

11; x = 29 1

3, so 29 falcons is a good estimate.

6-29. 2. Associative Property, 3. Combining like terms, 4. Additive Property of Equality, 5. Combining like terms

6-30. No; 2 is a prime number and it is even. 6-31. a. H: !x = 8 , C: x = !8 ; Yes it is true because the equality is maintained when you

take the opposite of both sides of the equation b. H: 3x + y = !11 , C: 6x + 2y = !22 ; Yes it is true because the equality is

maintained when both sides of an equation are multiplied by the same number c. H: “Tomas runs at a constant rate of 4 meters every five seconds”, C: “he will run

50 meters in 1 minute”; No, this statement is false.

Answer Key 37

Lesson 6.2.1

6-32. (–11, 4)

6-33. a. Yes; the two quantities are equal

b. Yes; again, we can switch these values because the top equation indicates that they are equal

c. x = !11 , y = 4

6-34. a. x = 4 , y = 12

b. x = 3 , y = !1

c. no solution d. b = !3 , c = !8

6-35. Yes, she is correct. To test, substitute the values for x and y into both equations to see if they are correct solutions.

6-36. There are 28 red and 56 green marbles.

6-37. a. 0 b. 1617

c. 10 d. 2

6-38. a. #2

b. 4 touchdowns and 9 field goals.

6-39. a. The graphed line should be y = !2x ! 3 .

b. Yes; (–3, 3) and (–2, 1) both make this rule true.

6-40. Katy is correct because the “6x – 1” should be substituted for y because they are equal.

6-41. upside-down parabola with x-intercepts (-2, 0) and (5, 0) and y-intercept (0, 10)

6-42. No. When -2 is substituted into the equation, the equation is false.

Algebra Connections 38

Lesson 6.2.2 6-44. a. 2y +1x = 40

b. Answers vary and could include: 10 yodelers and 20 xylophones, 15 yodelers and 10 xylophones

c. Answers could include (20,10), (10,15), (4,18), (30,5), (12,14), (14,13) etc. d. No. e. Yes 6-45. a. y = 2x b. (1,2), (2,4), (3,6), (4,8), etc.

c. (8,16) d. it makes both equations true

6-46. a. a line; 2y + x = 40 or y = !1

2x + 20

b. When each point on the line is substituted into the equation, it makes the equation true

c. The line y = 2x should be graphed.

d. (8, 16); It makes both equations true e. Answers vary. Common methods: as a point (x, y), as a statement (such as “x = …

and y = …”), or as a sentence (such as, “The club had 16 yodelers and 8 xylophones.”)

6-47. a. x = 6, y = 3

b. See table below. c. The line should have a y-intercept of (0, 9) and slope of –1.

d. (6, 3) e. Yes

6-48. They would be written in the form (a, b, c), or x = #, y = #, and z = #. 6-50. Yes, each point makes the equation true. 6-51. a. no solution b. x = 5, y = 2

6-52. a. h = 2c ! 3 b. 3h +1.5c = 201 c. 28 corndogs were sold. 6-53. three dice and two jacks 6-54. Yes. Adding equal values to both sides of an equality preserves the equality.

6-55. 3y(y ! 4) = 3y2 !12y b. (3y ! 4)(y + 5) = 3y2 ! 7y ! 20

x y

-2 11

-1 10

0 9

1 8

2 7

3 6

Answer Key 39

Lesson 6.2.3 6-56. (2, -2)

6-57 b. Equal amounts are being added to both sides. Therefore, both sides remain equal.

c. 7x = 14 , x = 2 , the y-terms were eliminated when simplified.

d. y = !2 , yes

f. x = 1 , y = 4

6-58. a. If b represents the number of bass and t represents the number of trout, then

3b + t = 30 , 5b ! t = 42 b. Yes – one variable (the variable representing trout) will be eliminated when the

equations are combined.

c. Pat caught 9 bass and 3 trout

6-59. x = !1

2, y = 2

6-60 a. (3, 4) b. (11, 2) c. (-1, 1)

6-61. a. (-5, 1) b. (3, 1) c. no solution

6-62. a. infinite solutions b. lines coincide

6-63. a. Let p represent the number of pizza slices and b represent the number of burritos sold. Then 2.50p + 3b = 358 and p = b ! 20

b. 31 pizza slices were sold.

6-64. $24.58

6-65. y = 3x + 3

6-66. a. x2 ! 3x !10 b. y2 + 5xy + 6x2

c. !3xy + 3y2 + 8x ! 8y d. x2 ! 9y2

Algebra Connections 40

Lesson 6.2.4 6-67. a. !3x = !15 , x = 5 , y = !3

6-68. a. No variable is eliminated.

c. (3, 4)

d. No – Multiplying the top equation by 2 created a zero with the y-terms. 6-69. a. m = !4 ,n = 5 b. a = 2 , b = 9 c. x = !1 , y = 6 d. infinite solutions

6-70. Answers vary, but one possible strategy is to multiply the top equation by 4 and the bottom equation by 3. Once strategies are presented, solve the system with the class. Solution: (1, 2).

6-71. a. (3, 1) b. (0, 4) c. (10, 2) d. (-4, 5) 6-72. These lines coincide. 6-73. 17, 18, and 19

6-74. a. H: y = 2

3x ! 5 , C: the point (6, –1) is a solution; yes this is true because

!1 = 23(6) ! 5 .

b. H: Figure 2 has 13 tiles and Figure 4 has 15 tiles, C: the pattern grows by 2 tiles each figure; No, this is not correct. The tile pattern grows by one tile each figure.

c. H: (3x +1)(x ! 2) = 4 , C: 3x2 ! 5x ! 2 = 4 ; yes, this is correct, as can be shown

with a generic rectangle. 6-75. They are both correct. The lines coincide. 6-76. y = 2x + 5 , 105 tiles

Answer Key 41

Lesson 6.2.5

6-77. a. If c is the number of capped bottles and b is the number of broken bottles, then

c + b = 15 and 4c ! 2b = 6 .

b. Erica has capped 6 bottles and broken 9.

6-78. a. substitution b. elimination c. equal values d. substitution e. equal values f. elimination g. elimination h. substitution

6-79. a. (2, 1) b. (1, -2) c. (-0.5, 0.5) d. infinite solutions e. (2, 5) f. no solution g. (0, -2) h. (10, 7)

6-81. a. (0, 13) b. (-6, 2) c. No solution d. (11, -5)

6-82. 2n = p and n + p = 168 , 56 nectarines are needed. 6-83. a. Yes, because these expressions are equal.

b. 5(3y) + y = 32 , y = 2 , x = 3.5

6-84. x

8=8

18, x = 32

9

6-85. y = 3x 6-86. It can be concluded that line l is parallel to (or coincides with) line n because all three

lines must have the same slope.

Algebra Connections 42

Lesson 6.3.1 6-87.

b. This rule is correct. c. When x = 1mm, y = 2.5mm

6-88. Answers vary 6-89. (2x ! 3)(y + 3x ! 5) = 2xy + 6x2 !19x ! 3y +15

6-90. a. TeleTalk, 40¢

b. TeleTalk: y = 8x, Americall: y = 30 + 5x, CellTime: y = 60 + 3x

c. 10 min, 15 min d. between 10 and 15 minutes 6-91. a. $18 b. She sold 19 brownies. 6-92. a. none b. one ( t = 3 ) c. one (m = 0 ) d. infinite

6-93. He should get no solution. Lines A and B are parallel, while B and C coincide. That means that A and C are also parallel.

6-94. Stevie is 6, Joan is 11, and Julio is 14.50. The possible contexts are varied. For

example, this could be the price of CD’s by famous artists.

Stevie Joan Julio Total 31.50?

3 5 8.50 16.50 Too low

10 19 22.50 51.50 Too high

7.50 14 17.50 39.00 Too high

6-95.

a. $15.25 b. $6.25 · 20 hours = $125

6-96. (a), (b), and (c) all create equivalent equations. Part (d) is not legal because unless

x = 1, -x + 1 " 0. 6-97. 76 roses were sold.

6-98. x2+ 5 = x

2+ 2x +1 , x = 2 and y = 9

6-99. a. all numbers b. (13, !32) c. (1, 2) d. (8, 7)

Ant Beetle Grass Hopper

Antenna: 2 mm 6 mm 20 mm

Leg: 4 mm 10 mm 31 mm

Number of years at company 1 3 6 7

Salary per hour $7.00 $8.50 $10.75 $11.50

Answer Key 43

6-100. a. Line

b. Answers will vary. Possible solutions: (0, 2), (1, 5), (2, 8), …

c. y = 3x + 2 , Yes, the points are the same.

6-101. y = 2x + 6 , 206 tiles

6-102. (-1, 0) and (2, 0)

6-103. Mr. Greer incorrectly distributed. The correct solution is x = 2.

6-104. n + d = 30 and 0.05n + 0.10d = 2.60 , so n = 8. There are 8 nickels.

6-105. a, b, and d are correct.

6-106. y = !5x + 3

6-107. a. x = 2.2 b. x = 8 c. x = –10.5 d. x = 0

6-108. Answers vary, but the answer should have the same number of x-terms on both sides

of the equation and the constants on each side should not be equal.

6-109. It can be concluded that y = –2, because 2(0) ! 3(!2) = 6 .

6-110. C