13.1.4: Chapter 4

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Be Prepared

4.1

ⓐ yes; ⓑ no

4.2

$m = 3; b = - 12$

4.3

$(6, 0), (0, - 4)$

4.4

$11$

4.5

Jon earned $22,000 and Ron earned $74,000.

4.6

The speed of the local train is 48 mph and the speed of the express train is 60 mph.

4.7

$6,287.05$

4.8

$0.082$

4.9

$$5.85$

4.10

$7$

4.11

conditional; one solution

4.12

conditional; one solution

4.13

$x = 1$

4.14

$p = 8.4$

4.15

$41$

4.16

$- 6$

4.17

$- 4$

4.18

$\frac{3}{2}$

4.19

$- 4 < a$

4.20

Try It

4.1

ⓐ yes ⓑ no

4.2

ⓐ no ⓑ yes

4.3

$(3, 2)$

4.4

$(2, 3)$

4.5

$(3, 4)$

4.6

$(5, −4)$

4.7

no solution

4.8

no solution

4.9

infinitely many solutions

4.10

infinitely many solutions

4.11

ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent

4.12

ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent

4.13

$(6, 1)$

4.14

$(−3, 5)$

4.15

$(2, \frac{3}{2})$

4.16

$(- \frac{1}{2}, −2)$

4.17

$(2, −1)$

4.18

$(−2, 3)$

4.19

$(1, 3)$

4.20

$(4, −3)$

4.21

$(6, 2)$

4.22

$(1, −2)$

4.23

infinitely many solutions

4.24

infinitely many solutions

4.25

ⓐ Since both equations are in standard form, using elimination will be most convenient. ⓑ Since one equation is already solved for x, using substitution will be most convenient.

4.26

ⓐ Since one equation is already solved for y, using substitution will be most convenient. ⓑ Since both equations are in standard form, using elimination will be most convenient.

4.27

3, 7

4.28

2, $−8$

4.29

160 policies

4.30

1000 suits

4.31

Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks.

4.32

Erin burned 11 calories for each minute on the rowing machine and 5 calories for each minute of weight lifting.

4.33

The angle measures are 55 and 35.

4.34

The angle measures are 5 and 85.

4.35

The angle measures are 42 and 138.

4.36

The angle measures are 66 and 114.

4.37

22, 68

4.38

36, 54

4.39

The length is 60 feet and the width is 35 feet.

4.40

The length is 60 feet and the width is 38 feet.

4.41

It will take Clark 4 hours to catch Mitchell.

4.42

It will take Sally $1 \frac{1}{2}$ hours to catch up to Charlie.

4.43

The rate of the boat is 11 mph and the rate of the current is 1 mph.

4.44

The speed of the canoe is 7 mph and the speed of the current is 1 mph.

4.45

The speed of the jet is 236 mph and the speed of the wind is 29 mph.

4.46

The speed of the jet is 408 mph and the speed of the wind is 24 mph.

4.47

206 adults, 347 children

4.48

42 adults, 105 children

4.49

13 dimes and 29 quarters

4.50

19 quarters and 51 nickels

4.51

3 pounds peanuts and 2 pounds cashews

4.52

10 pounds of beans, 10 pounds of ground beef

4.53

120 ml of 25% solution and 30 ml of 50% solution

4.54

125 ml of 10% solution and 125 ml of 40% solution

4.55

$42,000 in the stock fund and $8000 in the savings account

4.56

$1750 at 11% and $5250 at 13%

4.57

Bank $4,000; Federal $14,000

4.58

$41,200 at 4.5%, $24,000 at 7.2%

4.59

ⓐ $C (x) = 15 x + 25, 500$

ⓑ $R (x) = 32 x$

ⓒ

$Figure shows a graph with two intersecting lines. One of them passes through the origin. The other crosses the y axis at point 25,687.$

ⓓ $1,500$ ; when 1,500 benches are sold, the cost and revenue will be both 48,000

4.60

ⓐ $C (x) = 120 x + 150,000$

ⓑ $R (x) = 170 x$

ⓒ

$Figure shows a graph with two intersecting lines. One of them passes through the origin.$

ⓓ $3,000$ ; when 3,000 benches are sold, the revenue and costs are both $510,000

4.61

ⓐ yes ⓑ no

4.62

ⓐ no ⓑ yes

4.63

$(2, −1, 3)$

4.64

$(−2, 3, 4)$

4.65

$(−3, 4, −2)$

4.66

$(−2, 3, −1)$

4.67

no solution

4.68

no solution

4.69

infinitely many solutions $(x, 3, z)$ where $x = z - 3; y = 3; z$ is any real number

4.70

infinitely many solutions $(x, y, z)$ where $x = 5 z - 2; y = 4 z - 3; z$ is any real number

4.71

The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.

4.72

The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.

4.73

ⓐ $[\begin{matrix} 3 & 8 & −3 \\ 2 & 5 & −3 \end{matrix}]$
ⓑ $[\begin{matrix} 2 & −5 & 3 & 8 \\ 3 & −1 & 4 & 7 \\ 1 & 3 & 2 & −3 \end{matrix}]$

4.74

ⓐ $[\begin{matrix} 11 & 9 & −5 \\ 7 & 5 & −1 \end{matrix}]$
ⓑ $[\begin{matrix} 5 & −3 & 2 & −5 \\ 2 & −1 & −1 & 4 \\ 3 & −2 & 2 & −7 \end{matrix}]$

4.75

${\begin{matrix} x - y + 2 z = 3 \\ 2 x + y - 2 z = 1 \\ 4 x - y + 2 z = 0 \end{matrix}$

4.76

${\begin{matrix} x + y + z = 4 \\ 2 x + 3 y - z = 8 \\ x + y - z = 3 \end{matrix}$

4.77

ⓐ $[\begin{matrix} −2 & 3 & 0 & −2 \\ 4 & −1 & −4 & 4 \\ 5 & −2 & −2 & −2 \end{matrix}]$
ⓑ $[\begin{matrix} −2 & 3 & 0 & −2 \\ 4 & −1 & −4 & 4 \\ 15 & −6 & −6 & −6 \end{matrix}]$
ⓒ $[\begin{matrix} −2 & 3 & 0 & −2 \\ 3 & 4 & −13 & −16 & −8 \\ 15 & −6 & −6 & −6 \end{matrix}]$

4.78

ⓐ $[\begin{matrix} 4 & 1 & −3 & 2 \\ 2 & −3 & −2 & −4 \\ 5 & 0 & 4 & −1 \end{matrix}]$
ⓑ $[\begin{matrix} 8 & 2 & −6 & 4 \\ 2 & −3 & −2 & −4 \\ 5 & 0 & 4 & −1 \end{matrix}]$
ⓒ $[\begin{matrix} 14 & −7 & −12 & −8 \\ 2 & −3 & −2 & −4 \\ 5 & 0 & 4 & −1 \end{matrix}]$

4.79

$[\begin{matrix} 1 & −1 & 2 \\ 0 & −3 & −4 \end{matrix}]$

4.80

$[\begin{matrix} 1 & −1 & 3 \\ 0 & −5 & 8 \end{matrix}]$

4.81

The solution is $(4, −1) .$

4.82

The solution is $(−2, 0) .$

4.83

$(6, −1, −3)$

4.84

$(5, 7, 4)$

4.85

no solution

4.86

no solution

4.87

infinitely many solutions $(x, y, z),$ where $x = z - 3; y = 3; z$ is any real number.

4.88

infinitely many solutions $(x, y, z),$ where $x = 5 z - 2; y = 4 z - 3; z$ is any real number.

4.89

ⓐ $−14;$ ⓑ $−28$

4.90

ⓐ 2 ⓑ $−15$

4.91

ⓐ 3 ⓑ 11 ⓒ 2

4.92

ⓐ $−3$ ⓑ 2 ⓒ 3

4.93

4.94

4.95

$−11$

4.96

$−12$

4.97

$(- \frac{15}{7}, \frac{24}{7})$

4.98

$(−2, 0)$

4.99

$(−9, 3, −1)$

4.100

$(−6, 3, −2)$

4.101

no solution

4.102

infinite solutions

4.103

yes

4.104

yes

4.105

ⓐ no ⓑ yes

4.106

ⓐ yes ⓑ no

4.107

$The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.$

The solution is the grey region.

4.108

$The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.$

The solution is the grey region.

4.109

$The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.$

The solution is the grey region.

4.110

$The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.$

The solution is the grey region.

4.111

$The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution$

The solution is the grey region.

4.112

$The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.$

The solution is the grey region.

4.113

$The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.$

No solution.

4.114

$The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.$

No solution.

4.115

$The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.$

The solution is the grey region.

4.116

$The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.$

The solution is the grey region.

4.117

ⓐ ${\begin{matrix} 30 m + 20 p \leq 160 \\ 2 m + 3 p \leq 15 \end{matrix}$
ⓑ

$The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.$

ⓒ yes
ⓓ no

4.118

ⓐ ${\begin{matrix} a \geq p + 5 \\ a + 2 p \leq 400 \end{matrix}$
ⓑ

$The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.$

ⓒ no
ⓓ no

4.119

ⓐ ${\begin{matrix} 0.75 d + 2 e \leq 25 \\ 360 d + 110 e \geq 1000 \end{matrix}$
ⓑ

$The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.$

ⓒ yes
ⓓ no

4.120

ⓐ ${\begin{matrix} 140 p + 125 j \geq 1000 \\ 1.80 p + 1.25 j \leq 12 \end{matrix}$
ⓑ

$The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.$

ⓒ yes
ⓓ no

Section 4.1 Exercises

ⓐ yes ⓑ no

$(−2, 3)$

$(0, 2)$

$(2, 4)$

11.

$(−2, −2)$

13.

$(3, 3)$

15.

$(6, −4)$

17.

no solution

19.

no solution

21.

infinite solutions

23.

infinite solutions

25.

No solution, Inconsistent, Independent

27.

1 point, consistent and independent

29.

infinite solutions, consistent, dependent

31.

$(1, −4)$

33.

$(−3, 2)$

35.

$(- \frac{1}{2}, \frac{5}{2})$

37.

$(−5, 4)$

39.

$(0, 10)$

41.

$(4, −2)$

43.

$(4, 0)$

45.

no solution

47.

$(4, 5)$

49.

$(7, 12)$

51.

$(−3, −5)$

53.

$(2, −3)$

55.

$(−11, 2)$

57.

$(6, - 9)$

59.

$(−3, 2)$

61.

infinitely many

63.

infinitely many

65.

ⓐ substitution ⓑ elimination

67.

ⓐ elimination ⓑ substituion

69.

Answers will vary.

71.

Answers will vary.

Section 4.2 Exercises

73.

13 and 17

75.

$−7$ and $−19$

77.

14 and 23

79.

22 and $−67$

81.

Eighty cable packages would need to be sold to make the total pay the same.

83.

Mitchell would need to sell 120 stoves for the companies to be equal.

85.

8 and 40 gallons

87.

1000 calories playing basketball and 400 calories canoeing

89.

Oranges cost $2 per pound and bananas cost $1 per pound

91.

Package of paper $4, stapler $7

93.

Hot dog 150 calories, cup of cottage cheese 220 calories

95.

Owen will need 80 quarts of water and 20 quarts of concentrate to make 100 quarts of lemonade.

97.

$53.5$ degrees and $36.5$ degrees

99.

16 degrees and 74 degrees

101.

134 degrees and 46 degrees

103.

37 degrees and 143 degrees

105.

16 degrees and 74 degrees

107.

45 degrees and 45 degrees

109.

Width is 41 feet and length is 118 feet.

111.

Width is 10 feet and length is 40 feet.

113.

12 hours

115.

1.83 hour

117.

Boat rate is 16 mph and current rate is 4 mph.

119.

Boat rate is 18 mph and current rate is 2 mph.

121.

Jet rate is 265 mph and wind speed is 22 mph.

123.

Jet rate is 415 mph and wind speed is 25 mph.

125.

Answers will vary.

Section 4.3 Exercises

127.

110 adult tickets, 190 child tickets

129.

6 good seats, 10 cheap seats

131.

92 adult tickets, 220 children tickets

133.

13 nickels, 3 dimes

135.

42 dimes, 8 quarters

137.

17 $10 bills, 37 $20 bills

139.

80 pounds nuts and 40 pounds raisins

141.

9 pounds of Chicory coffee, 3 pounds of Jamaican Blue Mountain coffee

143.

10 bags of M&M’s, 15 bags of Reese’s Pieces

145.

$7.5$ liters of each solution

147.

80 liters of the 25% solution and 40 liters of the 10% solution

149.

240 liters of the 90% solution and 120 liters of the 75% solution

151.

$1600 at 8%, 960 at 6%

153.

$28,000 at 9%, $36,000 at $5.5 %$

155.

$8500 CD, $1500 savings account

157.

$55,000 on loan at 6% and $30,000 on loan at $4.5 %$

159.

ⓐ $C (x) = 5 x + 6500$

ⓑ $R (x) = 10 x$

ⓒ

$Figure shows a graph with two intersecting lines. One of them passes through the origin. The other crosses the y axis at point 6560.$

ⓓ 1,300; when 1,300 water bottles are sold, the cost and the revenue equal $13,000

161.

Answers will vary.

Section 4.4 Exercises

163.

ⓐ no ⓑ yes

165.

ⓐ no ⓑ no

167.

$(4, 5, 2)$

169.

$(7, 12, −2)$

171.

$(−3, −5, 4)$

173.

$(2, −3, −2)$

175.

$(6, −9, −3)$

177.

$(3, −4, −2)$

179.

$(−3, 2, 3)$

181.

$(−2, 0, −3)$

183.

no solution

185.

$x = \frac{203}{16}; y = \frac{–25}{16}; z = \frac{–231}{16};$

187.

$(x, y, z)$ where $x = 5 z + 2; y = −3 z + 1; z$ is any real number

189.

$(x, y, z)$ where $x = 5 z - 2; y = 4 z - 3; z$ is any real number

191.

45 degrees, 60 degrees, 75 degrees

193.

$20, $5, $10

195.

Answers will vary.

Section 4.5 Exercises

197.

ⓐ $[\begin{matrix} 2 & 4 & −5 \\ 3 & −2 & 2 \end{matrix}]$
ⓑ $[\begin{matrix} 3 & −2 & −1 & −2 \\ −2 & 1 & 0 & 5 \\ 5 & 4 & 1 & −1 \end{matrix}]$

199.

ⓐ $[\begin{matrix} 2 & −5 & −3 \\ 4 & −3 & −1 \end{matrix}]$
ⓑ $[\begin{matrix} 4 & 3 & −2 & −3 \\ −2 & 1 & −3 & 4 \\ −1 & −4 & 5 & −2 \end{matrix}]$

201.

${\begin{matrix} 2 x - 4 y = −2 \\ 3 x - 3 y = −1 \end{matrix}$

203.

${\begin{matrix} 2 x - 2 y = −1 \\ 2 y - z = 2 \\ 3 x - z = −2 \end{matrix}$

205.

ⓐ $[\begin{matrix} 3 & 2 & 1 \\ 4 & −6 & −3 \end{matrix}]$
ⓑ $[\begin{matrix} 12 & 8 & 4 \\ 4 & −6 & −3 \end{matrix}]$
ⓒ $[\begin{matrix} 12 & 8 & 4 \\ 24 & −10 & −5 \end{matrix}]$

207.

ⓐ $[\begin{matrix} 6 & - 5 & 2 & 3 \\ 3 & - 3 & 1 & - 1 \\ 2 & 1 & - 4 & 5 \end{matrix}]$
ⓑ $[\begin{matrix} 6 & - 5 & 2 & 3 \\ 15 & - 15 & 5 & - 5 \\ 2 & 1 & - 4 & 5 \end{matrix}]$
ⓒ $[\begin{matrix} 2 & - 7 & 10 & - 7 \\ 15 & - 15 & 5 & - 5 \\ 2 & 1 & - 4 & 5 \end{matrix}]$

209.

$[\begin{matrix} 1 & −2 & 3 & −4 \\ 0 & 5 & −11 & 17 \\ 0 & 1 & −10 & 7 \end{matrix}]$

211.

$(1, −1)$

213.

$(3, 3)$

215.

$(−2, 5, 2)$

217.

$(−3, −5, 4)$

219.

$(−3, 2, 3)$

221.

$(−2, 0, −3)$

223.

no solution

225.

no solution

227.

infinitely many solutions $(x, y, z)$ where $x = \frac{1}{2} z + 4; y = \frac{1}{2} z - 6; z$ is any real number

229.

infinitely many solutions $(x, y, z)$ where $x = 5 z + 2; y = −3 z + 1; z$ is any real number

231.

Answers will vary.

Section 4.6 Exercises

233.

235.

237.

239.

ⓐ 9 ⓑ $−3$ ⓒ 8

241.

$−77$

243.

245.

$−24$

247.

249.

$(7, 6)$

251.

$(−2, 0)$

253.

$(−3, 2)$

255.

$(−9, 3)$

257.

$(−3, −5, 4)$

259.

$(2, −3, −2)$

261.

$(−3, 2, 3)$

263.

$(−2, 0, −3)$

265.

infinitely many solutions

267.

inconsistent

269.

inconsistent

271.

infinitely many solutions

273.

yes

275.

yes

277.

Answers will vary.

279.

Answers will vary.

Section 4.7 Exercises

281.

ⓐ false ⓑ true

283.

ⓐ false ⓑ true

285.

ⓐ false ⓑ true

287.

$The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.$

The solution is the grey region.

289.

$The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.$

The solution is the grey region.

291.

$The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.$

The solution is the grey region.

293.

$The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

295.

$The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

297.

$The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

299.

$The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

301.

$The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

303.

$The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

305.

$The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

307.

$The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.$

The solution is the grey region.

309.

$The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.$

No solution.

311.

$The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.$

No solution.

313.

$The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.$

The solution is the grey region.

315.

$The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.$

The solution is the grey region.

317.

$The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.$

The solution is the grey region.

319.

ⓐ ${\begin{matrix} f \geq 0 \\ p \geq 0 \\ f + p \leq 20 \\ 2 f + 5 p \leq 50 \end{matrix}$
ⓑ

$The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.$

ⓒ yes
ⓓ no

321.

ⓐ ${\begin{matrix} c \geq 0 \\ a \geq 0 \\ c + a \leq 24 \\ a \geq 3 c \end{matrix}$
ⓑ

$The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.$

ⓒ yes
ⓓ no

323.

ⓐ ${\begin{matrix} w \geq 0 \\ b \geq 0 \\ 27 w + 16 b > 80 \\ 3.20 w + 1.75 b \leq 10 \end{matrix}$
ⓑ

$The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.$

ⓒ no
ⓓ yes

325.

ⓐ ${\begin{matrix} w \geq 0 \\ r \geq 0 \\ w + r \geq 4 \\ 270 w + 650 r \geq 1500 \end{matrix}$
ⓑ

$The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.$

ⓒ no
ⓓ yes

327.

Answers will vary.

Review Exercises

329.

ⓐ yes ⓑ no

331.

$The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.$

$(3, −1)$

333.

$The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.$

no solution

335.

one solution, consistent system, independent equations

337.

$(4, 5)$

339.

$(3, 1)$

341.

infinitely many solutions

343.

$(4, −1)$

345.

$(6, 2)$

347.

elimination

349.

50 irises and 150 tulips

351.

10 calories jogging and 10 calories cycling

353.

119 degrees and 61 degrees

355.

35 degrees and 55 degrees

357.

Length = 502 feet, Width = 212 feet

359.

$\frac{1}{2}$ an hour

361.

the rate of the jet is 395 mph, the rate of the wind is 7 mph

363.

41 dimes and 11 pennies

365.

$46 \frac{2}{3}$ liters of 30% solution, $23 \frac{1}{3}$ liters of 60% solution

367.

$29,000 for the federal loan, $14,000 for the private loan

369.

ⓐ no ⓑ no

371.

$(−3, 2, −4)$

373.

no solution

375.

25, 20, 15

377.

$[\begin{matrix} 4 & 3 & 0 & −2 \\ 1 & −2 & −3 & 7 \\ 2 & −1 & 2 & −6 \end{matrix}]$

379.

${\begin{matrix} x - 3 z = −1 \\ x - 2 y = −2 \\ − y + 2 z = 3 \end{matrix}$

381.

ⓐ $[\begin{matrix} 1 & −3 & −2 & 4 \\ 4 & −2 & −3 & −1 \\ 2 & 2 & −1 & −3 \end{matrix}]$
ⓑ $[\begin{matrix} 2 & −6 & −4 & 8 \\ 4 & −2 & −3 & −1 \\ 2 & 2 & −1 & −3 \end{matrix}]$
ⓒ $[\begin{matrix} 2 & −6 & −4 & 8 \\ 4 & −2 & −3 & −1 \\ 0 & −6 & −1 & 5 \end{matrix}]$

383.

$(−2, 5, 2)$

385.

no solution

387.

$−4$

389.

391.

$(−3, 2)$

393.

$(−3, 2, 3)$

395.

inconsistent

397.

ⓐ yes ⓑ yes

399.

$The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.$

The solution is the grey region.

401.

$The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.$

The solution is the grey region.

403.

$The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.$

No solution.

405.

ⓐ ${\begin{matrix} b \geq 0 \\ n \geq 0 \\ b + n \leq 40 \\ 12 b + 18 n \geq 500 \end{matrix}$
ⓑ

$The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.$

ⓒ yes
ⓓ no

Practice Test

407.

$The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.$

$(2, −3)$

409.

$(2, 1)$

411.

$(2, −2, 1)$

413.

$(5, 7, 4)$

415.

$7$

417.

15 liters of 1% solution, 5 liters of 5% solution

419.

The candy cost $20; the cookies cost $5; and the popcorn cost $10.

421.

ⓐ ${\begin{matrix} C \geq 0 \\ L \geq 0 \\ C + 0.5 L \leq 50 \\ L \geq 3 C \end{matrix}$
ⓑ

$The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.$

ⓒ no
ⓓ yes

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Section 4.1 Exercises

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Section 4.6 Exercises

Section 4.7 Exercises

Review Exercises

Practice Test