Intermediate Algebra

# Chapter 4

Intermediate AlgebraChapter 4

### Try It

4.1

yes no

4.2

no yes

4.3

$(3,2)(3,2)$

4.4

$(2,3)(2,3)$

4.5

$(3,4)(3,4)$

4.6

$(5,−4)(5,−4)$

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

$(6,1)(6,1)$

4.14

$(−3,5)(−3,5)$

4.15

$(2,32)(2,32)$

4.16

$(−12,−2)(−12,−2)$

4.17

$(2,−1)(2,−1)$

4.18

$(−2,3)(−2,3)$

4.19

$(1,3)(1,3)$

4.20

$(4,−3)(4,−3)$

4.21

$(6,2)(6,2)$

4.22

$(1,−2)(1,−2)$

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, $−8−8$

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 $112112$ 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 235 mph and the speed of the wind is 30 mph.

4.46

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

4.47

4.48

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

$C(x)=15x+25,500C(x)=15x+25,500$

$R(x)=32xR(x)=32x$

$1,5001,500$; when 1,500 benches are sold, the cost and revenue will be both 48,000

4.60

$C(x)=120x+150,000C(x)=120x+150,000$

$R(x)=170xR(x)=170x$

$3,0003,000$; when 3,000 benches are sold, the revenue and costs are both $510,000 4.61 yes no 4.62 no yes 4.63 $(2,−1,3)(2,−1,3)$ 4.64 $(−2,3,4)(−2,3,4)$ 4.65 $(−3,4,−2)(−3,4,−2)$ 4.66 $(−2,3,−1)(−2,3,−1)$ 4.67 no solution 4.68 no solution 4.69 infinitely many solutions$(x,3,z)(x,3,z)$ where $x=z−3;y=3;zx=z−3;y=3;z$ is any real number 4.70 infinitely many solutions $(x,y,z)(x,y,z)$ where$x=5z−2;y=4z−3;zx=5z−2;y=4z−3;z$ 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 $[38−325−3][38−325−3]$ $[2−5383−147132−3][2−5383−147132−3]$ 4.74 $[119−575−1][119−575−1]$ $[5−32−52−1−143−22−7][5−32−52−1−143−22−7]$ 4.75 ${x−y+2z=32x+y−2z=14x−y+2z=0{x−y+2z=32x+y−2z=14x−y+2z=0$ 4.76 ${x+y+z=42x+3y−z=8x+y−z=3{x+y+z=42x+3y−z=8x+y−z=3$ 4.77 $[−230−24−1−445−2−2−2][−230−24−1−445−2−2−2]$ $[−230−24−1−4415−6−6−6][−230−24−1−4415−6−6−6]$ $[−230−234−13−16−815−6−6−6][−230−234−13−16−815−6−6−6]$ 4.78 $[41−322−3−2−4504−1][41−322−3−2−4504−1]$ $[82−642−3−2−4504−1][82−642−3−2−4504−1]$ $[14−7−12−82−3−2−4504−1][14−7−12−82−3−2−4504−1]$ 4.79 $[1−120−3−4][1−120−3−4]$ 4.80 $[1−130−58][1−130−58]$ 4.81 The solution is $(4,−1).(4,−1).$ 4.82 The solution is $(−2,0).(−2,0).$ 4.83 $(6,−1,−3)(6,−1,−3)$ 4.84 $(5,7,4)(5,7,4)$ 4.85 no solution 4.86 no solution 4.87 infinitely many solutions $(x,y,z),(x,y,z),$ where $x=z−3;y=3;zx=z−3;y=3;z$ is any real number. 4.88 infinitely many solutions $(x,y,z),(x,y,z),$ where $x=5z−2;y=4z−3;zx=5z−2;y=4z−3;z$ is any real number. 4.89 $−14;−14;$ $−28−28$ 4.90 2 $−15−15$ 4.91 3 11 2 4.92 $−3−3$ 2 3 4.93 37 4.94 7 4.95 $−11−11$ 4.96 8 4.97 $(−157,247)(−157,247)$ 4.98 $(−2,0)(−2,0)$ 4.99 $(−9,3,−1)(−9,3,−1)$ 4.100 $(−6,3,−2)(−6,3,−2)$ 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 solution is the grey region. 4.108 The solution is the grey region. 4.109 The solution is the grey region. 4.110 The solution is the grey region. 4.111 The solution is the grey region. 4.112 The solution is the grey region. 4.113 No solution. 4.114 No solution. 4.115 The solution is the grey region. 4.116 The solution is the grey region. 4.117 ${30m+20p≤1602m+3p≤15{30m+20p≤1602m+3p≤15$ yes no 4.118 ${a≥p+5a+2p≤400{a≥p+5a+2p≤400$ no no 4.119 ${0.75d+2e≤25360d+110e≥1000{0.75d+2e≤25360d+110e≥1000$ yes no 4.120 ${140p+125j≥10001.80p+1.25j≤12{140p+125j≥10001.80p+1.25j≤12$ yes no ### Section 4.1 Exercises 1. yes no 3. yes no 5. $(−2,3)(−2,3)$ 7. $(0,2)(0,2)$ 9. $(2,4)(2,4)$ 11. $(−2,2)(−2,2)$ 13. $(3,3)(3,3)$ 15. $(6,−4)(6,−4)$ 17. no solution 19. no solution 21. infinite solutions 23. infinite solutions 25. No solutions, inconsistent, independent 27. 1 point, consistent and independent 29. infinite solutions, consistent, dependent 31. $(1,−4)(1,−4)$ 33. $(−3,2)(−3,2)$ 35. $(−1/2,5/2)(−1/2,5/2)$ 37. $(−5,4)(−5,4)$ 39. $(0,10)(0,10)$ 41. $(4,−2)(4,−2)$ 43. $(4,0)(4,0)$ 45. none 47. $(4,5)(4,5)$ 49. $(7,12)(7,12)$ 51. $(−3,−5)(−3,−5)$ 53. $(2,−3)(2,−3)$ 55. $(−11,2)(−11,2)$ 57. $(6/−9,24/7)(6/−9,24/7)$ 59. $(−3,2)(−3,2)$ 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. $−7−7$ and $−19−19$ 77. 14 and 23 79. 22 and $−67−67$ 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. $53.553.5$ degrees and $36.536.5$ degrees 99. 16 degrees and 74 degrees 101. 134 degrees and 46 degrees 103. 37 degrees and 143 degrees 105. $16°16°$ and $74°74°$ 107. $45°45°$ and $45°45°$ 109. Width is 41 feet and length is 118 feet. 111. Width is 10 feet and length is 40 feet. 113. 11 hours 115. $1.51.5$ 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. $7.57.5$ 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 $5.5%5.5%$

155.

$8500 CD,$1500 savings account

157.

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

159.

$C(x)=5x+6500C(x)=5x+6500$

$R(x)=10xR(x)=10x$

195.

### Section 4.5 Exercises

197.

$[24−53−22][24−53−22]$
$[3−2−1−2−2105541−1][3−2−1−2−2105541−1]$

199.

$[2−5−34−3−1][2−5−34−3−1]$
$[43−2−3−21−34−1−45−2][43−2−3−21−34−1−45−2]$

201.

${2x−4y=−23x−3y=−1{2x−4y=−23x−3y=−1$

203.

${2x−2y=−12y−z=23x−z=−2{2x−2y=−12y−z=23x−z=−2$

205.

$[3214−6−3][3214−6−3]$
$[12844−6−3][12844−6−3]$
$[128424−10−5][128424−10−5]$

207.

$[21−456−5233−31−1][21−456−5233−31−1]$
$[21−456−5233−31−1][21−456−5233−31−1]$
$[21−456−523−47−67][21−456−523−47−67]$

209.

$[1−23−405−111701−107][1−23−405−111701−107]$

211.

$(1,−1)(1,−1)$

213.

$(3,3)(3,3)$

215.

$(−2,5,2)(−2,5,2)$

217.

$(−3,−5,4)(−3,−5,4)$

219.

$(−3,2,3)(−3,2,3)$

221.

$(−2,0,−3)(−2,0,−3)$

223.

no solution

225.

no solution

227.

infinitely many solutions $(x,y,z)(x,y,z)$ where $x=12z+4;y=12z−6;zx=12z+4;y=12z−6;z$ is any real number

229.

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

231.

### Section 4.6 Exercises

233.

4

235.

10

237.

6 $−14−14$ $−6−6$

239.

9 $−3−3$ 8

241.

$−77−77$

243.

49

245.

$−24−24$

247.

25

249.

$(7,6)(7,6)$

251.

$(−2,0)(−2,0)$

253.

$(−3,2)(−3,2)$

255.

$(−9,3)(−9,3)$

257.

$(−3,−5,4)(−3,−5,4)$

259.

$(2,−3,−2)(2,−3,−2)$

261.

$(−3,2,3)(−3,2,3)$

263.

$(−2,0,−3)(−2,0,−3)$

265.

infinitely many solutions

267.

inconsistent

269.

inconsistent

271.

infinitely many solutions

273.

yes

275.

no

277.

279.

### Section 4.7 Exercises

281.

false true

283.

false true

285.

false true

287.

The solution is the grey region.

289.

The solution is the grey region.

291.

The solution is the grey region.

293.

The solution is the grey region.

295.

The solution is the grey region.

297.

The solution is the grey region.

299.

The solution is the grey region.

301.

The solution is the grey region.

303.

The solution is the grey region.

305.

The solution is the grey region.

307.

The solution is the grey region.

309.

No solution.

311.

No solution.

313.

The solution is the grey region.

315.

The solution is the grey region.

317.

The solution is the grey region.

319.

${f≥0p≥0f+p≤202f+5p≤50{f≥0p≥0f+p≤202f+5p≤50$

yes
no

321.

${c≥0a≥0c+a≤24a≥3c{c≥0a≥0c+a≤24a≥3c$

yes
no

323.

${w≥0b≥027w+16b>803.20w+1.75b≤10{w≥0b≥027w+16b>803.20w+1.75b≤10$

no
yes

325.

${w≥0r≥0w+r≥4270w+650r≥1500{w≥0r≥0w+r≥4270w+650r≥1500$

no
yes

327.

### Review Exercises

329.

yes no

331.

$(3,−1)(3,−1)$

333.

no solution

335.

one solution, consistent system, independent equations

337.

$(4,5)(4,5)$

339.

$(3,1)(3,1)$

341.

infinitely many solutions

343.

$(4,−1)(4,−1)$

345.

$(6,2)(6,2)$

347.

elimination

349.

50 irises and 150 tulips

351.

10 calories jogging and 10 calories cycling

353.

119, 61

355.

$35°35°$ and $55°55°$

357.

the length is 450 feet, the width is 264 feet

359.

$1212$ 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.

$46234623$ liters of 30% solution, $23132313$ liters of 60% solution

367.

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

369.

no yes

371.

$(−3,2,−4)(−3,2,−4)$

373.

no solution

375.

25, 20, 15

377.

$[430−21−2−372−12−6][430−21−2−372−12−6]$

379.

${x−3z=−1x−2y=−27−y+2z=3{x−3z=−1x−2y=−27−y+2z=3$

381.

$[1−3−244−2−3−122−1−3][1−3−244−2−3−122−1−3]$
$[2−6−484−2−3−122−1−3][2−6−484−2−3−122−1−3]$
$[2−6−484−2−3−10−6−15][2−6−484−2−3−10−6−15]$

383.

$(−2,5,−2)(−2,5,−2)$

385.

no solution

387.

$−4−4$

389.

21

391.

$(−3,2)(−3,2)$

393.

$(−3,2,3)(−3,2,3)$

395.

inconsistent

397.

yes no

399.

The solution is the grey region.

401.

The solution is the grey region.

403.

No solution.

405.

${b≥0n≥0b+n≤4012b+18n≥500{b≥0n≥0b+n≤4012b+18n≥500$

yes
no

### Practice Test

407.

$(2,−3)(2,−3)$

409.

$(2,1)(2,1)$

411.

$(2,−2,1)(2,−2,1)$

413.

$(5,7,4)(5,7,4)$

415.

$99$

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.

${C≥0L≥0C+0.5L≤50L≥3C{C≥0L≥0C+0.5L≤50L≥3C$

no
yes

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