Elementary Algebra

# Chapter 4

Elementary AlgebraChapter 4

### Try It

4.5

A: $(5,1)(5,1)$ B: $(−2,4)(−2,4)$ C: $(−5,−1)(−5,−1)$ D: $(3,−2)(3,−2)$ E: $(0,−5)(0,−5)$ F: $(4,0)(4,0)$

4.6

A: $(4,2)(4,2)$ B: $(−2,3)(−2,3)$ C: $(−4,−4)(−4,−4)$ D: $(3,−5)(3,−5)$ E: $(−3,0)(−3,0)$ F: $(0,2)(0,2)$

4.7

a, c

4.8

b, c

4.9

b

4.10

a, b

4.11
 $y=3x−1y=3x−1$ $xx$ $yy$ $(x,y)(x,y)$ 0 $−1−1$ $(0,−1)(0,−1)$ $−1−1$ $−4−4$ $(−1,−4)(−1,−4)$ 2 5 $(2,5)(2,5)$
4.12
 $y=6x+1y=6x+1$ $xx$ $yy$ $(x,y)(x,y)$ 0 1 $(0,1)(0,1)$ 1 7 $(1,7)(1,7)$ $−2−2$ $−11−11$ $(−2,−11)(−2,−11)$
4.13
 $2x−5y=202x−5y=20$ $xx$ $yy$ $(x,y)(x,y)$ 0 $−4−4$ $(0,−4)(0,−4)$ 10 0 $(10,0)(10,0)$ $−5−5$ $−6−6$ $(−5,−6)(−5,−6)$
4.14
 $3x−4y=123x−4y=12$ $xx$ $yy$ $(x,y)(x,y)$ 0 $−3−3$ $(0,−3)(0,−3)$ 4 0 $(4,0)(4,0)$ $−4−4$ $−6−6$ $(−4,−6)(−4,−6)$
4.15

4.16

4.17

4.18

4.19

yes, yes  yes, yes

4.20

no, no  yes, yes

4.37

x- intercept: $(2,0)(2,0)$; y- intercept: $(0,−2)(0,−2)$

4.38

x- intercept: $(3,0)(3,0)$, y- intercept: $(0,2)(0,2)$

4.39

x- intercept: $(4,0)(4,0)$, y- intercept: $(0,12)(0,12)$

4.40

x- intercept: $(8,0)(8,0)$, y- intercept: $(0,2)(0,2)$

4.41

x- intercept: $(4,0)(4,0)$, y- intercept: $(0,−3)(0,−3)$

4.42

x- intercept: $(4,0)(4,0)$, y- intercept: $(0,−2)(0,−2)$

4.49

$4 3 4 3$

4.50

$1 4 1 4$

4.51

$− 2 3 − 2 3$

4.52

$− 4 3 − 4 3$

4.57

$2 5 2 5$

4.58

$3 4 3 4$

4.59

$− 4 3 − 4 3$

4.60

$− 3 5 − 3 5$

4.61

$5 4 5 4$

4.62

$3 2 3 2$

4.63

undefined

4.64

0

4.65

1

4.66

1

4.67

$−1 −1$

4.68

10

4.75

$7 12 7 12$

4.76

$5 12 5 12$

4.77

$− 1 36 − 1 36$

4.78

$− 1 48 − 1 48$

4.79

slope $m=23m=23$ and y-intercept $(0,−1)(0,−1)$

4.80

slope $m=12m=12$ and y-intercept $(0,3)(0,3)$

4.81

$2 5 ; ( 0 , −1 ) 2 5 ; ( 0 , −1 )$

4.82

$− 4 3 ; ( 0 , 1 ) − 4 3 ; ( 0 , 1 )$

4.83

$− 1 4 ; ( 0 , 2 ) − 1 4 ; ( 0 , 2 )$

4.84

$− 3 2 ; ( 0 , 6 ) − 3 2 ; ( 0 , 6 )$

4.95

intercepts  horizontal line  slope–intercept  vertical line

4.96

vertical line  slope–intercept  horizontal line  intercepts

4.97
1. 50 inches
2. 66 inches
3. The slope, 2, means that the height, h, increases by 2 inches when the shoe size, s, increases by 1. The h-intercept means that when the shoe size is 0, the height is 50 inches.

4.98
1. 40 degrees
2. 65 degrees
3. The slope, $1414$, means that the temperature Fahrenheit (F) increases 1 degree when the number of chirps, n, increases by 4. The T-intercept means that when the number of chirps is 0, the temperature is $40°40°$.

4.99
1. $60 2.$185
3. The slope, 0.5, means that the weekly cost, C, increases by $0.50 when the number of miles driven, n, increases by 1. The C-intercept means that when the number of miles driven is 0, the weekly cost is$60

4.100
1. $35 2.$170
3. The slope, 1.8, means that the weekly cost, C, increases by $1.80 when the number of invitations, n, increases by 1.80. The C-intercept means that when the number of invitations is 0, the weekly cost is$35.;

4.101

parallel

4.102

parallel

4.103

parallel

4.104

parallel

4.105

parallel

4.106

parallel

4.107

not parallel; same line

4.108

not parallel; same line

4.109

perpendicular

4.110

perpendicular

4.111

not perpendicular

4.112

not perpendicular

4.113

$y = 2 5 x + 4 y = 2 5 x + 4$

4.114

$y = − x − 3 y = − x − 3$

4.115

$y = 3 5 x + 1 y = 3 5 x + 1$

4.116

$y = 4 3 x − 5 y = 4 3 x − 5$

4.117

$y = 5 6 x − 2 y = 5 6 x − 2$

4.118

$y = 2 3 x − 4 y = 2 3 x − 4$

4.119

$y = − 2 5 x − 1 y = − 2 5 x − 1$

4.120

$y = − 3 4 x − 4 y = − 3 4 x − 4$

4.121

$y = 8 y = 8$

4.122

$y = 4 y = 4$

4.123

$y = 5 2 x − 13 2 y = 5 2 x − 13 2$

4.124

$y = − 2 5 x + 22 5 y = − 2 5 x + 22 5$

4.125

$y = 1 3 x − 10 3 y = 1 3 x − 10 3$

4.126

$y = − 2 5 x − 23 5 y = − 2 5 x − 23 5$

4.127

$x = 5 x = 5$

4.128

$x = −4 x = −4$

4.129

$y = 3 x − 10 y = 3 x − 10$

4.130

$y = 1 2 x + 1 y = 1 2 x + 1$

4.131

$y = − 1 3 x + 10 3 y = − 1 3 x + 10 3$

4.132

$y = −2 x + 16 y = −2 x + 16$

4.133

$y = −5 y = −5$

4.134

$y = −1 y = −1$

4.135

$x = −5 x = −5$

4.136

$x = −4 x = −4$

4.137

yes  yes  yes  yes  no

4.138

yes  yes  no  no  yes

4.139

$y ≥ −2 x + 3 y ≥ −2 x + 3$

4.140

$y < 1 2 x − 4 y < 1 2 x − 4$

4.141

$x − 4 y ≤ 8 x − 4 y ≤ 8$

4.142

$3 x − y ≤ 6 3 x − y ≤ 6$

### Section 4.1 Exercises

1.
3.
5.
7.
9.

A: $(−4,1)(−4,1)$ B: $(−3,−4)(−3,−4)$ C: $(1,−3)(1,−3)$ D: $(4,3)(4,3)$

11.

A: $(0,−2)(0,−2)$ B: $(−2,0)(−2,0)$ C: $(0,5)(0,5)$ D: $(5,0)(5,0)$

13.

a, b

15.

a, c

17.

b, c

19.

a, b

21.
 $xx$ $yy$ $(x,y)(x,y)$ 0 $−4−4$ $(0,−4)(0,−4)$ 2 0 $(2,0)(2,0)$ $−1−1$ $−6−6$ $(−1,−6)(−1,−6)$
23.
 $xx$ $yy$ $(x,y)(x,y)$ 0 5 $(0,5)(0,5)$ 3 2 $(3,2)(3,2)$ $−2−2$ 7 $(−2,7)(−2,7)$
25.
 $xx$ $yy$ $(x,y)(x,y)$ 0 1 $(0,1)(0,1)$ 3 2 $(3,2)(3,2)$ 6 3 $(6,3)(6,3)$
27.
 $xx$ $yy$ $(x,y)(x,y)$ 0 $−2−2$ $(0,−2)(0,−2)$ 2 $−5−5$ $(2,−5)(2,−5)$ $−2−2$ 1 $(−2,1)(−2,1)$
29.
 $xx$ $yy$ $(x,y)(x,y)$ 0 2 $(0,2)(0,2)$ 3 4 $(3,1)(3,1)$ 6 0 $(6,0)(6,0)$
31.
 $xx$ $yy$ $(x,y)(x,y)$ 0 $−2−2$ $(0,−2)(0,−2)$ 10 2 $(10,2)(10,2)$ 5 0 $(5,0)(5,0)$
33.

35.

37.

39.

41.

43.

45.

47.

49.

Age and weight are only positive.

51.

53.

### Section 4.2 Exercises

55.

yes; no  no; no  yes; yes  yes; yes

57.

yes; yes  yes; yes  yes; yes  no; no

59.
61.
63.
65.
67.
69.
71.
73.
75.
77.
79.
81.
83.
85.
87.
89.
91.
93.
95.
97.
99.
101.
103.
105.
107.
109.
111.
113.
115.
117.
119.
121.
123.
125.
127.
129.
131.
133.
135.

$722,$850, $978 137. Answers will vary. ### Section 4.3 Exercises 139. $(3,0),(0,3)(3,0),(0,3)$ 141. $(5,0),(0,−5)(5,0),(0,−5)$ 143. $(−2,0),(0,−2)(−2,0),(0,−2)$ 145. $(−1,0),(0,1)(−1,0),(0,1)$ 147. $(6,0),(0,3)(6,0),(0,3)$ 149. $(0,0)(0,0)$ 151. $(4,0),(0,4)(4,0),(0,4)$ 153. $(−2,0),(0,−2)(−2,0),(0,−2)$ 155. $(5,0),(0,−5)(5,0),(0,−5)$ 157. $(−3,0),(0,3)(−3,0),(0,3)$ 159. $(8,0),(0,4)(8,0),(0,4)$ 161. $(2,0),(0,6)(2,0),(0,6)$ 163. $(12,0),(0,−4)(12,0),(0,−4)$ 165. $(2,0),(0,−8)(2,0),(0,−8)$ 167. $(5,0),(0,2)(5,0),(0,2)$ 169. $(4,0),(0,−6)(4,0),(0,−6)$ 171. $(3,0),(0,1)(3,0),(0,1)$ 173. $(−10,0),(0,2)(−10,0),(0,2)$ 175. $(0,0)(0,0)$ 177. $(0,0)(0,0)$ 179. 181. 183. 185. 187. 189. 191. 193. 195. 197. 199. 201. 203. 205. $(0,1000),(15,0)(0,1000),(15,0)$ At $(0,1000)(0,1000)$, he has been gone 0 hours and has 1000 miles left. At $(15,0)(15,0)$, he has been gone 15 hours and has 0 miles left to go. 207. Answers will vary. 209. Answers will vary. ### Section 4.4 Exercises 211. $1 4 1 4$ 213. $2 3 2 3$ 215. $−3 2 = − 3 2 −3 2 = − 3 2$ 217. $− 2 3 − 2 3$ 219. 221. 223. 225. 227. $2 5 2 5$ 229. $5 4 5 4$ 231. $− 1 3 − 1 3$ 233. $− 3 4 − 3 4$ 235. $3 4 3 4$ 237. $− 5 2 − 5 2$ 239. $− 2 3 − 2 3$ 241. $1 4 1 4$ 243. 0 245. undefined 247. 0 249. undefined 251. $5 2 5 2$ 253. $3 4 3 4$ 255. $− 5 2 − 5 2$ 257. $− 8 7 − 8 7$ 259. $7 3 7 3$ 261. $−1 −1$ 263. 265. 267. 269. 271. 273. 275. 277. 279. $1313$ 4 12 pitch or 4-in-12 pitch 281. $350350$; rise = 3, run = 50 283. 288 inches (24 feet) Models will vary. 285. When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right. 287. A vertical line has 0 run and since division by 0 is undefined the slope is undefined. ### Section 4.5 Exercises 289. slope $m=4m=4$ and y-intercept $(0,−2)(0,−2)$ 291. slope $m=−3m=−3$ and y-intercept $(0,1)(0,1)$ 293. slope $m=−25m=−25$ and y-intercept $(0,3)(0,3)$ 295. $−9 ; ( 0 , 7 ) −9 ; ( 0 , 7 )$ 297. $4 ; ( 0 , −10 ) 4 ; ( 0 , −10 )$ 299. $−4 ; ( 0 , 8 ) −4 ; ( 0 , 8 )$ 301. $− 8 3 ; ( 0 , 4 ) − 8 3 ; ( 0 , 4 )$ 303. $7 3 ; ( 0 , −3 ) 7 3 ; ( 0 , −3 )$ 305. 307. 309. 311. 313. 315. 317. 319. 321. horizontal line 323. vertical line 325. slope–intercept 327. intercepts 329. slope–intercept 331. horizontal line 333. intercepts 335. slope–intercept 337. 1.$28
2. $66.10 3. The slope, 2.54, means that Randy’s payment, P, increases by$2.54 when the number of units of water he used, w, increases by 1. The P–intercept means that if the number units of water Randy used was 0, the payment would be $28. 339. 1.$15
2. $143 3. The slope, 0.32, means that the cost, C, increases by$0.32 when the number of miles driven, m, increases by 1. The C-intercept means that if Janelle drives 0 miles one day, the cost would be $15. 341. 1.$750
2. $2418.60 3. The slope, 0.09, means that Patel’s salary, S, increases by$0.09 for every $1 increase in his sales. The S-intercept means that when his sales are$0, his salary is $750. 343. 1.$2850
2. $4950 3. The slope, 42, means that the cost, C, increases by$42 for when the number of guests increases by 1. The C-intercept means that when the number of guests is 0, the cost would be $750. 345. parallel 347. parallel 349. not parallel 351. parallel 353. parallel 355. parallel 357. parallel 359. parallel 361. not parallel 363. not parallel 365. not parallel 367. not parallel 369. not parallel 371. perpendicular 373. perpendicular 375. not perpendicular 377. not perpendicular 379. perpendicular 381. perpendicular 383. 1. For every increase of one degree Fahrenheit, the number of chirps increases by four. 2. There would be $−160−160$ chirps when the Fahrenheit temperature is $0°0°$. (Notice that this does not make sense; this model cannot be used for all possible temperatures.) 385. Answers will vary. ### Section 4.6 Exercises 387. $y = 4 x + 1 y = 4 x + 1$ 389. $y = 8 x − 6 y = 8 x − 6$ 391. $y = − x + 7 y = − x + 7$ 393. $y = −3 x − 1 y = −3 x − 1$ 395. $y = 1 5 x − 5 y = 1 5 x − 5$ 397. $y = − 2 3 x − 3 y = − 2 3 x − 3$ 399. $y = 2 y = 2$ 401. $y = −4 x y = −4 x$ 403. $y = −2 x + 4 y = −2 x + 4$ 405. $y = 3 4 x + 2 y = 3 4 x + 2$ 407. $y = − 3 2 x − 1 y = − 3 2 x − 1$ 409. $y = 6 y = 6$ 411. $y = 3 8 x − 1 y = 3 8 x − 1$ 413. $y = 5 6 x + 2 y = 5 6 x + 2$ 415. $y = − 3 5 x + 1 y = − 3 5 x + 1$ 417. $y = − 1 3 x − 11 y = − 1 3 x − 11$ 419. $y = 4 y = 4$ 421. $y = −7 y = −7$ 423. $y = − 5 2 x − 22 y = − 5 2 x − 22$ 425. $y = −4 x − 11 y = −4 x − 11$ 427. $y = −8 y = −8$ 429. $y = −4 x + 13 y = −4 x + 13$ 431. $y = x + 5 y = x + 5$ 433. $y = − 1 3 x − 14 3 y = − 1 3 x − 14 3$ 435. $y = 7 x + 22 y = 7 x + 22$ 437. $y = − 6 7 x + 4 7 y = − 6 7 x + 4 7$ 439. $y = 1 5 x − 2 y = 1 5 x − 2$ 441. $x = 4 x = 4$ 443. $x = −2 x = −2$ 445. $y = 2 y = 2$ 447. $y = −3 y = −3$ 449. $y = 4 x y = 4 x$ 451. $y = 1 2 x + 3 2 y = 1 2 x + 3 2$ 453. $y = 5 y = 5$ 455. $y = 3 x − 1 y = 3 x − 1$ 457. $y = −3 x + 3 y = −3 x + 3$ 459. $y = 2 x − 6 y = 2 x − 6$ 461. $y = − 2 3 x + 5 y = − 2 3 x + 5$ 463. $x = −3 x = −3$ 465. $x = 4 x = 4$ 467. $y = −4 y = −4$ 469. $y = −1 y = −1$ 471. $y = x y = x$ 473. $y = − 3 2 x − 1 y = − 3 2 x − 1$ 475. $y = − 3 4 x − 1 4 y = − 3 4 x − 1 4$ 477. $y = 5 4 x y = 5 4 x$ 479. $x = −5 x = −5$ 481. $y = 1 y = 1$ 483. $y = x + 5 y = x + 5$ 485. $y = 5 6 x + 2 y = 5 6 x + 2$ 487. $y = − 2 3 x + 5 y = − 2 3 x + 5$ 489. $y = − 3 5 x + 1 y = − 3 5 x + 1$ 491. $y = 2 y = 2$ 493. $y = x + 2 y = x + 2$ 495. $x = −3 x = −3$ 497. $y = − 1 3 x − 14 3 y = − 1 3 x − 14 3$ 499. $y = 3 4 x y = 3 4 x$ 501. $y = 1.2 x + 5.2 y = 1.2 x + 5.2$ 503. Answers will vary. ### Section 4.7 Exercises 505. yes no no yes no 507. yes no no yes yes 509. no no no yes yes 511. $y < 2 x − 4 y < 2 x − 4$ 513. $y ≤ − 1 3 x − 2 y ≤ − 1 3 x − 2$ 515. $x + y ≥ 3 x + y ≥ 3$ 517. $x + 2 y ≥ −2 x + 2 y ≥ −2$ 519. $2 x − y < 4 2 x − y < 4$ 521. $4 x − 3 y > 12 4 x − 3 y > 12$ 523. 525. 527. 529. 531. 533. 535. 537. 539. 541. 543. 545. 547. 549. 551. 553. 1. Answers will vary. ### Review Exercises 557. 559. 561. $(2,0)(2,0)$ $(0,−5)(0,−5)$ $(−4.0)(−4.0)$ $(0,3)(0,3)$ 563. a, b 565.  $xx$ $yy$ $(x,y)(x,y)$ 0 3 $(0,3)(0,3)$ 4 1 (4, 1) $−2−2$ 4 $(−2,4)(−2,4)$ 567.  $xx$ $yy$ $(x,y)(x,y)$ 0 $−3−3$ $(0,−3)(0,−3)$ 2 0 $(2,0)(2,0)$ $−2−2$ $−6−6$ $(−2,−6)(−2,−6)$ 569. Answers will vary. 571. Answers will vary. 573. yes; yes yes; no 575. 577. 579. 581. 583. 585. $( 3 , 0 ) , ( 0 , 3 ) ( 3 , 0 ) , ( 0 , 3 )$ 587. $( −1 , 0 ) , ( 0 , 1 ) ( −1 , 0 ) , ( 0 , 1 )$ 589. $( 6 , 0 ) , ( 0 , 4 ) ( 6 , 0 ) , ( 0 , 4 )$ 591. $( 0 , 0 ) ( 0 , 0 )$ 593. 595. 597. 599. $4 3 4 3$ 601. $− 2 3 − 2 3$ 603. 605. 607. 1 609. $− 1 2 − 1 2$ 611. undefined 613. 0 615. $−6 −6$ 617. $5 2 5 2$ 619. 621. 623. $1 10 1 10$ 625. slope $m=−23m=−23$ and y-intercept $(0,4)(0,4)$ 627. $5 3 ; ( 0 , −6 ) 5 3 ; ( 0 , −6 )$ 629. $4 5 ; ( 0 , − 8 5 ) 4 5 ; ( 0 , − 8 5 )$ 631. 633. 635. horizontal line 637. intercepts 639. plotting points 641. −$250  $450 The slope, 35, means that Marjorie’s weekly profit, P, increases by$35 for each additional student lesson she teaches. The P–intercept means that when the number of lessons is 0, Marjorie loses \$250.

643.

not parallel

645.

perpendicular

647.

$y = −5 x − 3 y = −5 x − 3$

649.

$y = −2 x y = −2 x$

651.

$y = −3 x + 5 y = −3 x + 5$

653.

$y = −4 y = −4$

655.

$y = 3 5 x y = 3 5 x$

657.

$y = −2 x − 5 y = −2 x − 5$

659.

$y = 1 2 x − 5 2 y = 1 2 x − 5 2$

661.

$y = 2 y = 2$

663.

$y = − 2 5 x + 8 y = − 2 5 x + 8$

665.

$y = 3 y = 3$

667.

$y = − 3 2 x − 6 y = − 3 2 x − 6$

669.

$y = 1 y = 1$

671.

yes  no  yes  yes  no

673.

$y > 2 3 x − 3 y > 2 3 x − 3$

675.

$x − 2 y ≥ 6 x − 2 y ≥ 6$

677.
679.
681.

### Practice Test

683.

yes  yes  no

685.

$( 3 , 0 ) , ( 0 , −4 ) ( 3 , 0 ) , ( 0 , −4 )$

687.

undefined

689.

$1 1$

691.
693.
695.
697.

$y = − 3 4 x − 2 y = − 3 4 x − 2$

699.

$y = 1 2 x − 4 y = 1 2 x − 4$

701.

$y = − 4 5 x − 5 y = − 4 5 x − 5$

703.
705.
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