A ry_?:2

y —x + 2x — y < 4 —y < —x + 4 y > x — 4

2.5)Hy>2 y —5x + 2

2( + y 1 y —2x + 1

X

EXAMPLE Solve.0 0

2x + yLs 4 3x -4- 2y_. 6

Answer

x?_ 0

2x+y54

y -2x 4

3x+ 2y s_ 6 2y -3x + 6 k.

3x +3

The solution is all points in the darkest shaded region.

Or* A. Exercises Graph each system of inequalities.

4. 2x — y < 5 —y < —2x + 5 y> 2x — 5

3x + y 4 y —3x + 4 6. 3x + y> 9

y> —3x+ 9

2x y 6 y —2x + 6

3. 3x — 2y 6 x + y 3 —2y —3x + 6 y —x + 3

y 3 — 3

1. x+y_2

x— y < 4

2. 5x+y>2

2x + y5_ 1

6. 3x + y> 9

2x + y 6

7. x+2y_ ._ —4

x+ 2y < 6

3. 3x — 2y 6 x+ y 3

4. 2x — y< 5 3x + y 4

9. x

y 0

x + 4y 8

2x+y4

► B. Exercises Graph each system of inequalities.

5. x+ 5y < 15 3x + 2y_. 8

11. 0

yS2

x+y>1

10. 0

8. 0

y>-0

Y" 0

3x + y 8

5x+ 6y 30

x+2y . 6

5. x + 5y < 15 3x + 2y -5 8 5y < —x+ 15 2y —3x + 8 y < 51 x + 3

Y 23

x + 4

316 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

316 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

CLIrrluiative Review16. 2x + y 4

y 5. -2x + 4 x+y2

Find the equation of each line if - x + 2 18. the slope is - 3 and the y- intercept is (0, 2). = -3x + 2 x>_ 0 19. the slope is 3 and it passes through (2, 5). = 3 x + [6.7] y>_0

20. it passes through (1, 5) and ( - 2, 2). = ± 4'6.31

21. it is vertical and passes through ( - 2, - 5). = - _15.3]

22. it is parallel to y = z x - 4 and passes through (1, 5). y = + [6.7]

Is eact. point a solution to the system of inequalities? Do not graph. 12. 5x + 3y 2 x - 2y 4 2 1 - 2(-2) 4

1 + 4 4 5 4 True 2x + 6y < 3 2(3) + 6(1) < 3 12 < 3 False 3y + 4 > 2x

-2 3(-2) + 4 > 2(-1) -2 > -2 False

3x + 4y < 12 3(-3) + 4(5) < 12 11 < 12 True

12. 5.X x

13. 3

• C. Exerci!es Graph

2x

+ 3y 5 2; (1, -2) 14. 5 - 2x y; (-1, -2) -2y? 4 a.; 3y + 4 > 2x - y > 4; (3, 1) 15. y 2 - x; (-3, 5)

+ 6y < 3 lc 3x + 4y < 12 .'as

the system of inequalities and then name (a) a point that satisfies the

5(1) + 3(-2) 5 - 6 2 -1 5_ 2 True

13. 3x - y> 4 3(3) - 1 > 4 8 > 4 True

14. 5 - 2x y

systern, (b) a point that does not satisfy the system, and (c) a point that lies 5 - 2(-1)

on the boundary of the region. 5 + 2 -2

16.X 0 17. x + y 3 7 -2 True 0 2x < 3y + 1 15. y 2 - x

+ y 5 4 y 0 2.0swers 5 2 - (-3) y 2 -mill vary. y < 2 a, 'a, 7. will vary. 5 5 True

17. x+-x + 3

2x < 3y + 1 2x- 1 <3y 2

y> T x -

7.10 SYSTEMS OF INEQUALITIES 317 19. y- 5 = - 2) y - 5 = - 3(y - 5) = - 4)3 3y - 15 = 2x - 4 3y = 2x + 11

2 11 y = T x + -3-

20.m = 2 -3 -2 -51 = = 1

y - 5 = 1(x- 1 ) y- 5 = x - 1 y = x + 4

22. y - 5 = 4-(x - 1)

y - 5 = x - 2(y - 5) = - 4-)2 2y - 10 = x - 1 2y = x + 9

1 9 y= TX T

7.10 SYSTEMS OF INEQUALITIES 317

.:gr_0913 marlsed_ off by irs

Chapter 7

Review"4112223147.7MTSM

Objective To help students prepare for evaluation

Vocabulary See Appendix A.

Assignment • Minimum: 3-18 multiples of 3. 26-33, 35 • Average: 1-23 odd, 26-32, 34-38 even • Extended: 2-16 even. 20-33, 36-38

Resources • ActIvities Manual, Cumulative Review. • Test Packet, Chapter 7 Exam.

Aris l iver.s Chapter 7 R'EAAVve 1. 31x + 4y = 6 y = 1 - x

1-2) + 4(3) = 6 3 = 1 - ( - 2) 6 = 6 3 = 3

2. 5,( - 2y = 7 x + y = 1 5(-2) - 2(3) = 7 -2 + 3 = 1 -L 10 - 6 = 7 1 = 1 -16 = 7 False

3. xpl- 2y = 4 3x + y = -3 -2 + 2(3) = 4 3(-2) + 3 = - 3 4 = 4 -3 = -3

4. x,+ y< 2 x + 2y 4 - 2 + 3 < 2 -2 + 2(3) 4 1 < 2 4>4

5. x -F y= 11 y= -x + 11 2x - y = - 5 -y = -2x - 5 y 2x + 5

6. 3x - y = 14 x + 3y = -12 = -3x 14

3y = -x - 12

y 3x - 14 y= 31 x 4

Chapter Review

Determine whether ( - 2, 3) is a solution to each system. 1. 3x + 4y = 6

3. x 2y = 4

y = 1 - x

3x + y = - 3 2. 5x - 2y = 7

4. x + y < 2 x + y = 1 x 2y 4

Use the graphing method to solve each system.

5. x+y= 11 7. x + y = - 2

2x - y = -5 3x + y = -10 . , - 4, 2 6. 3x - y = 14 8. x + 3y = 5

x + 3y = -12 y = 2 -

Use the substitution method to solve each system.

9. 2x + 3y = 8

11. 4x + 3y = -20

x + 2y -= 4

5x - 2y = -2

10. 5x - 3y = 12

12. 5x + y = 8

3x + y = -2 -10x - 2y = -3 nc

Use the addition method to solve each system.

13. x - 2y = -11 15. 5x - 3y = -25

3x + y = 16 6x + 2y = - 2 14. 2x + 3y =- 9 16. x + y = -2

4x - y 2 [ I 2x + 2y = -4 en:i!'E,

Solve each system using any method.

17. y = 2x - 3 19. 2x + 5y = 8 y = -g-x + 7,1, x - 5y = 4

18. x - 2y = - 42 20. 0.5x - 2.1y = 7.2

3x - y = - 31 "-4, 1.4x + 0.3y = 7.8

Solve each system using any method. Tell whether each is consistent or inconsistent, dependent or independent. 21. x + y = 5 nc solutior..!

x = 2 - y incorisiste!-.. 22. 3x + 5y = 1 ;2, -1); cc:-.:L•

4x - 3y = 11 ndepende:. 23. 3x + y = 6

2 -3-y = 5 - 2x incons::- .-

320 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

Tips •

Have three students solve the following system at the board. each using a different method (graphing, substitution, and addition). You may want to have three other students explain their work.

3x - y = 3 (2, 3) 2x + y = 7 Be sure the students who are not at the

board also work the problem on paper for practice. You may want to have a student graph a system of inequalities at this time as well.

Common Student Error. Students often confuse types of word problems. Help them by discussing characteristics and then prac-tice determining the type on several prob-lems. Be sure to include several of each type.

24. 3x + y = 4 ertire 2y = 8 - 6x 7-,7--Zr.rident

25. x + y2 = 8 4 2); x = y2 consistent,. independent

Ex. 28. Requires I = Prt, which results in the answer / = 0.07d(1). Students may leavt the last factor in their answer. n

320 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

6 ;. 2x + y < 6

y < -2x +3x - y 4

-3x + 4

y 3x - 4

y

35. 3x + 5y 12 2x - 3y < -6 5y -3x + 12 -3y < -2x -- 6

3 2 2 y + y > -Tx + 2

34. 5x + 2y 8 2y -5x + 8 y<_ 25 x +4

x + y > -5 y > - x - 5

Give a symbolic expression for each. 26. The distance traveled by a jet in 12 hours at x mph 27. The upstream speed of a kayaker that goes 3 mph in still water if

paddling in a current of c mph t - : 28. The annual interest received from an investment of d dollars at 7% = 29. The amount of salt when m gallons of a 20%-salt solution is mixed with

n I gallons of a 23%-salt solution 0.2m +

Solve algebraically. 30. Sugar Tooth Candy Company needs 300 gallons of a 32% sucrose solu-

tion for a certain kind of candy. The company has a solution that is 60% A la:. sucrose and a solution that is 25% sucrose. How many gallons of each .;i.icrose; 240 should the company mix together to obtain the desired solution? 3f 25% sucrosa

31. Mr. Arnold is going to invest $1550 in two separate accounts. One account pays 7.5%, and the other account pays 8.25%. How much should he invest in each account so that his annual return on his investment will be $121.20? .1390 at 7.3%; 3:360 et 3,2511

32. Two fishing boats leave Sandy Cove at the same time traveling in the same direction. One boat is traveling three times as fast as the other boat. Af:er five hours the faster boat is 80 miles ahead of the slower boat. What is the speed of each boat? slow an y. last one, 24 mph

Graph each system of inequalities. 33. 2x + y < 6 35. 3x + 5y 12

3x - y 4 2x - 3y < -6 34. 5x + 2y 8 36. y 3x + 4

x y > -5 y > 2

37. What does the graph of an inconsistent linear system look like? 2 parallel Ines 38. Explain the mathematical significance of II Corinthians 5:21. 2hrist died

as t, substitute in our *ca. His payment for Our debt is an illustration of the priticiple of substitution.

30. gal. of % of Total sucrose sucrose sucrose

x 0.6 0.6x

y 0.25 0.25y

300 0.32 96

25(x + y) = (300)25

100(0.6x + 0.25y) = (96)100 60x + 25y = 9600 25x + 25y = 7500 35x = 2100 60 + y = 300 x = 60 y = 240 60 gal. 60%. 240 gal. 25%

31. P r t

x 0.075 1 0.075x

y 0.0825 1 0.0825y

750(x + y) = (1550)750 x + 660 = 1550 x = 890

10,000(0.075x 0.0825y)-(121.20)10,000 750x + 825y = 1,212,000 750x + 750y = 1,162,500 75y = 49,500 y = 660 $890 at 7.5%, $660 at 8.25% 32. r t d

fast x 5 5x

slow y 5 5y

60% sucrose

25% sucrose

mix

x = 3y

5x - 80 = 5y 5(3y) - 80 = 5y 15y - 80 = 5y

x = 3(8) Oy = 80 x = 24 y = 8 slow 8 mph. fast 24 mph

CHAPTER REVIEW 321

CHAPTER REVIEW 321

16. 3x + y = 5 y -3x + 5

y = 3x - 4

1 . 3 - 5y = -15 - y = -3x - 15

x + 3 5 :onsistent; ndependent

20. 12x + 4y = 8 4y = -12x 8 y= -3x - 2

3y = 15 - 9x y = 5 - 3x y = -3x + 5

25.5x -3x = x = 5

26. x +

160(312 5x + 12 5x = 58

58 x= 27.4(x - 3) Ls

4x - 12 - 4x 55- 12 x -3

28. 12x + 1 > 2x + 1 > 19 2x > 18 x > 9

160

2x + 1 < -19 2x < -20 x < -10

3

3x 66x 66x

8 = 176

22 67

Ans. (22 —225

29.35 + 3'+y = 25x 30y =s x — 35

35

= 3 - 30

5 7 y= -6-

14 10x

10x _ 7 )

10x - 10x 14 = 14 14 14 Tru An entire lin

16.32 9 = 20 32 = 20x —

20 9 y 32 x 32

5 9 y 8 x 32

5x - 24y = 7 5x - 24(ix - =7

15x - 15x + 47 7 27 = 7 False

Ans. no solution

= 14

x + 2y -4 2y -x - 4 y 21 x-2

x + 2y < 6 2y < -x + 6 y< —21 x 3

11. 3x + y = -2 y = -3x - 2

3(4) ± Y = —27 26 ± Y =

- 27 + 26y 26y = -25

—25 —9 —25 `1

Y — 26 Ans. 26 26

12. x - 5y = 6 4x+ =9 x = 5y + 4(5y + 6) + 3y = 9

20y+ 4 + 3y = 9 23y = -15

—15 Y 23

x 3 x + T3- = 6 23x + 5 = 138 23x - 63 x = Ans. (63 , —2135

13. y2 - 9 4x + y2 = 209 4x + (x + 9) = 2 9 5x = 200 x = 40

y2 _-__= 40 + 9 \ y2 = 49 Y = =7 Ans. (40, 7). (40, - 7)

14.3x 4- 5y = 8 3x = 8 - Sy

8 5 x = T - Ty

x - 4y =7 - ÷y) - 4y = 7

6 310 y 4y = 7 1 - lOy - 2y = 21

2y = 5 —5

2x - 8y = 7 2x — 8( - 3x - 2) = 7 2x + 24x + 16 = 7 6x = -9

—9 — 26

608 ANSWERS

10. 3x ..- yam 8 —3x + 8

x 0 y

x + 2y s 6 2y —x + 6 y 21

x + 3

8. 5x + 6y s 30 6y s —5x + 30 y x + 5

9. x 4y Is 8 4y —x + 8

—1 y s —(- x + 2

x 0 0

2x+ys 4 y s —2x + 4 x>_ 0 y>0

y

—9 14 y— 7 — 7

3 _ —2 Y — 7

11. 5x — 2y = —2 —2y = —5x — 2 y= 5

4- 1

y = —3(4) )-2 14x = 6 =

Ans. (3, 7 —23)

14. 2x + 3y = 9 4x — y = 2 2x + 3y = 9

12x — 3y = 6 14x = 15

_ 15 X 14

Ans. 15 16 14 • 7

11.10 eviiiri 120\CLV

7. x + y = —2

3x + y —10

y —x — 2 y= —3x — 10

8. x + 3y = 5 3y = —x + 5

—1 y= 3 x+ 5

Z

9.x + 2y = 4 2x + 3y = 8 x = 4 — 2y 2(4 — 2y) + 3y = 8

8 — 4y + 3y = 8 x = 4 — 2(0) 8 — y = 8 x = 4 —y = 0 Ans (4, 0) y = 0

10.3x + y = —2 5x — 3y = 12 y=y=-3x---2 5x — 3(-3x — 2) = 12

5x + 9x + 6 = 12

4x + 3y = —20 4x + 3(lx + 1) = —20 4x + 2

5x + 3 = —20

• 8x -+ 15x + 6 = —40 23x= —46 5

Y = 2 (2) ± 1 x = —2 y = —5 + 1 Ans. (-2, —4) y = —4

12. 5x + y = 8 —10x— 2y= —3 y= 8 — 5x —10x —2(8 — 5x) = —3

—10x — 16 + 10x= —3 Ans. no sol. —16 = —3 False

13. x — 2y = —11 x — 2y = —11 2(3x 4 y)= (16)2 6x + 2y = 32

7x = 21 3(3) + y = 16 x = 3 Y = 7 Ans (3, 7)

4(4) — y = 2

60 14 — = 2

60 — 14y = 28 —14y= —32

16 Y = T

15. 5x — 3y = —25 10x — 6y = —50 6x + 2y = —2 18x + 6y = —6

28x = —56 6(-2) + 2y = —2 x = —2 —12 + 2y = —2 2y = 10 Y = 5 Ans. (-2, 5)

16. x + y = —2 2x + 2y = —4 2x + 2y = —4 2x + 2y = —4 Ans. entire line 0 = 0 True

17. y = 2x — 3 y 3x+7 2x — 3 = +x + 7

y = 2(6) — 3 6x — 9 = x + 21 y= 9 5x = 30 Ans. (6, 9) x = 6

18. x — 2y = —42 —3x + 6y = 126 3x — y = —31 3x — y = —31

5y = 95 x — 2(19) = —42 y = 19 x — 38 = —42 x = —4 Ans. (-4, 19)

19. 2x + 5y = 8 4 — 5y = 4 x — 5y = 4 —5y = 0

3x= 12 y = 0 x = 4 Ans. (4, 0)

20. 10(0.5x — 2.1y) = (7.2)10 70(1.4x + 0.3y) = (7.8)70

5x — 21y = 72 5(6) — 21y = 72 98x + 21y= 546 30 — 21y = 72

103x = 618 —21y = 42 x = 6 y = —2

Ans. (6, —2) 21. x + y = 5

x + y = 2 0 = 3 False Ans. no solution

22. 3(3x + 5y) = (1)3 5(4x — 3y) = (11)5 9x + 15y = 3 3(2) + 5y = 1

20x — 15y 55 6 + 5y = 1

29x = 58 5y = —5 x= 2 y= —1

Ans. (2, —1) 23. 2(3x + y) = (6)2 6x + 2y = 12

3(2x + 3 = (5)3 6x + 2y = 15 Ans. no sol. 0 = —3

False 24. 2(3x + y)= (4)2 6x + 2y = 8

2y = 8 — 6x 6x + 2y = 8 Ans. entire line 0 = 0 True

y = 2

ANSWERS 60S