CHAPTER System of Equations and Inequalities 5 Solutions Key

System of Equations and InequalitiesSolutions Key

ArE you rEAdy?

1. B 2. F

3. A 4. E

5. D

6. Plot (0, 1). Count 3 units up and 4 units right and plot another point. Draw the line connecting the two points.

2

-2

x

y

0 2

7. Plot (0, 5). Count 3 units down and 1 unit right and plot another point. Draw the line connecting the two points.

2

4

x

y

0 4

8. Plot (0, -6). Count 1 unit up and 1 unit right and plot another point. Draw the line connecting the two points.

-2

-4

-6

x y

0 2 4

9. x + y = 4 ______ -x ___ -x y = -x + 4 Plot (0, 4). Count 1 unit down and 1 unit right and

plot another point. Draw the line connecting the two points.

2

4

x

y

0 2 4

10. Plot (0, 4). Count 2 units down and 3 units right and plot another point. Draw the line connecting the two points.

2

x

y

0 2 4 6

11. Plot (0, -5). Count 0 units up and 1 unit right and plot another point. Draw the line connecting the two points.

-2

-4

x y

0 2 -2

12. -7x - 18 = 3 ________ + 18 ____ +18 -7x = 21

-7x ____ -7

= 21 ___ -7

x = -3

13. 12 = -3n + 6 ___ -6 _______ - 6 6 = -3n

6 ___ -3

= -3n ____ -3

-2 = n

14. 1 __ 2 d + 30 = 32

_______ - 30 ____ -30

1 __ 2 d = 2

2 ( 1 __ 2 d) = 2(2)

d = 4

15. -2p + 9 = -3 _______ - 9 ___ -9 -2p = -12

-2p

____ -2

= -12 ____ -2

p = 6

16. 33 = 5y + 8 ___ -8 ______ - 8 25 = 5y

25 ___ 5 =

5y ___

5

5 = y

17. -3 + 3x = 27 _______ +3 ___ +3 3x = 30

3x ___ 3

= 30 ___ 3

x = 10

18. 7x + y = 4 _______ -7x ____ -7x y = -7x + 4

19. y + 2 = -4x ____ - 2 ____ -2 y = -4x - 2

20. 8 = x - y ___ + y _____ + y 8 + y = x ______ -8 ___ -8 y = x - 8

21. x + 2 = y - 5 ____ + 5 _____ + 5 x + 7 = y y = x + 7

22. 2y - 3 = 12x _____ + 3 ___ +3 2y = 12x + 3

2y

___ 2 = 12x + 3 _______

2

y = 6x + 3 __ 2

23. y + 3 __ 4

x = 4

______

- 3 __ 4

x ____

- 3 __ 4

x

y = - 3 __ 4

x + 4

24. t - 5 = 7 - 5 = 2

25. 9 - 2a = 9 - 2(4) = 9 - 8 = 1

26. 1 __ 2 x - 2

= 1 __ 2 (14) - 2

= 7 - 2 = 5

27. n + 15 = 37 + 15 = 52

153 Holt McDougal Algebra 1

5CHAPTER

CS10_A1_MESK710372_C05.indd 153 3/30/11 11:02:25 PM

28. 9c + 4

= 9 ( 1 __ 3 ) + 4

= 3 + 4 = 7

29. 16 + 3d = 16 + 3(5) = 16 + 15 = 31

30. b - 9 ≥ 1 _____ + 9 ___ +9 b ≥ 10

8 4 0 12 16 20 -4

31. -2x < 10

-2x ____ -2

> 10 ___ -2

x > -5

0 -4 -3 -2 -1 -5 -6

32. 3y ≤ -3

3y

___ 3 ≤ -3 ___

3

y ≤ -1

0 2 -6 -4 -2 -8 -10

33. 1 __ 3 y ≤ 5

3 ( 1 __ 3 y) ≤ 3(5)

y ≤ 15

5 0 10 15 20 -5 -10

SolvIng SyStEmS by grAphIng

CHECK IT OUT!

1a. _____________ 2x + y = 5 2(1) + (3) 5 2 + 3 5 5 5 3

_______________ -2x + y = 1 -2(1) + (3) 1 -2 + 3 1 1 1 3

(1, 3) is a solution of the system.

b. _____________ x - 2y = 4 2 - 2(-1) 4 2 + 2 4 4 4 3

_____________ 3x + y = 6 3(2) - (1) 6 6 - 1 6 5 6 7

(2, -1) is not a solution of the system.

2a. 7

-2

x

y

0 3 -8

y = - 2x - 1 y = x + 5

(-2, 3)

The solution appears to be at (-2, 3).

b. 2x + y = 4 _______ -2x ____ -2x y = -2x + 4

4

-2

x

y

0 12 -4

y = -2x + 4

(3, -2)

y = 1 _ 3

x - 3

The solution appears to be at (3, -2).

3. Let x represent the number of movie rentals and y represent the total cost.

Video club A: y = 3x + 10 Video club B: y = 2x + 15 Graph y = 3x + 10 and y = 2x + 15.

(5, 25)

Cost of Movie Rentals

0 1 3 5 2 4 6 7

5 10 15 20 25 30

Movies

Cost

( $) Club B

Club A

The lines appear to intersect at (5, 25). So, the cost at both places will be the same for 5 movie rentals and that cost will be $25.

THINK AND DISCUSS

1. Locate the point of intersection. The ordered pair is a solution of the system.

2. Substitute the x- and y-values of the ordered pair into all of the equations. If all equations are true, the solution is correct.

3.

1. Graph the first and second equation.

2. Identify the point of intersection.

3. Check the solution.

Solving a Linear System by Graphing

ExErCISESGUIDED PRACTICE

1. an ordered pair that satisfies both equations

2. ______________ 3x + y = 4 3(2) + (-2) 4 6 - 2 4 4 4 3

______________ x - 3y = 4 (2) - 3(-2) 4 2 + 6 4 8 4 7


3. ______________ x - 2y = 5 (3) - 2(-1) 5 3 + 2 5 5 5 3

______________ 2x - y = 7 2(3) - (-1) 7 6 + 1 7 7 7 3

(3, -1) is a solution of the system.

4. _______________ -x + y = 6 -(-1) + (5) 6 1 + 5 6 6 6 3

________________ 2x + 3y = 13 2(-1) + 3(5) 13 -2 + 15 13 13 13 3

(-1, 5) is a solution of the system.

5. 5

-3

x

y

0 6

y = -x + 3

(2, 1)

y = 1 _ 2 x

The solution appears to be at (2, 1).


5-1

CS10_A1_MESK710372_C05.indd 154 3/30/11 11:02:29 PM

6. 2x + y = 1 _______ -2x ____ -2x y = -2x + 1

-5

x

y

0 6

y = -2x + 1

y = x - 2

(1, -1) -2


7. x + y = 3 ______ -x ___ -x y = -x + 3

9

x

y

0 1 -9

y = -x + 3

y = -2x - 1

(-4, 7)


8. Let x represent the number of cubic yards and let y represent the total cost.

Lawn and Garden: y = 30x + 30 Yard Depot: y = 25x + 55 Graph y = 30x + 30 and y = 25x + 55. Mulch Delivery Charges

0 1 3 5 2 4 6

40 80

120 160 200

Yard Depot

Lawn and Garden

Mulch (yd3)

Cost

( $) (5, 180)

The lines appear to intersect at (5, 180). So, the cost at both places will be the same for 5 cubic yards and that cost will be $180.

PRACTICE AND PROBLEM SOLVING

9. ______________ x - 2y = 8 (1) - 2(-4) 8 1 + 8 8 9 8 7

______________ 4x - y = 8 4(1) - (-4) 8 4 + 4 8 8 8 3


10. _________________ 2x - 3y = -7 2(-2) - 3(1) -7 -4 - 3 -7 -7 -7 3

________________ 3x + y = -5 3(-2) + (1) -5 -6 + 1 -5 -5 -5 3


11. ______________ 2x + y = 12 2(5) + (2) 12 10 + 2 12 12 12 3

_________________ -3y - x = -11 -3(2) - (5) -11 -6 - 5 -11 -11 -11 3


12. 5

-4

x

y

0 1 -8

y = -x - 1

(-2, 1)

y = 1 _ 2 x + 2


13.

5

x

y

0 6

y = x y = -x + 6

(3, 3)


14. x = -y + 3 ___ + y ______ +y x + y = 3 ______ -x ___ -x y = -x + 3

9

x

y

0 1 -9

y = -x + 3

y = -2x - 1

(-4, 7)


15. x + y = 2 _______ -x ___ -x y = -x + 2

2

-4

x

y

0 6 -2

(3, -1)



CS10_A1_MESK710372_C05.indd 155 3/30/11 11:02:33 PM

16. Let x represent the number of weeks and let y represent the distance run each week.

Angelo: y = x + 7 Marc: y = 2x + 4 Graph y = x + 7 and y = 2x + 4. Running Distance

0 1 3 5 2 4

3

6

9

12

Weeks

Dis

tanc

e ( m

iles)

Marc

Angelo

The lines appear to intersect at (3, 10). So the distance run by Angelo and Marc will be the same in 3 weeks and that distance will be 10 mi.

17a. Let x represent the number of carnations and y represent the amount of money.

Selling: y = 2x Buying: y = 0.50x + 16

⎧

⎨

⎩ y = 2x

y = 0.50x + 16

b. Carnation Sales

0 4 8 12

8

16

24

Carnations

Cost

($) Florist’s price

School band’s price

The solution represents the number of carnations that need to be sold for the band will break even.

c. The solution appears to be at (10.6, 21.3); no; you cannot sell part of a carnation; 11 carnations.

18a. Let h represent the number of hats and let c represent the total cost.

Hats Off: c = 5h + 50 Top Stuff: c = 6h + 25

b. Hat Charges

0 8 16 24

75

150

Hats

Cost

of h

ats

($)

Hats Off

Top Stuff

25 hats; $175

c. fewer than 25 hats: Top Stuff; more than 25 hats: Hats Off

19. (-2.4, -9.3)

20. 4.8x + 0.6y = 4 ___________ -4.8x _____ -4.8x 0.6y = -4.8x + 4

0.6y

____ 0.6

= -4.8x + 4 _________ 0.6

y = -8x + 20 ___ 3

(0.8, 0.1)

21. 8 __ 3 x + y = 5 __

9

________

- 8 __ 3 x

____ - 8 __

3 x

y = - 8 __ 3 x + 5 __

9

(0.3, -0.3)

22. (-1.6, 1.3)

23. Let x represent the number of white hydrangeas in the first bed and let y represent the number of pink hydrangeas in the second bed.

First bed: x + y = 45 Second bed: 2x + 3y = 120 Graph x + y = 45 and 2x + 3y = 120. x + y = 45 ______ -x ___ -x y = -x + 45

2x + 3y = 120 ________ -2x ____ -2x 3y = -2x + 120

3y

___ 3 = -2x + 120 _________

3

y = - 2 __ 3

x + 40

Number of Hydrangeas

0 5 10 15

10 20 30 40 50 60

White Hydrangeas

Pink

Hyd

rang

eas

Second Bed

First Bed (15, 30)

The lines appear to intersect at (15, 30). So 15 white hydrangeas and 30 pink hydrangeas should be planted in the first bed. The gardener should buy 15 + 2(15) = 15 + 30 = 45 white hydrangeas and 30 + 3(30) = 30 + 90 = 120 pink hydrangeas.

24. Morning: y = 5x + 200 Afternoon: y = 11x Graph y = 5x + 200 and y = 11x.

Burning Calories

0 30 10 20

80 160 240 320 400

Time (min)

Calo

ries

(33.3, 366.7)

Morning

Afternoon

The lines appear to intersect at (33.3, 366.7). So, Rusty shoud jog for at least 34 min to burn off more calories in the afternoon.


CS10_A1_MESK710372_C05.indd 156 3/30/11 11:02:36 PM

25. Let x represent the number of years and y represent the height.

First tree: y = x + 2 Second tree: y = 0.5x + 6 Graph y = x + 2 and y = 0.5x + 6

Height of Trees

0 12 4 8 10 2 6

2 4 6 8

10

Years

Hei

ght

( ft)

12 14

(8, 10) Second Tree

First Tree

The lines appear to intersect at (8, 10). So, the trees will be the same height in 8 years.

26. Possible answer: Store A rents carpet cleaners for a fee of $10, plus $3 per day. Store B rents carpet cleaners for a fee of $20, plus $5 per day. For how many days rental will the costs be the same?

27. The coordinates of the intersection point of a system give an ordered pair that satisfies both equations.

TEST PREP

28. B; If x represents the number of miles and y represents the total cost, the cost for Taxi company A is y = 0.50x + 4 and the cost for Taxi company B is y = 0.25x + 5, so B is correct.

29. H; One line has a positive slope and a y-intercept of 1, and the other has a positive slope and a y-intercept of -3. The only system of equations with two such lines is H.

30. Check (2, 6) is on y = 2x + 2. ___________ y = 2x + 2 (6) 2(2) + 2 6 4 + 2 6 6 3

Solve for b. y = 2.5x + b (6) = 2.5(2) + b 6 = 5 + b ___ -5 ______ -5 1 = b

ChALLENGE AND ExTEND

31. Let x represent the number of months after month 1 and let y represent the number sold.

VCRs: y = -10x + 500 DVD Players: y = 15x + 250 Graph y = -10x + 500 and y = 15x + 250.

VCR & DVD Sales

0 12 4 8 10 2 6

100 200 300 400 500

Months

Num

ber

Sold

(10, 400)

VCR

DVD

The solution appears to be at (10, 400). So, the total number sold will be the same for both items in month 10 + 1 = 11 and that will be 400 of each item.

32a. Let x represent the number of minutes and y represent the cost.

Long Distance Inc.: y = 0.03x + 1.45 Far Away Calls: y = 0.02x + 1.52 Graph y = 0.03x + 1.45 and y = 0.02x + 1.45. Cost of Long Distance Calls

0 3 6 9

1.4

1.5

1.6

1.7

Minutes

Cost

($)

Far Away Calls

Long Distance Inc.

(7, 1.66)

The solution appears to be at (7, 1.66). So, the cost for both companies will be the same for 7 min and that cost will be $1.66.

b. It is better to use Long Distance Inc. if the call is under 7 minutes because it costs less. If the call is over 7 minutes, it is better to use Far Away Calls because it costs less.

c. The line will intersect the y-axis at 1.5. Far Away Calls is cheaper because it costs less per minute.

SolvIng SyStEmS by SubStItutIon

CHECK IT OUT!

1a. y = x + 3 2x + 5 = x + 3 ______ -x ______ -x x + 5 = 3 _____ - 5 ___ -5 x = -2

y = x + 3 y = -2 + 3 y = 1 (-2, 1)

b. x + 8y = 16 (2y - 4) + 8y = 16 10y - 4 = 16 ______ + 4 ___ +4 10y = 20

10y

____ 10

= 20 ___ 10

y = 2

x = 2y - 4 x = 2(2) - 4 x = 4 - 4 x = 0 (0, 2)

c. 2x + y = -4 _______ -2x ____ -2x y = -2x - 4

x + y = -7 x + (-2x - 4) = -7 -x - 4 = -7 ______ + 4 ___ +4 -x = -3 -1(-x) = -1(-3) x = 3

x + y = -7 3 + y = -7 ______ -3 ___ -3 y = -10 (3, -10)


5-2

CS10_A1_MESK710372_C05.indd 157 3/30/11 11:02:38 PM

2. -2x + y = 8 _______ +2x ____ +2x y = 2x + 8

3x + 2y = 9 3x + 2(2x + 8) = 9 3x + 2(2x) + 2(8) = 9 3x + 4x + 16 = 9 7x + 16 = 9 _______ - 16 ____ -16 7x = -7

7x ___ 7 = -7 ___

7

x = -1

-2x + y = 8 -2(-1) + y = 8 2 + y = 8 ______ -2 ___ -2 y = 6 (-1, 6)

3a. Let m represent the number of months and let t represent the total cost.

Option 1: t = 60 + 80m Option 2: t = 160 + 70m

t = 60 + 80m 160 + 70m = 60 + 80m _________ - 70m ________ - 70m 160 = 60 + 10m ____ -60 __________ -60 100 = 10m

100 ____ 10

= 10m ____ 10

10 = m

t = 60 + 80m = 60 + 80(10) = 60 + 800 = 860 In 10 months, the total cost for each option will be

the same, $860.

b. The first option; the first option is cheaper for the first 9 months; the second option is cheaper after 10 months.

THINK AND DISCUSS

1. at the point (5, 10)

2. The solution for each is (5, 3).

x + y = 8 x = - y + 8

( - y + 8) - y = 2 - 2 y + 8 = 2

- 2 y = - 6 y = 3

x + 3 = 8 x = 5

(5, 3)

x + y = 8 y = - x + 8

x - ( - x + 8) = 2 x + x - 8 = 2

2 x - 8 = 2 2 x = 10

x = 5

5 + y = 8 y = 3

(5, 3)

x - y = 2 x = y + 2

( y + 2) + y = 8 2 y + 2 = 8

2 y = 6 y = 3

x - 3 = 2 x = 5

(5, 3)

x - y = 2 - y = - x + 2

y = x - 2

x + ( x - 2) = 8 2 x - 2 = 8

2 x = 10 x = 5

5 - y = 2 - y = - 3

y = 3 (5, 3)

Solve x + y = 8 for x. Solve x + y = 8 for y.

Solve x - y = 2 for x. Solve x - y = 2 for y.

x + y = 8x - y = 2


1. y = 5x - 10 3x + 8 = 5x - 10 _______ -3x ________ -3x 8 = 2x - 10 ____ +10 _______ + 10 18 = 2x

18 ___ 2 = 2x ___

2

9 = x

y = 5x - 10 y = 5(9) - 10 y = 45 - 10 y = 35 (9, 35)

2. 3x + y = 2 _______ -3x ____ -3x y = -3x + 2

4x + y = 20 4x + (-3x + 2) = 20 x + 2 = 20 _____ - 2 ___ -2 x = 18

3x + y = 2 3(18) + y = 2 54 + y = 2 ________ -54 ____ -54 y = -52 (18, -52)

3. 4x + y = 20 4x + (x + 5) = 20 5x + 5 = 20 ______ - 5 ___ -5 5x = 15

5x ___ 5 = 15 ___

5

x = 3

y = x + 5 y = 3 + 5 y = 8 (3, 8)


CS10_A1_MESK710372_C05.indd 158 3/30/11 11:02:40 PM

4. x - 2y = 10 _____ + 2y ____ +2y x = 2y + 10

1 __ 2 x - 2y = 4

1 __ 2 (2y + 10) - 2y = 4

1 __ 2 (2y) + 1 __

2 (10) - 2y = 4

y + 5 - 2y = 4 5 - y = 4 ______ -5 ___ -5 -y = -1 -(-y) = -1(-1) y = 1

x - 2y = 10 x - 2(1) = 10 x - 2 = 10 _____ + 2 ___ +2 x = 12 (12, 1)

5. y - 4x = 3 _____ + 4x ____ +4x y = 4x + 3

2x - 3y = 21 2x - 3(4x + 3) = 21 2x - 3(4x) - 3(3) = 21 2x - 12x - 9 = 21 -10x - 9 = 21 ________ + 9 ___ +9 -10x = 30

-10x _____ -10

= 30 ____ -10

x = -3

y - 4x = 3 y - 4(-3) = 3 y + 12 = 3 ______ - 12 ____ -12 y = -9 (-3, -9)

6. -x - y = 0 -(y - 8) - y = 0 -(y) - (-8) - y = 0 -y + 8 - y = 0 8 - 2y = 0 _______ -8 ___ -8 -2y = -8

-2y

____ -2

= -8 ___ -2

y = 4

x = y - 8 x = 4 - 8 x = -4 (-4, 4)


Green Lawn: t = 49 + 29m Grass Team: t = 25 + 37m

t = 49 + 29m 25 + 37m = 49 + 29m ________ - 29m ________ - 29m 25 + 8m = 49 _________ -25 ____ -25 8m = 24

8m ___ 8 = 24 ___

8

m = 3

t = 49 + 29m = 49 + 29(3) = 49 + 87 = 136 In 3 months, the total cost for each company will be

the same, $136.

b. Green Lawn; for 6 months, Green Lawn’s service costs only 49 + 29(6) = 49 + 174 = $223, while Grass Team’s costs 25 + 37(6) = 25 + 222 = $247.


8. y = x + 3 2x + 4 = x + 3 ______ -x ______ -x x + 4 = 3 _____ - 4 ___ -4 x = -1

y = x + 3 y = -1 + 3 y = 2 (-1, 2)

9. y = 2x + 10 -2x - 6 = 2x + 10 _______ +2x ________ +2x -6 = 4x + 10 ____ -10 _______ - 10 -16 = 4x

-16 ____ 4

= 4x ___ 4

-4 = x

y = 2x + 10 y = 2(-4) + 10 y = -8 + 10 y = 2 (-4, 2)

10. x + 2y = 8 _____ - 2y ____ -2y x = -2y + 8

x + 3y = 12 (-2y + 8) + 3y = 12 y + 8 = 12 _____ - 8 ___ -8 y = 4

x + 2y = 8 x + 2(4) = 8 x + 8 = 8 _____ - 8 ___ -8 x = 0 (0, 4)


CS10_A1_MESK710372_C05.indd 159 3/30/11 11:02:41 PM

11. 2x + 2y = 2 ______ - 2y ____ -2y 2x = -2y + 2

2x ___ 2 =

-2y + 2 _______

2

x = -y + 1

-4x + 4y = 12 -4(-y + 1) + 4y = 12 -4(-y) - 4(1) + 4y = 12 4y - 4 + 4y = 12 8y - 4 = 12 ______ + 4 ___ +4 8y = 16

8y

___ 8 = 16 ___

8

y = 2

2x + 2y = 2 2x + 2(2) = 2 2x + 4 = 2 ______ - 4 ___ -4 2x = -2

2x ___ 2

= -2 ___ 2

x = -1 (-1, 2)

12. -y = -2x + 4 -(0.5x + 2) = -2x + 4 -(0.5x) - (2) = -2x + 4 -0.5x - 2 = -2x + 4 _________ +2x _______ +2x 1.5x - 2 = 4 _______ + 2 ___ +2 1.5x = 6

1.5x ____ 1.5

= 6 ___ 1.5

x = 4

y = 0.5x + 2 y = 0.5(4) + 2 y = 2 + 2 y = 4 (4, 4)

13. -x + y = 4 ______ +x ___ +x y = x + 4

3x - 2y = -7 3x - 2(x + 4) = -7 3x - 2(x) - 2(4) = -7 3x - 2x - 8 = -7 x - 8 = -7 _____ + 8 ___ +8 x = 1

-x + y = 4 -(1) + y = 4 -1 + y = 4 ______ +1 ___ +1 y = 5 (1, 5)

14. 3x + y = -8 _______ -3x ____ -3x y = -3x - 8

-2x - y = 6 -2x - (-3x - 8) = 6 -2x - (-3x) - (-8) = 6 -2x + 3x + 8 = 6 x + 8 = 6 _____ - 8 ___ -8 x = -2

3x + y = -8 3(-2) + y = -8 -6 + y = -8 ______ +6 ___ +6 y = -2 (-2, -2)

15. x + 2y = -1 _____ - 2y ____ -2y x = -2y - 1

4x - 4y = 20 4(-2y - 1) - 4y = 20 4(-2y) + 4(-1) - 4y = 20 -8y - 4 - 4y = 20 -12y - 4 = 20 ________ + 4 ___ +4 -12y = 24

-12y

_____ -12

= 24 ____ -12

y = -2

x + 2y = -1 x + 2(-2) = -1 x - 4 = -1 _____ + 4 ___ +4 x = 3 (3, -2)

16. 4x = y - 1 ___ +1 _____ + 1 4x + 1 = y

6x - 2y = -3 6x - 2(4x + 1) = -3 6x - 2(4x) - 2(1) = -3 6x - 8x - 2 = -3 -2x - 2 = -3 _______ + 2 ___ +2 -2x = -1

-2x ____ -2

= -1 ___ -2

x = 1 __ 2

4x = y - 1

4 ( 1 __ 2 ) = y - 1

2 = y - 1 ___ +1 _____ + 1 3 = y

( 1 __ 2 , 3)


CS10_A1_MESK710372_C05.indd 160 3/30/11 11:02:43 PM


Option 1: t = 150 + 35m Option 2: t = 60m

t = 150 + 35m 60m = 150 + 35m _____ -35m _________ - 35m 25m = 150

25m ____ 25

= 150 ____ 25

m = 6

t = 60m = 60(6) = 360 In 6 months, the total cost for each option will be

the same, $360.

b. the second option; for 5 months, it will cost only 60(5) = $300, while the other option costs

150 + 35(5) = 150 + 175 = $325.

18. x = 5

x + y = 8 5 + y = 8 ______ -5 ___ -5 y = 3 (5, 3)

19. x = 2y + 6 x = 2(-3x + 4) + 6 x = 2(-3x) + 2(4) + 6 x = -6x + 8 + 6 x = -6x + 14 ____ +6x ________ +6x 7x = 14

7x ___ 7 = 14 ___

7

x = 2

y = -3x + 4 y = -3(2) + 4 y = -6 + 4 y = -2 (2, -2)

20. 3x - y = 11 _______ -3x ____ -3x -y = -3x + 11 -1(-y) = -1(-3x + 11) y = -1(-3x) - 1(11) y = 3x - 11

5y - 7x = 1 5(3x - 11) - 7x = 1 5(3x) + 5(-11) - 7x = 1 15x - 55 - 7x = 1 8x - 55 = 1 _______ + 55 ____ +55 8x = 56

8x ___ 8 = 56 ___

8

x = 7

3x - y = 11 3(7) - y = 11 21 - y = 11 _______ -21 ____ -21 -y = -10 -1(-y) = -1(-10) y = 10 (7, 10)

21. x - y = 2 ____ + y ___ +y x = y + 2

1 __ 2 x + 1 __

3 y = 6

1 __ 2 (y + 2) + 1 __

3 y = 6

1 __ 2 (y) + 1 __

2 (2) + 1 __

3 y = 6

1 __ 2 y + 1 + 1 __

3 y = 6

5 __ 6 y + 1 = 6

______ - 1 ___ -1

5 __ 6 y = 5

6 __ 5 ( 5 __

6 y) = 6 __

5 (5)

y = 6

x - y = 2 x - 6 = 2 ____ + 6 ___ +6 x = 8 (8, 6)


CS10_A1_MESK710372_C05.indd 161 3/30/11 11:02:44 PM

22. 2x + y = 5 2(7 - 2y) + y = 5 2(7) + 2(-2y) + y = 5 14 - 4y + y = 5 14 - 3y = 5 ________ -14 ____ -14 -3y = -9

-3y

____ -3

= -9 ___ -3

y = 3

x = 7 - 2y x = 7 - 2(3) x = 7 - 6 x = 1 (1, 3)

23. 2.2x + 5 = y 2.2x + 5 = 1.2x - 4 _________ -1.2x _________ -1.2x x + 5 = -4 _____ - 5 ___ -5 x = -9

y = 1.2x - 4 y = 1.2(-9) - 4 y = -10.8 - 4 y = -14.8 (-9, -14.8)

24. Let x represent the first number and let y represent the second number.

Sum is 50: x + y = 50 First is 43 less than twice second: x = 2y - 43

⎧

⎨

⎩ x + y = 50

x = 2y - 43

x + y = 50 (2y - 43) + y = 50 3y - 43 = 50 _______ + 43 ____ +43 3y = 93

3y

___ 3 = 93 ___

3

y = 31

x = 2y - 43 x = 2(31) - 43 x = 62 - 43 x = 19 The two numbers are 19 and 31.

25. 20 coins: n + d = 20 Value $1.40: 0.05n + 0.10d = 1.40

n + d = 20 _______ -n ___ -n d = -n + 20

0.05n + 0.10d = 1.40 0.05n + 0.10(-n + 20) = 1.40 0.05n + 0.10(-n) + 0.10(20) = 1.40 0.05n - 0.10n + 2.00 = 1.40 -0.05n + 2.00 = 1.40 ____________ - 2.00 _____ -2.00 -0.05n = -0.60

-0.05n _______ -0.05

= -0.60 ______ -0.05

n = 12

n + d = 20 12 + d = 20 _______ -12 ____ -12 d = 8 There are 12 nickels and 8 dimes in the jar.

26. Let p represent the price of the popcorn and d represent the price of the drinks.

Customer #3598: 3p + 2d = 21 Customer #3599: 2p + 4d = 22

⎧

⎨

⎩ 3p + 2d = 21

2p + 4d = 22

2p + 4d = 22 _______ - 4d ____ -4d 2p = -4d + 22

2p

___ 2 = -4d + 22 _________

2

p = -2d + 11

3p + 2d = 21 3(-2d + 11) + 2d = 21 3(-2d) + 3(11) + 2d = 21 -6d + 33 + 2d = 21 33 - 4d = 21 ________ -33 ____ -33 -4d = -12

-4d ____ -4

= -12 ____ -4

d = 3

3p + 2d = 21 3p + 2(3) = 21 3p + 6 = 21 ______ - 6 ___ -6 3p = 15

3p

___ 3 = 15 ___

3

p = 5 The price of a large bag of popcorn is $5 and the

price of a small drink is $3.


CS10_A1_MESK710372_C05.indd 162 3/30/11 11:02:45 PM

27. Let x represent the amount in the 5% account and let y represent the amount in the 6% account.

Total of $1000: x + y = 1000 $58 in interest: 0.05x + 0.06y = 58

⎧

⎨

⎩ x + y = 1000

0.05x + 0.06y = 58

x + y = 1000 ____ - y _____ -y x = -y + 1000

0.05x + 0.06y = 58 0.05(-y + 1000) + 0.06y = 58 0.05(-y) + 0.05(1000) + 0.06y = 58 -0.05y + 50 + 0.06y = 58 50 + 0.01y = 58 ___________ -50 ____ -50 0.01y = 8

0.01y

_____ 0.01

= 8 ____ 0.01

y = 800

x + y = 1000 x + 800 = 1000 ______ - 800 _____ -800 x = 200 Helen invested $200 in the 5% account and $800 in

the 6% account.

28. x + y = 90 x + (4x - 10) = 90 5x - 10 = 90 _______ + 10 ____ +10 5x = 100

5x ___ 5 = 100 ____

5

m∠x = 20°

y = 4x - 10 y = 4(20) - 10 y = 80 - 10 m∠y = 70°

29. x + y = 90 2y + y = 90 3y = 90

3y

___ 3 = 90 ___

3

m∠y = 30°

x = 2y x = 2(30) m∠x = 60°

m∠y = 30°

30. x + y = 90 x + 2(x - 15) = 90 x + 2(x) + 2(-15) = 90 x + 2x - 30 = 90 3x - 30 = 90 _______ + 30 ____ +30 3x = 120

3x ___ 3 = 120 ____

3

m∠x = 40°

y = 2(x - 15) y = 2(40 - 15) y = 2(25) m∠y = 50°

31a. rate · Time = Distance

With Headwind p - w · 3 = 240

With Tailwind p + w · 2 = 240

b. (p - w)3 = 240 (p)3 + (-w)3 = 240 3p - 3w = 240

(p + w)2 = 240 (p)2 + (w)2 = 240 2p + 2w = 240

⎧

⎨

⎩ 3p - 3w = 240

2p + 2w = 240

c. 3p - 3w = 240 _______ + 3w ____ +3w 3p = 3w + 240

3p

___ 3 = 3w + 240 ________

3

p = w + 80

2p + 2w = 240 2(w + 80) + 2w = 240 2(w) + 2(80) + 2w = 240 2w + 160 + 2w = 240 4w + 160 = 240 ________ - 160 -160 4w = 80

4w ___ 4 = 80 ___

4

w = 20

2p + 2w = 240 2p + 2(20) = 240 2p + 40 = 240 _______ - 40 ____ -40 2p = 200

2p

___ 2 = 200 ____

2

p = 100 The speed of the plane is 100 mi/h and the speed

of the wind is 20 mi/h.

32. Possible answer: Solve one of the equations for either x or y. Then substitute the expression equal to x or y into the other equation. This creates a one-variable equation that can be solved. When you get the value of one variable, substitute it into one of the original equations and solve to find the value of the other variable.

33. The solution of a system solved by graphing is the same as the solution of a system solved by substitution.

34a. Let x represent the cost of a book and let y represent the cost of a backpack.

Total $26: 2x + y = 26 Book costs $8 less than backpack: x = y - 8

⎧

⎨

⎩ 2x + y = 26

x = y - 8


CS10_A1_MESK710372_C05.indd 163 3/30/11 11:02:47 PM

b. 2x + y = 26 2(y - 8) + y = 26 2(y) + 2(-8) + y = 26 2y - 16 + y = 26 3y - 16 = 26 _______ + 16 ____ +16 3y = 42

3y

___ 3 = 42 ___

3

y = 14

x = y - 8 x = 14 - 8 x = 6 book: $6; backpack: $14

c. 2x + y = 26 _______ -2x ____ -2x y = -2x + 26

x = y - 8 ___ + 8 _____ + 8 x + 8 = y

Juanita’s Purchase

0 4 8 12 14 2 6 10

8

16

24

32

4

12

20

28

Cost of Book ($)

Cost

of B

ackp

ack

($)

(6, 14)

The solution appears to be at (6, 14). Possible answer: Substitution works well since x

is already isolated. Graphing requires solving both equations for y.

35. Possible estimate: (1.75, -2.5) x + y = -0.6 ______ -x _____ -x y = -x - 0.6

2x - y = 6 2x - (-x - 0.6) = 6 2x - (-x) - (-0.6) = 6 2x + x + 0.6 = 6 3x + 0.6 = 6 _______ - 0.6 ____ -0.6 3x = 5.4

3x ___ 3 = 5.4 ___

3

x = 1.8

x + y = -0.6 1.8 + y = -0.6 ________ -1.8 ____ -1.8 y = -2.4 (1.8, -2.4)

TEST PREP

36. D; Since the total number of cousins is 24, m + f = 24 or f = 24 - m. Since the number of males was 6 less than twice the number of females, m = 2f - 6, so D is correct.

37. F; If d is the number of dimes and n is the number of nickels, then d + n represents the number of coins, so Roger has 12 coins. Since d is 5 greater than n, there are 5 more dimes than nickels, so F is correct.


38. Let n represent the number of new cars and let u represent the number of used cars.

Total of 378 cars: n + u = 378

Ratio of new to used is 5:4: 5 __ 4 = n __ u

4n = 5u

⎧

⎨ ⎩ n + u = 378

4n = 5u

n + u = 378 _______ -n ___ -n u = -n + 378

4n = 5u 4n = 5(-n + 378) 4n = 5(-n) + 5(378) 4n = -5n + 1890 ____ +5n __________ +5n 9n = 1890

9n ___ 9 = 1890 _____

9

n = 210

n + u = 378 210 + u = 378 _________ -210 _____ -210 u = 168 The car dealership has 210 new cars and 168 used

cars.

39. t = 4

s + 3t = 10 s + 3(4) = 10 s + 12 = 10 ______ - 12 ____ -12 s = -2

2r - 3s - t = 12 2r - 3(-2) - 4 = 12 2r + 6 - 4 = 12 2r + 2 = 12 _____ - 2 ___ -2 2r = 10

2r __ 2 = 10 ___

2

r = 5

r = 5; s = -2; t = 4


CS10_A1_MESK710372_C05.indd 164 3/30/11 11:02:49 PM

40. y + z = 5 ______ -y ___ -y z = -y + 5

2y - 4z = -14 2y - 4(-y + 5) = -14 2y - 4(-y) - 4(5) = -14 2y + 4y - 20 = -14 6y - 20 = -14 _______ + 20 ____ +20 6y = 6

6y

___ 6 = 6 __

6

y = 1

y + z = 5 1 + z = 5 ______ -1 ___ -1 z = 4

x + y + z = 7 x + 1 + 4 = 7 x + 5 = 7 _____ - 5 ___ -5 x = 2

x = 2; y = 1; z = 4

41. -b + c = -5 ______ +b ___ +b c = b - 5

3b + 2c = 15 3b + 2(b - 5) = 15 3b + 2(b) + 2(-5) = 15 3b + 2b - 10 = 15 5b - 10 = 15 _______ + 10 ____ +10 5b = 25

5b ___ 5 = 25 ___

5

b = 5

-b + c = -5 -5 + c = -5 ______ +5 ___ +5 c = 0

a + 2b + c = 19 a + 2(5) + 0 = 19 a + 10 = 19 ______ - 10 ____ -10 a = 9

a = 9; b = 5; c = 0

SolvIng SyStEmS by ElImInAtIon

CHECK IT OUT!

1. y + 3x = -2 _____________ + 2y - 3x = 14 3y + 0 = 12 3y = 12

3y

___ 3 = 12 ___

3

y = 4

y + 3x = -2 4 + 3x = -2 _______ -4 ___ -4 3x = -6

3x ___ 3 = -6 ___

3

x = -2 (-2, 4)

2. 3x + 3y = 15 ________________ - (-2x + 3y = -5)

3x + 3y = 15 ____________ + 2x - 3y = 5 5x + 0 = 20 5x = 20

5x ___ 5

= 20 ___ 5

x = 4

3x + 3y = 15 3(4) + 3y = 15 12 + 3y = 15 ________ -12 ____ -12 3y = 3

3y

___ 3

= 3 __ 3

y = 1 (4, 1)

3a. 3x + 2y = 6 _______________ + 3(-x + y = -2)

3x + 2y = 6 ________________ + (-3x + 3y = -6) 0 + 5y = 0 5y = 0

5y

___ 5 = 0 __

5

y = 0

-x + y = -2 -x + 0 = -2 -x = -2 -1(-x) = -1(-2) x = 2 (2, 0)

b. 3(2x + 5y = 26) __________________ + 2(-3x - 4y = -25)

6x + 15y = 78 ________________ + (-6x - 8y = -50) 0 + 7y = 28 7y = 28

7y

___ 7

= 28 ___ 7

y = 4

2x + 5y = 26 2x + 5(4) = 26 2x + 20 = 26 _______ - 20 ____ -20 2x = 6

2x ___ 2

= 6 __ 2

x = 3 (3, 4)

4. Let ℓ represent the number of lilies and let t represent the number of tulips.

Total $14.85: 1.25ℓ + 0.90t = 14.85 13 flowers: ℓ + t = 13

1.25ℓ + 0.90t = 14.85 ____________________ + (-1.25)(ℓ + t = 13)

1.25ℓ + 0.90t = 14.85 ________________________ + (-1.25ℓ - 1.25t = -16.25) 0 - 0.35t = -1.40 -0.35t = -1.40

-0.35t ______ -0.35

= -1.40 ______ -0.35

t = 4

ℓ + t = 13 ℓ + 4 = 13 ____ - 4 ___ -4 ℓ = 9 Sally bought 9 lilies and 4 tulips.


5-3

CS10_A1_MESK710372_C05.indd 165 3/30/11 11:02:51 PM

THINK AND DISCUSS

1. Multiplying the second equation by -1 and adding is like adding the opposite, which is the same as subtracting; possible answer: 3x + y = 5; 3x + 2y = 6

2. No matter which variable you solve for first, the answer will be the same.

3.

Substitution: Elimination using addition or subtraction:

Elimination using multiplication:

Solving Systems of Linear Equations

x = 5 y x + y = 10

4 x + y = 1 2 x - y = 7

2 x + 6 y = 10 x + y = - 2


1. -x + y = 5 _____________ + x - 5y = -9 0 - 4y = -4 -4y = -4

-4y

____ -4

= -4 ___ -4

y = 1

x - 5y = -9 x - 5(1) = -9 x - 5 = -9 _____ + 5 ___ +5 x = -4 (-4, 1)

2. x + y = 12 __________ + x - y = 2 2x + 0 = 14 2x = 14

2x ___ 2 = 14 ___

2

x = 7

x + y = 12 7 + y = 12 ______ -7 ___ -7 y = 5 (7, 5)

3. 2x + 5y = -24 _____________ + 3x - 5y = 14 5x + 0 = -10 5x = -10

5x ___ 5 = -10 ____

5

x = -2

2x + 5y = -24 2(-2) + 5y = -24 -4 + 5y = -24 _______ +4 ____ +4 5y = -20

5y

___ 5 = -20 ____

5

y = -4 (-2, -4)

4. x - 10y = 60 ______________ - (x + 14y = 12)

x - 10y = 60 _________________ + (-x - 14y = -12) 0 - 24y = 48 -24y = 48

-24y

_____ -24

= 48 ____ -24

y = -2

x - 10y = 60 x - 10(-2) = 60 x + 20 = 60 ______ - 20 ____ -20 x = 40 (40, -2)

5. 5x + y = 0 ______________ - (5x + 2y = 30)

5x + y = 0 _________________ + (-5x - 2y = -30) 0 - y = -30 -y = -30 -1(-y) = -1(-30) y = 30

5x + y = 0 5x + 30 = 0 ______ - 30 ____ -30 5x = -30

5x ___ 5 = -30 ____

5

x = -6 (-6, 30)

6. -5x + 7y = 11 ________________ - (-5x + 3y = 19)

-5x + 7y = 11 ______________ + 5x - 3y = -19 0 + 4y = -8 4y = -8

4y

___ 4 = -8 ___

4

y = -2

-5x + 3y = 19 -5x + 3(-2) = 19 -5x - 6 = 19 _______ + 6 ___ +6 -5x = 25

-5x ____ -5

= 25 ___ -5

x = -5 (-5, -2)

7. 2x + 3y = 12 ______________ + 3(5x - y = 13)

2x + 3y = 12 ______________ + 15x - 3y = 39 17x + 0 = 51 17x = 51

17x ____ 17

= 51 ___ 17

x = 3

2x + 3y = 12 2(3) + 3y = 12 6 + 3y = 12 _______ -6 ___ -6 3y = 6

3y

___ 3 = 6 __

3

y = 2 (3, 2)

8. -3x + 4y = 12 _________________ + (-4)(2x + y = -8)

-3x + 4y = 12 ________________ + (-8x - 4y = 32) -11x + 0 = 44 -11x = 44

-11x _____ -11

= 44 ____ -11

x = -4

2x + y = -8 2(-4) + y = -8 -8 + y = -8 ______ +8 ___ +8 y = 0 (-4, 0)

9. 3(2x + 4y = -4) __________________ + (-2)(3x + 5y = -3)

6x + 12y = -12 _______________ + (-6x - 10y = 6) 0 + 2y = -6 2y = -6

2y

___ 2

= -6 ___ 2

y = -3

2x + 4y = -4 2x + 4(-3) = -4 2x - 12 = -4 _______ + 12 ____ +12 2x = 8

2x ___ 2 = 8 __

2

x = 4 (4, -3)


CS10_A1_MESK710372_C05.indd 166 3/30/11 11:02:53 PM

10. Let x represent the number of hamburger bun packages and let y represent the number of hot-dog bun packages.

$20 of buns: 2.00x + 1.50y = 20 12 packages: x + y = 12

2.00x + 1.50y = 20 __________________ + (-2.00)(x + y = 12)

2.00x + 1.50y = 20 ______________________ + (-2.00x - 2.00y = -24) 0 - 0.50y = -4 -0.50y = -4

-0.50y

______ -0.50

= -4 ______ -0.50

y = 8

x + y = 12 x + 8 = 12 ____ - 8 ___ -8 x = 4 The Harlin family bought 4 packages of hamburger

buns and 8 packages of hot dog buns.


11. -x + y = -1 ___________ + 2x - y = 0 x + 0 = -1 x = -1

-x + y = -1 -(-1) + y = -1 1 + y = -1 ______ -1 ___ -1 y = -2 (-1, -2)

12. -2x + y = -20 ____________ + 2x + y = 48 0 + 2y = 28 2y = 28

2y

___ 2 = 28 ___

2

y = 14

2x + y = 48 2x + 14 = 48 ______ - 14 ____ -14 2x = 34

2x ___ 2 = 34 ___

2

x = 17 (17, 14)

13. 3x - y = -2 _____________ + (-2x + y = 3) x + 0 = 1 x = 1

-2x + y = 3 -2(1) + y = 3 -2 + y = 3 ______ +2 ___ +2 y = 5 (1, 5)

14. x - y = 4 ______________ -(x - 2y) = -10

x - y = 4 ________________ + (-x + 2y = -10) 0 + y = -6 y = -6

x - y = 4 x - (-6) = 4 x + 6 = 4 _____ - 6 ___ -6 x = -2 (-2, -6)

15. x + 2y = 5 _______________ -(3x + 2y) = -17

x + 2y = 5 _________________ + (-3x - 2y = -17) -2x + 0 = -12 -2x = -12

-2x ____ -2

= -12 ____ -2

x = 6

x + 2y = 5 6 + 2y = 5 _______ -6 ___ -6 2y = -1

2y

___ 2 = -1 ___

2

y = - 1 __ 2

(6, - 1 __ 2 )

16. 3x - 2y = -1 ______________ -(3x - 4y) = -9

3x - 2y = -1 ________________ + (-3x + 4y = -9) 0 + 2y = -10 2y = -10

2y

___ 2

= -10 ____ 2

y = -5

3x - 2y = -1 3x - 2(-5) = -1 3x + 10 = -1 _______ - 10 ____ -10 3x = -11

3x ___ 3

= -11 ____ 3

x = - 11 ___ 3

(- 11 ___ 3

, -5)

17. 3(x - y = -3) ____________ + 5x + 3y = 1

3x - 3y = -9 ____________ + 5x + 3y = 1 8x + 0 = -8 8x = -8

8x ___ 8 = -8 ___

8

x = -1

5x + 3y = 1 5(-1) + 3y = 1 -5 + 3y = 1 _______ +5 ___ +5 3y = 6

3y

___ 3 = 6 __

3

y = 2 (-1, 2)

18. 9x - 3y = 3 ___________________ + (-3)(3x + 8y = -17)

9x - 3y = 3 ________________ + (-9x - 24y = 51) 0 - 27y = 54 -27y = 54

-27y

_____ -27

= 54 ____ -27

y = -2

3x + 8y = -17 3x + 8(-2) = -17 3x - 16 = -17 _______ + 16 ____ +16 3x = -1

3x ___ 3 = -1 ___

3

x = - 1 __ 3

(- 1 __ 3 , -2)


CS10_A1_MESK710372_C05.indd 167 3/30/11 11:02:55 PM

19. 3(5x + 2y = -1) _________________ + (-5)(3x + 7y = 11)

15x + 6y = -3 ___________________ + (-15x - 35y = -55) 0 - 29y = -58 -29y = -58

-29y

_____ -29

= -58 ____ -29

y = 2

3x + 7y = 11 3x + 7(2) = 11 3x + 14 = 11 _______ - 14 ____ -14 3x = -3

3x ___ 3 = -3 ___

3

x = -1 (-1, 2)

20. Let c represent the number of candles and let v represent the number of vases.

Spent $31.50: 3.00c + 4.25v = 31.50 8 tables: c + v = 8

3.00c + 4.25v = 31.50 ___________________ + (-3.00)(c + v = 8)

3.00c + 4.25v = 31.50 ________________________ + (-3.00c - 3.00v = -24) 0 + 1.25v = 7.50 1.25v = 7.50

1.25v _____ 1.25

= 7.50 ____ 1.25

v = 6

c + v = 8 c + 6 = 8 ____ - 6 ___ -6 c = 2 Mrs. Gonzalez bought 2 candles and 6 vases.

21. Difference is 2: ℓ - w = 2 Perimeter is 40: 2ℓ + 2w = 40

⎧

⎨ ⎩ ℓ - w = 2

2ℓ + 2w = 40

2(ℓ - w = 2) _____________ + 2ℓ + 2w = 40

2ℓ - 2w = 4 _____________ + 2ℓ + 2w = 40 4ℓ + 0 = 44 4ℓ = 44

4ℓ ___ 4 = 44 ___

4

ℓ = 11

2ℓ + 2w = 40 2(11) + 2w = 40 22 + 2w = 40 _________ -22 ____ -22 2w = 18

2w ___ 2 = 18 ___

2

w = 9 The length of the rectangle is 11 units and the width

of the rectangle is 9 units.

22. A; the student did not distribute the negative sign to all values in the parentheses and so mistakenly added 3 to -3.

23a. 1% Solution

+5%

Solution=

4 % Solution

Amount of Solution (mL)

x + y = 100

Amount of Acid (mL)

0.01x + 0.05y = 0.04(100)

b. 0.04(100) = 4

⎧

⎨ ⎩ x + y = 100

0.01x + 0.05y = 4

c. 0.01x + 0.05y = 4 ___________________ + (-0.01)(x + y = 100)

0.01x + 0.05y = 4 _____________________ + (-0.01x - 0.01y = -1) 0 + 0.04y = 3 0.04y = 3

0.04y

_____ 0.04

= 3 ____ 0.04

y = 75

x + y = 100 x + 75 = 100 _____ - 75 ____ -75 x = 25 The chemist should use 25 mL of the 1% solution

and 75 mL of the 5% solution.

24. (40, -2); Possible answer: elimination using subtraction; the coefficient of the x-values are the same.

25. (3, 3); Possible answer: elimination using multiplication; none of the variables have the same coefficient so they cannot be eliminated by addition or subtraction.

26. (3, -1); Possible answer: elimination using subtraction; the y-terms have the same coefficients.

27. ( 46 ___ 7 , 8 __

7 ) ; Possible answer: substitution; the second

equation is already solved for a variable.

28. (- 1 __ 2 , 2) ; Possible answer: graphing; both equations

are in slope-intercept form.

29. ( 15 ___ 7 , 9 __

7 ) ; Possible answer: elimination using

multiplication; when the first equation is multiplied by -2 the x-coefficients are opposites.


CS10_A1_MESK710372_C05.indd 168 3/30/11 11:02:57 PM

30. Since they did not sell any of the expensive mulch, they did not sell any cocoa.

Let h represent the number of bags of hardwood mulch and let p represent the number of bags of pine bark.

Sold 176 bags: h + p = 176 Total sales of $520: 3.50h + 2.75p = 520

3.50h + 2.75p = 520 ___________________ + (-3.50)(h + p = 176)

3.50h + 2.75p = 520 _______________________ + (-3.50h - 3.50p = -616) 0 - 0.75p = -96 -0.75p = -96

-0.75p

_______ -0.75

= -96 ______ -0.75

p = 128

h + p = 176 h + 128 = 176 _______ - 128 _____ -128 h = 48 They sold 48 bags of hardwood mulch and 128 bags

of pine bark mulch.

31a. Let A represent the price of an item on shelf A and let B represent the price of an item on shelf B.

Option 1: 3A + 2B = 16 Option 2: 2A + 3B = 14

⎧

⎨ ⎩ 3A + 2B = 16

2A + 3B = 14

b. 2(3A + 2B = 16) ___________________ + (-3)(2A + 3B = 14)

6A + 4B = 32 _________________ + (-6A - 9B = -42) 0 - 5B = -10 -5B = -10

-5B ____ -5

= -10 ____ -5

B = 2

2A + 3B = 14 2A + 3(2) = 14 2A + 6 = 14 ______ - 6 ___ -6 2A = 8

2A ___ 2 = 8 __

2

A = 4

A = 4; B = 2

c. Buying the first package will save 3(6 - 4) + 2(3 - 2) = 6 + 2 = $8; buying the

second package will save 2(6 - 4) + 3(3 - 2) = 4 + 3 = $7.

32. -4(3x + y = 1) _______________ + 2x + 4y = -6

-12x - 4y = -4 ______________ + 2x + 4y = -6 -10x + 0 = -10 -10x = -10

-10x _____ -10

= -10 ____ -10

x = 1

3x + y = 1 3(1) + y = 1 3 + y = 1 ______ -3 ___ -3 y = -2

(1, -2); possible answer: to check the solution, algebraically, substitute 1 for x and -2 for y in both equations. If both equations are true statements, the solution is correct. To check the solution graphically, graph both equations on the same coordinate plane and locate their intersection. If the intersection is (1, -2) then the solution is correct.

TEST PREP

33. A; Let x represent the number of 2-point problems and y represent the number of 3-point problems. So, since the total number of problems is 25, x + y = 25. Since the total number of points is 60, 2x + 3y = 60.

34. G; Notice that 15 + 11 (1 1 __ 4 ) = 15 + 13.75 = 28.75

and 10 + 15 (1 1 __ 4 ) = 10 + 18.75 = 28.75, so G is

correct.

35a. Let s represent the number of student tickets sold and let n represent the number of nonstudent tickets sold.

358 tickets sold: s + n = 358 School made $752.25: 1.50s + 3.25n = 752.25

⎧

⎨ ⎩ s + n = 358

1.50s + 3.25n = 752.25

b. 1.50s + 3.25n = 752.25 ______________________ + (-1.50)(s + n = 358)

1.50s + 3.25n = 752.25 _________________________ + (-1.50s - 1.50n = -537) 0 + 1.75n = 215.25 1.75n = 215.25

1.75n _____ 1.75

= 215.25 ______ 1.75

n = 123

s + n = 358 s + 123 = 358 ______ - 123 _____ -123 s = 235 s = 235; n = 123; 235 student tickets were sold

and 123 nonstudent tickets were sold.


CS10_A1_MESK710372_C05.indd 169 3/30/11 11:02:59 PM


36. x + 16 1 __ 2 = - 3 __

4 y

x + 16 1 __ 2 = - 3 __

4 ( 1 __

2 x)

x + 33 ___ 2 = - 3 __

8 x

________ -x ____ -x

33 ___ 2 = - 11 ___

8 x

- 8 ___ 11

( 33 ___ 2 ) = - 8 ___

11 (- 11 ___

8 x)

-12 = x

y = 1 __ 2 x

y = 1 __ 2 (-12)

y = -6

x = -12; y = -6

37. 1 __ 2 z = 5

2 ( 1 __ 2 z) = 2(5)

z = 10

2x + y + z = 17 2x + y + 10 = 17 __________ - 10 ____ -10 2x + y = 7

2x + y = 7 __________ + x - y = 5 3x + 0 = 12 3x = 12

3x ___ 3 = 12 ___

3

x = 4

x - y = 5 4 - y = 5 ______ -4 ___ -4 -y = 1 -1(-y) = -1(1) y = -1

x = 4; y = -1; z = 10

38. x - 2y - z = -1 _____________________ + (-x + 2y + 4z = -11) 0 + 0 + 3z = -12 3z = -12

3z ___ 3 = -12 ____

3

z = -4

x - 2y - z = -1 x - 2y - (-4) = -1 x - 2y + 4 = -1 _________ - 4 ___ -4 x - 2y = -5

2x + y + z = 1 2x + y + (-4) = 1 2x + y - 4 = 1 _________ + 4 ___ +4 2x + y = 5

x - 2y = -5 ____________ + 2(2x + y = 5)

x - 2y = -5 _____________ + 4x + 2y = 10 5x + 0 = 5 5x = 5

5x ___ 5 = 5 __

5

x = 1

2x + y + z = 1 2(1) + y + (-4) = 1 2 + y - 4 = 1 y - 2 = 1 _____ + 2 ___ +2 y = 3

x = 1; y = 3; z = -4

39. Let x represent the number in the tens’ column and let y represent the number in the ones’ column.

Sum is 5: x + y = 5 The number is 10x + y. Result of multiplying by 3: 3(10x + y) = 42

⎧

⎨ ⎩ x + y = 5

3(10x + y) = 42

-3(x + y = 5) _______________ + 3(10x + y) = 42

-3x - 3y = -15 ______________ + 30x + 3y = 42 27x + 0 = 27 27x = 27

27x ____ 27

= 27 ___ 27

x = 1

x + y = 5 1 + y = 5 ______ -1 ___ -1 y = 4 The number is 10(1) + 4 = 14.

SolvIng SpEcIAl SyStEmS

CHECK IT OUT!

1. Possible answer: Substitute -2x + 5 for y in the second equation. 2x + (-2x + 5) = 1 2x - 2x + 5 = 1 0 + 5 = 07

2. Possible answer: Substitute x - 3 for y in the second equation. -x(x - 3) - 3 = 0 x - x + 3 - 3 = 0 0 + 0 = 0 0 = 03

3a. x + 2y = -4 → y = - 1 __ 2 x - 2

-2(y + 2) = x → y = - 1 __ 2 x - 2

This system is consistent and dependent. It has infinitely many solutions.

b. y = -2(x - 1) → y = -2x + 2 y = -x + 3 → y = -x + 3 This system is consistent and independent. It has

one solution.

c. 2x - 3y = 6 → y = 2 __ 3 x - 2

y = 2 __ 3 x → y = 2 __

3 x

This system is inconsistent. It has no solution.

4. Yes. Let x represent the number of months and let y represent the total amount in the account.

Matt: y = 20x + 100 Ben: y = 30x + 80 The graphs of the two equations have different

slopes so they intersect. If the pattern continues, the amount of money in the two accounts will eventually be equal.


5-4

CS10_A1_MESK710372_C05.indd 170 3/30/11 11:03:01 PM

THINK AND DISCUSS

1. solving algebraically by substitution or elimination, comparing slopes and y-intercepts, or graphing

2.

Exactly one solution: independent;

possible graph:

Infinitely many solutions: dependent;

possible graph:

No solution: inconsistent;

possible graph:

Linear System of Equations

x

y

x

y

x

y


1. consistent

2. Possible answer: Substitute x + 1 for y in the second equation.-x + (x + 1) = 3 -x + x + 1 = 3 1 = 37

3. Possible answer: Substitute -3x + 2 for y in the first equation. 3x + (-3x + 2) = 6 3x - 3x + 2 = 6 2 = 67

4. Possible answer: Compare slopes and intercepts.-y = 4x + 1 → y = -4x - 14x + y = 2 → y = -4x - 2The lines have the same slope and different y-intercepts. Therefore the lines are parallel.

5. Possible answer: Substitute -x + 3 for y in the second equation. x + (-x + 3) - 3 = 0 x - x + 3 - 3 = 0 0 = 03

6. Possible answer: Substitute -2x - 4 for y in the second equation.2x - (2x - 4) - 4 = 0 2x - 2x + 4 - 4 = 0 0 = 03

7. Possible answer: Add the two equations. -7x + y = -2 ___________ + 7x - y = 2 0 = 03

8. y = 2(x + 3) → y = 2x + 6 -2y = 2x + 6 → y = -x - 3 This system is consistent and independent. It has

one solution.

9. y = -3x - 1 → y = -3x - 1 3x + y = 1 → y = -3x + 1 This system is inconsistent. It has no solution.

10. 9y = 3x + 18 → y = 1 __ 3 x + 2

1 __ 3 x - y = -2 → y = 1 __

3 x + 2


11. Yes. Let x represent the number of hours since Luke started to run and let y represent the number of miles.

Micah: y = 4x + 2 Luke: y = 6x The graphs of the two equations have different

slopes so they intersect. If their rates continue, Micah’s and Luke’s distance will eventually be equal.


12. Possible answer: Substitute 2x - 2 for y in the second equation.-2x + (2x - 2) = 1 -2x + 2x - 2 = 1 -2 = 17

13. Possible answer: Substitute -x - 1 for y in the first equation.x + (-x - 1) = 3 x - x - 1 = 3 -1 = 37

14. Possible answer: Substitute - 1 __ 2

x - 4 for y in the first equation.

x + 2(- 1 __ 2 x - 4) = -4

x - x - 8 = -4 -8 = -47

15. Possible answer: Compare slopes and intercepts.-6 + y = 2x → y = 2x + 6 → y = 2x - 36The lines have the same slope and different y-intercepts. Therefore the lines are parallel.

16. Possible answer: Substitute -2x + 3 for y in the second equation.2x + (-2x + 3) - 3 = 0 2x - 2x + 3 - 3 = 0 0 = 03

17. Possible answer: Substitute x - 2 for y in the second equation.x - (x - 2) - 2 = 0 x - x + 2 - 2 = 0 0 = 03

18. Possible answer: Substitute -x - 4 for y in the first equation.x + (-x - 4) = -4 x - x - 4 = -4 -4 = -43

19. Possible answer: Compare slopes and intercepts.-9x - 3y = -18 → y = -3x + 6 3x + y = 6 → y = -3x + 6

The lines have the same slope and the same y-intercept. Therefore the graphs are one line.


CS10_A1_MESK710372_C05.indd 171 3/30/11 11:03:02 PM

20. y = -x + 5 → y = -x + 5 x + y = 5 → y = -x + 5 This system is consistent and dependent. It has

infinitely many solutions.

21. y = -3x + 2 y = 3x This system is consistent and independent. It has

one solution.

22. y - 1 = 2x → y = 2x + 1 y = 2x - 1 → y = 2x - 1 This system is inconsistent. It has no solution.

23. Yes. Let x represent the number of hours and let y represent the number of miles.

Mandy: y = 5x Nikki: y = 6x - 1 The graphs of the two equations have different

slopes so they intersect. If the pattern continues, Nikki will eventually catch up to Mandy.

24. No. Let x represent the number of minutes after copier A started and let y represent the number of copies.

Copier A: y = 35x Copier B: y = 35x + 10 The graphs of the two equations are parallel lines,

so there is no solution. If the pattern continues, the number of copies made by each machine will never be equal.

25. Let x represent the number of DVDs rented and let y represent the number of video games rented.

First week: 4x + 2y = 18 → y = -2x + 9 Second week: 2x + y = 9 → y = -2x + 9 There are infinitely many answers for the cost

of each video game and DVD. The system is consistent and dependent.

26. Let x represent the number of pounds of cashews and let y represent the number of pounds of peanuts.

Rosa: x + 2y = 10 → y = - 1 __ 2 x + 5

Sabrina: 2x + y = 11 → y = -2x + 11 This system is consistent and independent. It has

one solution.

y = - 1 __ 2 x + 5

-2x + 11 = - 1 __ 2 x + 5

________ +2x _______ +2x

11 = 3 __ 2 x + 5

___ -5 ______ - 5

6 = 3 __ 2 x

2 __ 3 (6) = 2 __

3 ( 3 __

2 x)

4 = x

2x + y = 11 2(4) + y = 11 8 + y = 11 ______ -8 ___ -8 y = 3 The cashews cost $4/lb and the peanuts cost $3/lb.

27. Let x represent the number of years and let y represent the number of geodes.

Pam: y = 4x + 2 Tommy: y = 4x + 2 They will always have the same amount. Both

started with 2 and add 4 every year.

28a. 2(3) = 6 2(4) = 8 2(5) = 10 2(6) = 12

2(12) = 24 2(13) = 26 2(14) = 28 2(15) = 30

In both tables, the value of y is twice the value of x. y = 2x; y = 2x

b.

2

x

y

2 -2

It is one line in the coordinate plane.

c. The equations are the same.

d. The graphs will now be parallel lines. The equations will have the same slope but different y-intercepts, so you know they will be parallel when graphed.

29. The graph will be 2 parallel lines.

30a. Let b represent the number of buttons to be made and let c represent the total cost.

Buttons, etc.: c = 50 + 1.10b Logos: c = 40 + 1.10b

b. c = 50 + 1.10b → c = 1.10b + 50 c = 40 + 1.10b → c = 1.10b + 40 Never; the slope, or cost per button is the same,

and the initial cost, or y-intercepts, are different.

c. For the same price, the y-intercept would have to be changed to 40. Possible answer: The y-intercept may be lowered to a value less than 40, or the slope may be changed to a value less than 1.1.

31. Student A; the lines are not parallel, so they will intersect.

32. The graph of a system that is consistent and independent shows two lines that intersect at one point. The graph of a system that is consistent and dependent shows only one line.

TEST PREP

33. D; Since 2x - y = 3 → y = 2x - 3 and 6x - 3y = 9 → y = 2x -3, the system is consistent and dependent.

34. H; Since two graphs that have different slopes, they must intersect at only one point. So if the graphs have different slopes, the system must be consistent, and independent.


CS10_A1_MESK710372_C05.indd 172 3/30/11 11:03:04 PM


35. p = q; p ≠ q

36a. 3x + 4y = 0 → y = - 3 __ 4 x

4x + 3y = 0 → y = - 4 __ 3 x

This system is consistent and independent. Intersection is (0, 0).

b. 2x + 5y = 0 → y = - 2 __ 5 x

5x + 2y = 0 → y = - 5 __ 2 x

This system is consistent and independent. Intersection is (0, 0).

c. Those systems are consistent and independent, so they only have one solution.

Notice that: ax + by = 0 → y = - a __

b x

bx + ay = 0 → y = - b __ a x

Since both equations have a y-intercept of 0, they intersect at (0, 0). Therefore, systems of equations of this form only intersect at (0, 0).

rEAdy to go on? Section A Quiz

1. _______________ y = -2x - 3 (1) -2(-2) - 3 1 4 - 3 1 1 3

____________ y = x + 3 (1) (-2) + 3 1 1 3


2. _____________ x - 4y = 1 (9) - 4(2) 1 9 - 8 1 1 1 3

______________ 2x - 3y = 2 2(9) - 3(2) 2 18 - 6 2 12 2 7

(9, 2) is not a solution of the system.

3. ___________

y = - 1 __ 3 x

(-1) - 1 __ 3 (3)

-1 -1 3

_______________ y + 2x = 5 (-1) + 2(3) 5 -1 + 6 5 5 5 3


4.

x

y

0

6

1 -2

-8

(- 2, 3)

y = x + 5

y = 1 _ 2 x + 4


5.

x

y

0

4

-5 -2 (0,-2)

2x - y = 2

y = -x - 2

3


6.

x

y

0

7

-7

-5 (3,-5)

4x + y = 7

5

2 _ 3 x + y = - 3


7. Let x represent the number of months and let y represent the amount of money.

Christiana: y = 10x + 50 Marlena: y = 15x + 30 Graph y = 10x + 50 and y = 15x + 30.

Money in Account

0 1 3 5 2 4 6

20 40 60 80

100 Christina

Marlena

Months

Am

ount

( $)

(4, 90)

The lines appear to intersect at (4, 90). So both accounts will have the same amount deposited in 4 months and that amount will be $90.

8. 2x + y = 11 2x + (-x + 5) = 11 x + 5 = 11 _____ - 5 ___ -5 x = 6

y = -x + 5 y = -(6) + 5 y = -1 (6, -1)


CS10_A1_MESK710372_C05.indd 173 3/30/11 11:03:07 PM

9. 3x - y = -2 _______ -3x ____ -3x -y = -3x - 2 -1(-y) = -1(-3x - 2) y = 3x + 2

4x - 3y = -1 4x - 3(3x + 2) = -1 4x - 3(3x) - 3(2) = -1 4x - 9x - 6 = -1 -5x - 6 = -1 _______ + 6 ___ +6 -5x = 5

-5x ____ -5

= 5 ___ -5

x = -1

3x - y = -2 3(-1) - y = -2 -3 - y = -2 ______ +3 ___ +3 -y = 1 -1(-y) = -1(1) y = -1 (-1, -1)

10. y = -2x - 5 -x = -2x - 5 ____ +2x _______ +2x x = -5

y = -x y = -(-5) y = 5 (-5, 5)

11. x + 3y = 15 ______________ + 2x - 3y = -6 3x + 0 = 9 3x = 9

3x ___ 3 = 9 __

3

x = 3

x + 3y = 15 3 + 3y = 15 _______ -3 ___ -3 3y = 12

3y

___ 3 = 12 ___

3

y = 4 (3, 4)

12. x + y = 2 _____________ - (2x + y = -1)

x + y = 2 ______________ + (-2x - y = 1) -x + 0 = 3 -x = 3 -1(-x) = -1(3) x = -3

x + y = 2 -3 + y = 2 ______ +3 ___ +3 y = 5 (-3, 5)

13. 3(-2x + 5y = -1) _______________ + 2(3x + 2y = 11)

-6x + 15y = -3 ____________ + 6x + 4y = 22 0 + 19y = 19 19y = 19

19y

____ 19

= 19 ___ 19

y = 1

3x + 2y = 11 3x + 2(1) = 11 3x + 2 = 11 ______ - 2 ___ -2 3x = 9

3x ___ 3 = 9 __

3

x = 3 (3, 1)

14. Let x represent the number of black and white drawings and let y represent the number of color drawings. Note that 2 h = 120 min.

Drew for 2 hours: 10x + 25y = 120 9 drawings: x + y = 9

⎧

⎨ ⎩ 10x + 25y = 120

x + y = 9

10x + 25y = 120 _______________ + (-10)(x + y = 9)

10x + 25y = 120 ___________________ + (-10x - 10y = -90) 0 + 15y = 30 15y = 30

15y

____ 15

= 30 ___ 15

y = 2

x + y = 9 x + 2 = 9 ____ - 2 ___ -2 x = 7 Akira drew 7 black and white drawings and 2 color

drawings.

15. y = -2x - 6 → y = -2x - 6 2x + y = 5 → y = -2x + 5 This system has no solution so it is an inconsistent

system.

16. x + y = 2 → y = -x + 2 2x + 2y = -6 → y = -x - 3 This system has no solution so it is an inconsistent

system.

17. y = -2x + 4 → y = -2x + 4 2x + y = 4 → y = -2x + 4 If this system were graphed, the graphs would be

the same line. There are infinitely many solutions.

18. 3x = -6y + 3 → y = - 1 __ 2 x + 1 __

2

2y = -x + 1 → y = - 1 __ 2 x + 1 __

2


19. y = -4x + 2 → y = -4x + 2 4x + y = -2 → y = -4x - 2 This system is inconsistent. It has no solution.

20. 4x - 3y = 8 → y = 4 __ 3 x - 8 __

3

y = 4(x + 2) → y = 4x + 8 This system is consistent and independent. It has

one solution.

SolvIng lInEAr InEQuAlItIES

CHECK IT OUT!

1a. ________ y < x + 1 5 4 + 1 5 < 5 7 (4, 5) is not a solution.

b. _________ y > x - 7 1 1 - 7 1 > -6 3 (1, 1) is a solution.


5-5

CS10_A1_MESK710372_C05.indd 174 3/30/11 11:03:09 PM

2a. 4x - 3y > 12 ________ -4x ____ -4x -3y > -4x + 12

-3y

____ -3

< -4x + 12 ________ -3

y < 4 __ 3 x - 4

Graph the boundary line y = 4 __ 3 x - 4. Use a dashed

line for <. The inequality is <, so shade below the line.

2

-2

-4

x

y

0 2 -2

b. 2x - y - 4 > 0 _________ + y ___ +y 2x - 4 > y y < 2x - 4 Graph the boundary line y = 2x - 4. Use a dashed


-2

-4

x y

0 2 -2

c. Graph the boundary line y = - 2 __ 3 x + 1. Use a solid

line for ≥. The inequality is ≥, so shade above the line.

2

-2

x

y

0 2 -2

3a. Let b represent the number of black olives and let g represent the number of green olives.

2.5b + 2g ≤ 6

b. 2.5b + 2g ≤ 6 __________ -2.5b _____ -2.5b 2g ≤ -2.5b + 6

2g

___ 2 ≤ -2.5b + 6 _________

2

g ≤ -1.25b + 3 Since Dirk cannot buy a negative number of pounds

of olives, the system is graphed only in Quadrant I. Graph the boundary line y = -1.25x + 3. Use a solid line for ≤. The inequality is ≤, so shade below the line.

Olive Combinations

0 1 2 3 4

1

2

3

4

Black olives

Gre

en o

lives

c. Possible answers: (1 lb black, 1 lb green), (0.5 lb black, 2 lb green)

4a. y-intercept: 0; slope: -1 y = mx + b → y = -x The graph is shaded below a dashed boundary line.

Replace = with < to write the inequality y < -x.

b. y-intercept: -3; slope -2 y = mx + b → y = -2x - 3 The graph is shaded above a solid boundary line.

Replace = with ≥ to write the inequality y ≥ -2x - 3.

THINK AND DISCUSS

1. Possible answer: In both cases you will always graph a line. They are different because the graph of a linear inequality can be a dashed or solid line, and the graph must be shaded on one side of the line.

2. To write a linear inequality from a graph, find the equation of the boundary line. Use ≥ or ≤ for a solid boundary line. Use > or < for a dashed boundary line. If the half-plane is shaded above the line, use > or ≥, and if it is shaded below, use < or ≤.

3. Inequality

Symbol

Boundary Line

Shading

<

Dashed

Below

>

Dashed

Above

≤

Solid

Below

≥

Solid

Above

y < 5 x + 2 y > 7 x - 3 y ≤ 9 x + 1 y ≥ - 3 x - 2


CS10_A1_MESK710372_C05.indd 175 3/30/11 11:03:11 PM


1. No; a dashed line means that the ordered pairs are not solutions to the linear inequality.

2. ___________ y ≤ -x + 3 3 -(0) + 3 3 ≤ 3 3 (0, 3) is a solution.

3. _____________ y > -2x - 2 0 -2(2) - 2 0 -4 - 2 0 > -6 3 (2, 0) is a solution.

4. _____________ y < 2x + 4 1 2(-2) + 4 1 -4 + 4 1 < 0 7 (-2, 1) is not a solution.

5. Graph the boundary line y = -x. Use a solid line for ≤. The inequality is ≤, so shade below the line.

2

-2

x

y

0 2 -2

6. Graph the boundary line y = 3x + 1. Use a dashed line for >. The inequality is >, so shade above the line.

2

x

y

2 -2

7. -y < -x + 4 -1(-y) > -1(-x + 4) y > x - 4 Graph the boundary line y = x - 4. Use a dashed

line for >. The inequality is >, so shade above the line.

-2

-4

y

x 0 2 -2

8. -y ≥ x + 1 -1(-y) ≤ -1(x + 1) y ≤ -x - 1 Graph the boundary line y = -x - 1. Use a solid

line for ≤. The inequality is ≤, so shade below the line.

2

-2

-4

x

y

2 -2

9a. Let x represent the number of cups of orange juice and let y represent the number of cups of pineapple juice.

x + y ≤ 16

b. x + y ≤ 16 ______ -x ___ -x y ≤ -x + 16 Since Jack cannot make a negative number of cups,

the system is graphed only in Quadrant I. Graph the boundary line y = -x + 16. Use a solid line for ≤. The inequality is ≤ so shade below the line.

Punch Combinations

0 4 8 12 16

4

8

12

16

Orange juice (c)

Pine

appl

e ju

ice

(c)

c. Possible answer: (2 c orange , 2 c pineapple), (4 c orange, 10 c pineapple)

10. y-intercept: 3; slope: 0 y = mx + b → y = 3 The graph is shaded below a dashed boundary line.

Replace = with < to write the inequality y < 3.

11. y-intercept: 5; slope: 1 y = mx + b → y = x + 5 The graph is shaded above a solid boundary line.

Replace = with ≥ to write the inequality y ≥ x + 5.


12. ____________ y ≥ 2x + 3 3 2(3) + 3 3 6 + 3 3 ≥ 9 7 (2, 3) is not a solution.

13. ____________ y < 3x - 3 -1 3(1) - 3 -1 3 - 3 -1 < 0 3 (1, -1) is a solution.

14. ____________ y > 4x + 7 7 4(0) + 7 7 0 + 7 7 > 7 7 (0, 7) is not a solution.

15. Graph the boundary line y = -2x + 6. Use a dashed line for >. The inequality is >, so shade above the line.

4

-4

x

y

0 2


CS10_A1_MESK710372_C05.indd 176 3/30/11 11:03:14 PM

16. -y ≥ 2x -1(-y) ≤ -1(2x) y ≤ -2x Graph the boundary line y = -2x. Use a solid line

for ≤. The inequality is ≤, so shade below the line.

-2

x

y

0 2 -2

17. x + y ≤ 2 ______ -x ___ -x y ≤ -x + 2 Graph the boundary line y = -x + 2. Use a solid


2

-2

x

y

0 2 -2

18. x - y ≥ 0 ____ + y ___ +y x ≥ y y ≤ x Graph the boundary line y = x. Use a solid line for

≤. The inequality is ≤, so shade below the line.

2

-2

x

y

2 -2

19a. Let x represent the number of pounds of hamburgers and let y represent the number of pounds of hot dogs.

3x + 2y ≤ 30

b. 3x + 2y ≤ 30 ________ -3x ____ -3x 2y ≤ -3x + 30

2y

___ 2 ≤ -3x + 30 ________

2

y ≤ - 3 __ 2 x + 15

Since Beverly cannot buy a negative number of hamburgers or hot dogs, the system is graphed only in Quadrant I. Graph the boundary line

y = - 3 __ 2 x + 15. Use a solid line for ≤. The

inequality is ≤, so shade below the line.Food Combinations

0 4 8

6

12

Hamburger meat (lb)

Hot

dog

s (lb

)

c. Possible answer: (3, 2), (5, 6)

20. y-intercept: -1; slope: 2 __ 3

y = mx + b → y = 2 __ 3 x - 1

The graph is shaded above a dashed boundary line.

Replace = with > to write the inequality y > 2 __ 3

x - 1.

21. y-intercept: 3; slope: - 1 __ 5

y = mx + b → y = - 1 __ 5 x + 3

The graph is shaded below a solid boundary line. Replace = with ≤ to write the ineqaulity

y ≤ - 1 __ 5 x + 3.

22a. 125x + 100y ≥ 500

b. 125x + 100y ≥ 500 ____________ -125x ______ -125x 100y ≥ -125x + 500

100y

_____ 100

≥ -125x + 500 ___________ 100

y ≥ -1.25x + 5 Since the store cannot sell a negative number of

items, the system is graphed only in Quadrant I. Graph the boundary line y = -1.25x + 5. Use a solid line for ≥. The inequality is ≥ so shade above the line.

Electronics Sales

0 2 4 6 8

2

4

6

8

DVD players

CD p

laye

rs

c. x and y must be whole numbers.

d. Possible answer: (5 DVD, 1 CD), (7 DVD, 6 CD), (9 DVD, 2 CD)

23. y ≤ 2 - 3x y ≤ -3x + 2 Graph the boundary line y = -3x + 2. Use a solid


2

-2

x

y

0 2 -2


CS10_A1_MESK710372_C05.indd 177 3/30/11 11:03:17 PM

24. -y < 7 + x -1(-y) > -1(7 + x) y > -x - 7 Graph the boundary line y = -x - 7. Use a dashed


4

x

y

0 4

-4

-4

25. 2x - y ≤ 4 _______ -2x ____ -2x -y ≤ -2x + 4 -1(-y) ≥ -1(-2x + 4) y ≥ 2x - 4 Graph the boundary line y = 2x - 4. Use a solid line

for ≥. The inequality is ≥, so shade above the line.

-2

-4

x y

0 2 -2

26. 3x - 2y > 6 ________ -3x ____ -3x -2y > -3x + 6

-2y

____ -2

< -3x + 6 _______ -2

y < 3 __ 2 x - 3

Graph the boundary line y = 3 __ 2 x - 3. Use a dashed


-2

x y

0 2 -2

27a. Let x represent the length and let y represent the width.

2x + 2y ≤ 18

b. 2x + 2y ≤ 18 ________ -2x ____ -2x 2y ≤ -2x + 18

2y

___ 2 ≤ -2x + 18 ________

2

y ≤ -x + 9

Since Marvin cannot have an negative length or width, the system is graphed only in Quadrant I. Graph the boundary line y = -x + 9. Use a solid line for ≤. The inequality is ≤, so shade below the line.

Rectangular Garden

0 2 4 6 8

2

4

6

8

Length (yd)

Wid

th (y

d)

Possible answer: (3, 1), (1, 6), (6, 2)

c. 4 yd × 4 yd

28. 15x + 11y ≤ 77 __________ -15x _____ -15x 11y ≤ -15x + 77

11y

____ 11

≤ -15x + 77 _________ 11

y ≤ - 15 ___ 11

x + 7

Since Stephen can only buy a whole number of fish, the system is graphed only in Quadrant I. Graph the

boundary line y = - 15 ___ 11

x + 7. Use a solid line for ≤.

The inequality is ≤, so shade below the line.Fish Combinations

0 2 4 6

2

4

6

Yellow tangs

Clow

n fis

h

Possible answer: (2, 2.5); you cannot buy half a fish.

29. Graph the boundary line y = 1. Use a dashed line for >. The inequality is >, so shade above the line.

2

-2

x

y

0 2 -2

30. -2 < x x > -2 Graph the boundary line x = -2. Use a dashed line

for >. The inequality is >, so shade right of the line.

2

-2

x

y

0 2


CS10_A1_MESK710372_C05.indd 178 3/30/11 11:03:20 PM

31. Graph the boundary line x = -3. Use a solid line for ≥. The inequality is ≥, so shade right of the line.

2

-2

x

y

0 2 -2

32. Graph the boundary line y = 0. Use a solid line for ≤. The inequality is ≤, so shade below the line.

2

-2

x

y

0 2 -2

33. 0 ≥ x x ≤ 0 Graph the boundary line x = 0. Use a solid line for

≤. The inequality is ≤, so shade left of the line.

2

-2

x

y

0 2 -2

34. -12 + y > 0 _______ +12 ____ +12 y > 12 Graph the boundary line y = 12. Use a dashed line

for >. The inequality is >, so shade above the line.

4

8

x

y

0 2 -2

35. x + 7 < 7 ____ - 7 ___ -7 x < 0 Graph the boundary line x = 0. Use a dashed line

for <. The inequality is <, so shade left of the line.

2

-2

x

y

0 2 -2

36. -4 ≥ x - y ___ + y _____ + y -4 + y ≥ x ______ +4 ___ +4 y ≥ x + 4 Graph the boundary line y = x + 4. Use a solid line

for ≥. The inequality is ≥, so shade above the line.

2

4

x

y

0 -2 -4

37. Let a represent the number of adult tickets and let s represent the number of student tickets.

7a + 4s ≥ 280 ________ -7a ____ -7a 4s ≥ -7a + 280

4s ___ 4 ≥ -7a + 280 __________

4

s ≥ -1.75a + 70 Since only a positive number of tickets can be sold,

the system is graphed only in Quadrant I. Graph the boundary line y = -1.75x + 70. Use a solid line for ≥. The inequality is ≥, so shade above the line.

Tickets Sold

0 20 40 60

20

40

60

Adult tickets

Stud

ent

tick

ets

Possible answer: (40, 10), (40, 20), (20, 100)

38. Possible answer: It is not possible to list an infinite number of solutions, so shading is used to show all the solutions in the coordinate plane.

39. Possible answer: Sammy volunteers at the hospital and the library. The total number of hours he volunteers is more than 5; h + ℓ > 5; Check students’ graphs and solutions.

40a. x + y ≤ 50

b. x + y ≤ 50 ______ -x ___ -x y ≤ -x + 50


CS10_A1_MESK710372_C05.indd 179 3/30/11 11:03:23 PM

Since Gloria can only make a positive number of bears, the system is graphed only in Quadrant I. Graph the boundary line y = -x + 50. Use a solid line for ≤. The inequality is ≤, so shade below the line.

Teddy Bear Combinations

0 10 20 30 40

10

20

30

40

Girl bears

Boy

bear

s

c. Possible answer: (10, 20), (30, 10), (15, 15)

41. Student A; the graph is a solid line, so ≤ should be used, not <.

42. Possible answer: If the inequality is in slope-intercept form, then > or ≥ means you shade above the boundary line, and < or ≤ means you shade below the boundary line. The shading represents all the solutions of the inequality.

TEST PREP

43. B; Since 4 > 2 = -1 + 3, (1, 4) is a solution of y > -x + 3.

44. H; The boundary line is y = -2x + 3. Since the line is solid and the graph is shaded below the line, use ≤. So y ≤ -2x + 3, or 2x + y ≤ 3.

45. C; The equation is rewritten as x ≥ 3. The boundary line is solid because the inequaltiy is ≥. The graph is shaded to the right because all values of x greater than or equal to 3 are solutions.


46. 0 ≥ -6 - 2x - 5y ____ +5y ____________ + 5y 5y ≥ -6 - 2x

5y

___ 5 ≥ -6 - 2x _______

5

y ≥ - 2 __ 5 x - 6 __

5

Graph the boundary line y = - 2 __ 5 x - 6 __

5 . Use a solid


2

-2

x

y

0 -2 -4

47. Graph the boundary function y = |x|. Use a dashed line for >. The inequality is >, so shade above the function.

2

-2

x

y

0 2 -2

48. Graph the boundary function y = |x - 3|. Use a solid line for ≥. The inequality is ≥, so shade above the function.

2

-4

-2

x

y

0 6 4 2 -2

49. Find the boundary line.

m = y 2 - y 1

_______ x 2 - x 1 = 1.5 - 3 _______ -3 - 0

= -1.5 _____ -3

= 1 __ 2

y - y 1 = m(x - x 1 )

y - 1.5 = 1 __ 2 (x - (-3))

y - 1.5 = 1 __ 2 (x + 3)

y - 1.5 = 1 __ 2 x + 1.5

______ + 1.5 ________ + 1.5

y = 1 __ 2 x + 3

Since the points on the boundary line are solutions, use ≥ or ≤. Check which sign makes (1, 1) not a solution.

____________

y ≥ 1 __ 2 x + 3

1 1 __ 2 (1) + 3

1 1 __ 2 + 3

1 ≥ 7 __ 2 7

____________

y ≤ 1 __ 2 x + 3

1 1 __ 2 (1) + 3

1 1 __ 2 + 3

1 ≤ 7 __ 2 3

Since (1, 1) is not a solution of y ≥ 1 __ 2 x + 3, the

inequality is y ≥ 1 __ 2 x + 3.

50. The boundary lines must be x = 0 and y = 0. Since Quadrant I is not shaded, no points in that plane are solutions to either inequality, so the inequalities are x ≤ 0 and y ≤ 0.


CS10_A1_MESK710372_C05.indd 180 3/30/11 11:03:26 PM

SolvIng SyStEmS of lInEAr InEQuAlItIES

CHECK IT OUT!

1a. _____________ y < -3x + 2 1 -3(0) + 2 1 < 2 3

__________ y ≥ x - 1 1 0 - 1 1 ≥ -1 3


b. ___________ y > -x + 1 0 -0 + 1 0 > 1 7

__________ y > x - 1 0 0 - 1 0 > -1 3


2a.

-2

x

y

0 2

Possible answer: solutions: (3, 3), (4, 4); not solutions: (-3, 1), (-1, -4)

b. 3x + 6y ≤ 12 ________ -3x ____ -3x 6y ≤ -3x + 12

6y

___ 6 ≤ -3x + 12 ________

6

y ≤ - 1 __ 2 x + 2

4

-4

x

y

0 4 -4

Possible answer: solutions: (0, 0), (3, -2); not solutions: (4, 4), (1, -6)

3a.

2

-2

x

y

0

b.

2

x

y

2 -2

c.

-2

x

y

0 -2

4. Let x represent the number of pounds of pepper jack cheese and let y represent the number of pounds of cheddar cheese.

x ≥ 2 y ≥ 2 4x + 2y ≤ 20 The graph should only be in the first quadrant

because the number of pounds of cheese cannot be negative.

Cheese Combinations

0 2 4 6 8

2

4

6

8

Pepper jack cheese (lb)

Ched

dar

chee

se (l

b)

Alice could buy any combination represented by an ordered pair in the solution region. Answers do not need to be whole numbers because she can buy a real number of pounds of cheese.

Possible answer: (3 lb pepper jack, 2 lb cheddar), (2.5 lb pepper jack, 4 lb cheddar)

THINK AND DISCUSS

1. Possible answer: To write a system of linear inequalities from a graph, write the linear inequality for each graph that makes up the system.

2.

-4

y

0 -2 -4

Possible answer: (-4, -3)

Possible answer: (0, 0)

-4

x

y

0 -2 -4

y ≥ 2 x + 1

y > 1 __ 2 x - 2

y < 2 x + 1

y ≥ 1 __ 2 x - 2


1. all

2. ___________ y < -x + 3 0 -0 + 3 0 < 3 3

__________ y < x + 2 0 0 + 2 0 < 2 3


3. _______ y < 3 0 < 3 3

__________ y > x - 2 0 0 - 2 0 > -2 3


4. ________ y > 3x 0 3(1) 0 > 3 7

__________ y ≤ x + 1 0 1 + 1 2 ≤ 1 7



5-6

CS10_A1_MESK710372_C05.indd 181 3/30/11 11:03:29 PM

5. 4

x

y

0 2 4 -2

Possible answer: solutions: (3, 3), (4, 3); not solutions: (0, 0), (2, 1)

6.

2

4

-2

x

y

0 2 4 -2

Possible answer: solutions: (0, 0), (1, 1); not solutions: (2, -1), (3, 1)

7. 3x + y ≥ 3 _______ -3x ____ -3x y ≥ -3x + 3

2

x

y

0 2 -2


8. 2x - 4y ≤ 8 ________ -2x ____ -2x -4y ≤ -2x + 8

-4y

____ -4

≥ -2x + 8 _______ -4

y ≥ 1 __ 2 x - 2

-4

x

y

0 4

Possible answer: solutions: (0, 0), (1, 1); not solutiosn: (3, 0), (-3, -4)

9. no solutions

x

y

0 2

10. no solutions

-2

x

y

0 2 -2

11. all points between the parallel lines and on the solid line

2

x

y

2 -2

12. all points between the parallel lines

2

x

y

0 2

13. same solutions as y > 2x - 1

2

-4

x

y

0 2 -2

14. same solutions as y ≤ -3x - 3

4

-4

x

y

-2

2

15. Let x represent the number of cups of lemonade and let y represent the number of cupcakes.

x ≥ 5 y ≥ 5 2x + y ≥ 25 The graph should only be in the first quadrant

because sales are not negative.Sales Goals

0 8 16 24

8

16

24

Lemonade (c)

Cupc

akes

(6, 13)

(10, 10)

To meet the sales goal, Sandy could sell any combination represented by an ordered pair of whole numbers in the solution region. Answer must be whole numbers because Sandy cannot sell a part of a cup of lemonade or cupcake.

Possible answer: (6, 13), (10, 10)


16. ___________ y ≥ -x - 1 0 -0 - 1 0 ≥ -1 3

____________ y ≤ 2x + 4 0 2(0) + 4 0 ≤ 4 3


17. ___________ x + y < 3 0 + 0 3 0 < 3 3

____________ y > 3x - 4 0 3(0) - 4 0 > -4 3


18. ________ y > 3x 0 3(1) 0 > 3 7

____________ y > 3x + 1 0 3(1) + 1 0 > 4 7


19. 2

-2

x

y

0 2 -2

Possible answer: solutions (-2, 0), (-3, 1); not solutions: (0, 0), (1, 4)


CS10_A1_MESK710372_C05.indd 182 3/30/11 11:03:34 PM

20. 2

x

y

0

Possible answer: solutions (-2, -2), (-3, -2); not solutions: (0, 0), (1, -4)

21. 6x + 2y ≥ -2 ________ -6x ____ -6x 2y ≥ -6x - 2

2y

___ 2 ≥ -6x - 2 _______

2

y ≥ -3x - 1

4

x

y

0 2

Possible answer: solutions (-1, 3), (0, 5); not solutions: (0, 0), (1, 4)

22. 9x + 3y ≤ 6 ________ -9x ____ -9x 3y ≤ -9x + 6

3y

___ 3 ≤ -9x + 6 _______

3

y ≤ -3x + 2

2

-2

x

y

2 -2

Possible answer: solutions (-2, 0), (-3, 1); not solutions: (0, 0), (1, 4)

23. no solutions

2

4

x

y

0 2 -2


-4

x y

0 2 -2

25. all points are solutions

2

-2

x

y

0 -2


2

-2

x

y

2 -2

27. same solutions as y > 2

-2

x

y

0

28. same solutions as y ≤ 2x - 4

-4

x

y

0 2 4 -2

29. Let x represent the number of hours at the pharmacy and let y represent the number of hours baby-sitting.

15x + 10y ≥ 90 x + y ≤ 20 The graph should only be in the first quadrant

because time is not negative.

Linda’s Work Hours

0 6 12 18

6

12

18

Hours at pharmacy

Hou

rs b

aby-

sitt

ing

(8.5, 10)

(0, 9)

To meet her goals, Linda could work for any combination represented by an ordered pair in the solution region. Answers do not need to be whole numbers because she can work a real number of hours.

Possible answer: (0, 9), (8.5, 10)

30. Let x represent the number of acres of corn and let y represent the number of acres of soybeans.

x ≥ 40 y ≥ 50 x + y ≤ 200 The graph should only be in the first quadrant

because Tony cannot plant a negative number of acres.

Planting Combinations

0 50 100 150 200

50

100

150

200

Corn (acres)

Soyb

eans

(acr

es)

To meet his goals, Tony could plant any combination represented by an ordered pair in the solution region. Answers do not need to be whole numbers because he can plant a real number of acres.

Possible answer: (60, 80), (100, 60)

31.

-2

x

y

0 2 -2

32. 2

x

y

2 -2


CS10_A1_MESK710372_C05.indd 183 3/30/11 11:03:38 PM

33. 2

-2

x

y

0 2 -2

34.

2

x

y

0 2 -2

35. ⎧

⎨ ⎩ y > x + 1

y < x + 3

36. ⎧

⎨ ⎩ y ≥ 1

y ≤ 3

37. ⎧

⎨ ⎩ y < 2

x ≥ -2

38. Let x represent the ages in years and let y represent the heights in inches.

x ≥ 17 x < 23 y ≥ 60 y ≤ 80 The graph should only be in the first quadrant

because time and height cannot be negative.

0 10 20 30 40

20

40

60

80

Age (yr)

Hei

ght

(in.)

Possible answer: (20, 70), (19, 75), (18, 69)

39. Student B; the inequality symbols are incorrect.

40. Let x represent the length of the area and let y represent the width of the area.

x ≥ 30 2x + 2y ≤ 150 The graph should only be in the first quadrant

because length is not negative.Dimensions for Dog Area

0 20 40 60

20

40

60

Width (ft)

Leng

th (f

t)

41. Yes; the solutions of the system ⎧

⎨ ⎩ y ≥ x + 4

y ≤ x + 4

are

represented by all ordered pairs on the line y = x + 4.

42a. Let x represent the number of girl bears and let y represent the number of boy bears.

x + y ≤ 40

b. 15x + 12y ≥ 540 c. Teddy Bear Combinations

0 12 24 36

12

24

36

Girl bears

Boy

bear

s

43. The boundary lines must be parallel. If the boundary lines are not parallel, there will be overlap.

TEST PREP

44. D; Since 2(1) + 1 = 3 ≥ 3 and 1 ≥ -1 = -2(1) + 1, (1, 1) is a solution of both inequalities and therefore, is a solution of the system.

45. G; The solution region is between the two lines. Therefore, the solutions lie below y = 2x + 1 and above y = 2x - 3.

46. y + x > 2 ____ - x ___ -x y > -x + 2

2

4

x

y

0 2 -2

Possible answer: (0, 3), (-2, 6)


47.

2

x

y

0 2 -2

The solution region is a triangle. The height is about 3.5 units. The base is about 7 units.

A = 1 __ 2 bh

≈ 1 __ 2 (7)(3.5)

≈ 12 The answer is approximately 12 square units.

48. Possible answer: ⎧ ⎪

⎨

⎪ ⎩ y ≥ 3 __

2 x + 5 __

2

y > - 1 __ 2 x


CS10_A1_MESK710372_C05.indd 184 3/30/11 11:03:41 PM

49. |y| < 1 y < 1 AND y > -1 Graph y < 1 and y > -1. The solutions of |y| < 1 lie

in the solution region.

2

-2

x

y

0

50. ⎧

⎨ ⎩ x < 0

y < 0

rEAdy to go on? Section b Quiz

1. _______________ y < -2x + 1 -2 -2(3) + 1 -2 -6 + 1 -2 < -5 7 (3, -2) is not a solution.

2. ____________ y ≥ 3x - 5 1 3(2) - 5 1 6 - 5 1 ≥ 1 3 (2, 1) is a solution.

3. _______________ y ≤ 4x - 10 -6 4(1) - 10 -6 4 - 10 -6 ≤ -6 3 (1, -6) is a solution.

4. Graph the boundary line y = 4x - 3. Use a solid line for ≥. The inequalitiy is ≥, so shade above the line.

-4

-2

x y

0 2 -2

5. 3x - y < 5 _______ -3x ____ -3x -y < -3x + 5 -1(-y) > -1(-3x + 5) y > 3x - 5 Graph the boundary line y = 3x - 5. Use a dashed

line for >. The inequalitiy is >, so shade above the line.

-4

-2

x y 0 2 -2

6. 2x + 3y < 9 ________ -2x ____ -2x 3y < -2x + 9

3y

___ 3 < -2x + 9 _______

3

y < - 2 __ 3 x + 3

Graph the boundary line y = - 2 __ 3

x + 3. Use a

dashed line for <. The inequalitiy is <, so shade below the line.

2

-2

x

y

0 4 2

7. Graph the boundary line y = - 1 __ 2

x. Use a solid line

for ≤. The inequalitiy is ≤, so shade below the line.

4

2

x

y

0 4

8. Let x represent the number of pairs of pants and let y represent the number of shirts.

30x + 15y ≤ 150 __________ -30x _____ -30x 15y ≤ -30x + 150

15y

____ 15

≤ -30x + 150 __________ 15

y ≤ -2x + 10 Since Theo cannot buy a negative number of

clothes, the system is graphed only in Quadrant I. Graph the boundary line y = -2x + 10. Use a solid line for ≤. The inequality is ≤, so shade below the line.

Possible Clothing Combinations

Pants

0

4

8

Shir

ts

12

6 4 2

Possible answer: (4, 2), (3, 2), (2, 6)

9. y-intercept: -5; slope: 2 y = mx + b → y = 2x - 5 The graph is shaded below a dashed boundary line.

Replace = with < to write the inequality y < 2x - 5.

10. y-intercept: none; slope: undefined The graph is a vertical line at x = -3. The graph is shaded on the right side of a dashed

boundary line. Replace = with > to write the inequality x > -3.


CS10_A1_MESK710372_C05.indd 185 3/30/11 11:03:44 PM

11. y-intercept: 0; slope: -1 y = mx + b → y = -x The graph is shaded above a solid boundary line.

Replace = with ≥ to write the inequality y ≥ -x.

12. ___________ y > -2 -1 > -2 3

_____________ y < x + 4 -1 -3 + 4 -1 < 1 3

(-3, -1) is a solution of the system.

13. ___________ y ≤ x + 4 0 -3 + 4 0 ≤ 1 3

_______________ y ≥ -2x - 6 0 -2(-3) - 6 0 6 - 6 0 ≥ 0 3


14. ________ y ≥ 3x 0 3(0) 0 ≥ 0 3

_____________ 2x + y < -1 2(0) + 0 -1 0 + 0 -1 0 < -1 7


15.

-3

x

y

0 -2 -6

Possible answer: solutions: (0, 0), (2, 2); not solutions: (-6, 0), (-4, 4)

16. x + y ≤ 2 ______ -x ___ -x y ≤ -x + 2

2x + y ≥ -1 _______ -2x ____ -2x y ≥ -2x - 1

6

-6

x

y

0 6 -6

Possible answer: solutions: (2, 0), (2, -2); not solutions: (6, 0), (-4, 0)

17. 2x - 5y ≤ -5 ________ -2x ____ -2x -5y ≤ -2x - 5

-5y

____ -5

≥ -2x - 5 _______ -5

y ≥ 2 __ 5 x + 1

3x + 2y < 10 ________ -3x ____ -3x 2y < -3x + 10

2y

___ 2 < -3x + 10 ________

2

y < - 3 __ 2 x + 5

4

2

x

y

0 2 -2

Possible answer: solutions: (-6, 6), (-5, 0); not solutions: (2, 0), (4, 8)

18.

2

-2

x

y

2 -3

The solutions of the system are the solutions of y ≥ x + 1.

19. 2

x

y

2 -2

The system has no solutions.

20.

2

-2

x

y

3 -3

The solutions are between the graphs of y = -3x + 5 and y = -3x - 2.

21. Let x represent the number of pounds of mangos and let y represent the number of pounds of apples.

x ≤ 45 y ≤ 50 4x + 3y ≥ 300 The graph should only be in the first quadrant

because sales cannot be negative.

Fruit Combinations

Mangos (lb)

0

28

56

Appl

es (l

b) 84

36 24 12

To meet his goals, the grocer could sell any combination represented by an ordered pair in the solution region. Answers do not need to be whole numbers because he can sell real number pounds of fruit.

Possible answer: (45, 40), (42, 50)

Study guIdE: rEvIEw

1. independent system 2. system of linear equations

3. solution of a system of linear inequalities

4. inconsistent system 5. independent system


CS10_A1_MESK710372_C05.indd 186 3/30/11 11:03:48 PM

SOLvINg SySTEmS by grApHINg

6. ________________ y = -6x + 5 (-5) -6(0) + 5 -5 5 7

______________ x - y = 5 (0) - (-5) 5 5 5 3


7. ________________ x - 2y = -2 (4) - 2(3) -2 4 - 6 -2 -2 -2 3

____________

y = 1 __ 2 x + 1

(3) 1 __ 2 (4) + 1

3 2 + 1 3 3 3


8. ________________ x + y = 9

(1 3 __ 4 ) + (7 1 __

4 ) 9

9 9 3

________________ 2y = 6x + 4

2 (7 1 __ 4 ) 6 (1 3 __

4 ) + 4

2 ( 29 ___ 4 ) 6 ( 7 __

4 ) + 4

29 ___ 2 21 ___

2 + 8 __

2

29 ___ 2 29 ___

2 3

(1 3 __ 4 , 7 1 __

4 ) is a solution of the system.

9. ________________ y = -2x + 5 (-1) -2(-1) + 5 -1 2 + 5 -1 7 8

________________ 3y = 6x + 3 3(-1) 6(-1) + 3 -3 -6 + 3 -3 -3 3

(-1, -1) is not a solution of the system.

10. 5

-4

x

y

0 4 -5

y = 3x + 2

y = -2x - 3

(-1, -1)

The solution appears to be at (-1, -1).

11. 2x - 2y = -2 ________ -2x ____ -2x -2y = -2x - 2

-2y

____ -2

= -2x - 2 _______ -2

y = x + 1

7

7

x

y

0

y = - 1 _ 3

x + 5

y = x + 1

(3, 4)


12. Let x represent the number of hours and let y represent the cost.

Garage A: y = 0.50x + 6 Garage B: y = x + 2 Graph y = 0.5x + 6 and y = x + 2. Parking Cost

0 2 4 6

4

8

12

16

2

6

10

14

Time (hours)

Cost

($) Garage A

Garage B

(8, 10)

The lines appear to intersect at (8, 10). The cost at both places will be the same for 8 hours of parking and that cost will be $10.

SOLvINg SySTEmS by SUbSTITUTION

13. y = x + 3 2x + 12 = x + 3 _______ -x ______ -x x + 12 = 3 ______ - 12 ____ -12 x = -9

y = x + 3 y = -9 + 3 y = -6 (-9, -6)

14. y = -4x 2x - 3 = -4x _______ -2x ____ -2x -3 = -6x

-3 ___ -6

= -6x ____ -6

1 __ 2

= x

y = -4x

y = -4 ( 1 __ 2

)

y = -2

( 1 __ 2 , -2)

15. 2x + y = 4 _______ -2x ____ -2x y = -2x + 4

3x + y = 3 3x + (-2x + 4) = 3 x + 4 = 3 _____ - 4 ___ -4 x = -1

2x + y = 4 2(-1) + y = 4 -2 + y = 4 ______ +2 ___ +2 y = 6 (-1, 6)


5-1

5-2

CS10_A1_MESK710372_C05.indd 187 3/30/11 11:03:51 PM

16. x + y = -1 x + (-2x + 3) = -1 -x + 3 = -1 ______ - 3 ___ -3 -x = -4 -1(-x) = -1(-4) x = 4

y = -2x + 3 y = -2(4) + 3 y = -8 + 3 y = -5 (4, -5)

17. -y - 2x = 8 -y - 2(y - 7) = 8 -y - 2(y) - 2(-7) = 8 -y - 2y + 14 = 8 -3y + 14 = 8 ________ - 14 ____ -14 -3y = -6

-3y

____ -3

= -6 ___ -3

y = 2

x = y - 7 x = 2 - 7 x = -5 (-5, 2)

18. 1 __ 2 x + y = 9

________

- 1 __ 2 x

____ - 1 __

2 x

y = - 1 __ 2 x + 9

3x - 4y = -6

3x - 4 (- 1 __ 2 x + 9) = -6

3x - 4 (- 1 __ 2 x) - 4(9) = -6

3x + 2x - 36 = -6 5x - 36 = -6 _______ + 36 ____ +36 5x = 30

5x ___ 5 = 30 ___

5

x = 6

1 __ 2 x + y = 9

1 __ 2 (6) + y = 9

3 + y = 9 ______ -3 ___ -3 y = 6 (6, 6)

19. Let x represent the number of hours and let y represent the total cost.

Motor Works: y = 70x + 650 Jim’s Car Care: y = 55x + 800

y = 70x + 650 55x + 800 = 70x + 650 __________ -55x __________ -55x 800 = 15x + 650 _____ -650 ________ - 650 150 = 15x

150 ____ 15

= 15x ____ 15

10 = x

y = 70x + 650 y = 70(10) + 650 y = 700 + 650 y = 1350 In 10 hours, the total cost for each garage will be the

same, $1350. Motor Works; 8 hours will cost $30 less at Motor

Works.

SOLvINg SySTEmS by ELImINATION

20. 4x + y = -1 _____________ + 2x - y = -5 6x + 0 = -6 6x = -6

6x ___ 6 = -6 ___

6

x = -1

4x + y = -1 4(-1) + y = -1 -4 + y = -1 ______ +4 ___ +4 y = 3 (-1, 3)

21. x + 2y = -1 _____________ -(x + y) = -2

x + 2y = -1 ______________ + (-x - y = -2) 0 + y = -3 y = -3

x + y = 2 x - 3 = 2 ____ + 3 ___ +3 x = 5 (5, -3)

22. (-2)(x + y = 12) _____________ + 2x + 5y = 27

-2x - 2y = -24 _____________ + 2x + 5y = 27 0 + 3y = 3 3y = 3

3y

___ 3 = 3 __

3

y = 1

x + y = 12 x + 1 = 12 ____ - 1 ___ -1 x = 11 (11, 1)


5-3

CS10_A1_MESK710372_C05.indd 188 3/30/11 11:03:53 PM

23. 3x - 2y = -6

___________________

+ (-9) ( 1 __ 3 x + 3y = 9)

3x - 2y = -6 __________________ + (-3x - 27y = -81) 0 - 29y = -87 -29y = -87

-29y

_____ -29

= -87 ____ -29

y = 3

3x - 2y = -6 3x - 2(3) = -6 3x - 6 = -6 ______ + 6 ___ +6 3x = 0

3x ___ 3 = 0 __

3

x = 0 (0, 3)

24. 3x + y = 2 3x + (-4x) = 2 -x = 2 -1(-x) = -1(2) x = -2

y = -4x y = -4(-2) y = 8 (-2, 8); possible answer: substitution; the second

equation is already solved for y, and y has a coefficient of 1 in the first equation.

25.

1

- 7

7 x y

y = -2x + 1

y = 1 _ 3

x - 6 (3, -5)

(3, -5); possible answer: graphing; both equations are already in slope-intercept form.

26. 2y = -3x 2(-2x + 2) = -3x 2(-2x) + 2(2) = -3x -4x + 4 = -3x _______ +4x ____ +4x 4 = x

y = -2x + 2 y = -2(4) + 2 y = -8 + 2 y = -6 (4, -6); possible answer: substitution; the second

equation is already solved for y.

27. x - y = 0 ___________ + 3x + y = 8 4x + 0 = 8 4x = 8

4x ___ 4 = 8 __

4

x = 2

3x + y = 8 3(2) + y = 8 6 + y = 8 ______ -6 ___ -6 y = 2 (2, 2); possible answer: elimination; the coefficients

of the y-terms are opposites.

SOLvINg SpECIAL SySTEmS

28. This system is inconsistent. It has no solution.

29. y = -x + 4 → y = -x + 4 x + y = 4 → y = -x + 4 This system is consistent and dependent. It has


30. This system is consistent and independent. It has one solution.

y = 2x 3x + 2 = 2x _______ -3x ____ -3x 2 = -x -1(2) = -1(-x) -2 = x

y = 2x y = 2(-2) y = -4 (-2, -4)

31. -4x - y = 6 → y = -4x - 6

1 __ 2 y = -2x - 3 → y = -4x - 6


32. x + 2y = 8 → y = - 1 __ 2 x + 4

y = - 1 __ 2 x + 4 → y = - 1 __

2 x + 4



5-4

CS10_A1_MESK710372_C05.indd 189 3/30/11 11:03:54 PM

33. y - 2x = -1 → y = 2x - 1 y + 2x = -5 → y = -2x - 5 This system is consistent and independent. It has

one solution. y - 2x = -1 _____________ + y + 2x = -5 2y + 0 = -6 2y = -6

2y

___ 2

= -6 ___ 2

y = -3

y + 2x = -5 -3 + 2x = -5 _______ +3 ___ +3 2x = -2

2x ___ 2 = -2 ___

2

x = -1 (-1, -3)

34. Yes. Let x represent the number of months after month 1 and let y represent the number of DVDs.

Tristan: y = 5x + 2 Marco: y = 4x + 8 The graphs of the two equations have different

slopes so they intersect. If the pattern continues, Tristan will eventually have the same number of DVDs as Marco. In fact, at month 7 (x = 6), Tristan and Marco will have 32 DVDs each.

35. This system is consistent and independent. It has one solution.


37. 2x + y = 2 → y = -2x + 2 y - 2 = -2x → y = -2x + 2 This system is consistent and dependent. It has


38. -3x - y = -5 → y = -3x + 5 y = -3x - 5 → y = -3x - 5 This system is inconsistent. It has no solution.

39. 2x + 3y = 1 → y = - 2 __ 3 x + 1 __

3

3x + 2y = 1 → y = - 3 __ 2 x + 1 __

2

This system is consistent and independent. It has one solution.

40. x + 1 __ 2 y = 3 → y = -2x + 6

2x = 6 - y → y = -2x + 6 This system is consistent and dependent. It has



SOLvINg LINEAr INEqUALITIES

42. ______________ y < 2x - 3 -3 2(0) - 3 -3 < -3 7 (0, -3) is not a solution.

43. ___________ y ≥ x - 3 -1 2 - 3 -1 ≥ -1 3 (2, -1) is a solution.

44. _____________ y > -3x + 4 0 -3(6) + 4 0 -18 + 4 0 > -14 3 (6, 0) is a solution.

45. ___________ y ≤ x - 3 10 10 - 3 10 ≤ 7 7 (10, 10) is not a

solution.

46. Graph the boundary line y = -2x + 5. Use a dashed line for <. The inequality is <, so shade below the line.

2

-2

x

y

0 4 2

47. x - y ≥ 2 ______ -x ___ -x -y ≥ -x + 2 -1(-y) ≤ -1(-x + 2) y ≤ x - 2 Graph the boundary line y = x - 2. Use a solid line

for ≤. The inequality is ≤, so shade below the line.

2

-2

x

y

0 2 -2

48. -x + 2y ≥ 6 _______ +x ___ +x 2y ≥ x + 6

2y

___ 2 ≥ x + 6 _____

2

y ≥ 1 __ 2 x + 3

Graph the boundary line y = 1 __ 2 x + 3. Use a solid


4

2

x

y

0 -4 -6 -2

49. Graph the boundary line y = -4x. Use a dashed line for >. The inequality is >, so shade above the line.

-2

x

y

0 2 -2


5-5

CS10_A1_MESK710372_C05.indd 190 3/30/11 11:03:57 PM

50. x + y + 4 > 0 __________ -x ___ -x y + 4 > -x _____ - 4 ___ -4 y > -x - 4 Graph the boundary line y = -x - 4. Use a dashed


-4

-2

x y

0 -4 -2

51. 5 - y ≥ 2x ______ -5 ___ -5 -y ≥ 2x - 5 -1(-y) ≤ -(2x - 5) y ≤ -2x + 5 Graph the boundary line y = -2x + 5. Use a

dashed line for ≤. The inequality is ≤, so shade below the line.

4

2

x

y

0 2 -2

52. Let x represent the number of slices of pizza and let y represent the number of bottles of soda.

2x + y ≥ 450 _______ -2x ____ -2x y ≥ -2x + 450 Since the club cannot sell a negative number of

items, the system is graphed only in Quadrant I. Graph the boundary line y = -2x + 450. Use a solid line for ≥. The inequality is ≥, so shade above the line.

Fundraising Needs

Slices of pizza

0

150

300

Bott

les

of le

mon

ade

450

450 300 150

Possible answer: (200, 50), (150, 150)

SOLvINg SySTEmS Of LINEAr INEqUALITIES

53. ______________ y > -2x + 9 3 -2(3) + 9 3 -6 + 9 3 > 3 7

________ y ≥ x 3 ≥ 3 3


54. ________________ 2x - y > -5 2(-1) - 0 -5 -2 > -5 3

_______________ y ≤ -3x - 3 0 -3(-1) - 3 0 3 - 3 0 ≤ 0 3


55.

4

-4

x

y

0 4 -4

Possible answer: solutions: (-6, 6), (-10, 0); not solutions: (0, 0), (4, -4)

56.

4

-4

x

y

0 4

Possible answer: solutions: (0, 0), (-5, 0); not solutions: (8, 0), (3, -3)

57. -x + 2y > 6 _______ +x ___ +x 2y > x + 6

2y

___ 2 > x + 6 _____

2

y > 1 __ 2 x + 3

x + y < 4 ______ -x ___ -x y < -x + 4

6

-6

x

y

0 4 2

Possible answer: solutions: (-6, 2), (-8, 1); not solutions: (0, 0), (4, 1)

58. x - y > 7 _______ -x ___ -x -y > -x + 7 -1(-y) < -1(-x + 7) y < x - 7

x + 3y ≤ 15 _______ -x ___ -x 3y ≤ -x + 15

3y

___ 3

≤ -x + 15 _______ 3

y ≤ - 1 __ 3

x + 5

4

-4

y

0 8 4

Possible answer: solutions: (8, -8), (9, 0); not solutions: (0, 0), (0, -4)

59. 4

-6

x

y

4 -6 0


5-6

CS10_A1_MESK710372_C05.indd 191 3/30/11 11:04:01 PM

60. 4x + 2y ≥ 10 ________ -4x ____ -4x 2y ≥ -4x + 10

2y

___ 2

≥ -4x + 10 ________ 2

y ≥ -2x + 5

6x + 3y < -9 ________ -6x ____ -6x 3y < -6x - 9

3y

___ 3 < -6x - 9 _______

3

y < -2x - 3

4

-6

x

y

6 -6

chAptEr tESt

1. ____________ y = -4x (-4) -4(1) -4 -4 3

_____________ y = 2x - 2 (-4) 2(1) - 2 -4 0 7


2. _______________ 3x - y = 1 3(0) - (-1) 1 0 + 1 1 1 1 3

_______________ x + 5y = -5 0 + 5(-1) -5 -5 -5 3


3. _______________ x - 2y = -1 (3) - 2(2) -1 3 - 4 -1 -1 -1 3

________________ -3x + 2y = 5 -3(3) + 2(2) 5 -9 + 4 5 -5 5 7


4. x

y

0 6 -2

-6

(0,-3)

y = x - 3

y = -2x - 3


5. 2x + y = -8 _______ -2x ____ -2x y = -2x - 8

x y 0 4 -2

-8

(-3, -2)

y = -2x - 8

y = 1 _ 3 x - 1

The solution appears to be at (-3, -2).

6. x = y + 2 ___ - 2 _____ - 2 x - 2 = y

x

y

0

6

6 -2

-2

(3, 1)

y = -x + 4

y = x - 2


7. y = -6

y = -2x - 2 -6 = -2x - 2 ___ +2 _______ + 2 -4 = -2x

-4 ___ -2

= -2x ____ -2

2 = x (2, -6)

8. -x + y = -4 -x + (2x - 11) = -4 x - 11 = -4 ______ + 11 ____ +11 x = 7

y = 2x - 11 y = 2(7) - 11 y = 14 - 11 y = 3 (7, 3)

9. x - 3y = 3 _____ + 3y ____ +3y x = 3y + 3

2x = 3y 2(3y + 3) = 3y 2(3y) + 2(3) = 3y 6y + 6 = 3y _______ -6y ____ -6y 6 = -3y

6 ___ -3

= -3y

____ -3

-2 = y

x - 3y = 3 x - 3(-2) = 3 x + 6 = 3 _____ - 6 ___ -6 x = -3 (-3, -2)

10. Let x represent the number of days and let y represent the total cost.

Pet Care: y = 30x + 15 Fido’s: y = 28x + 27

y = 28x + 27 30x + 15 = 28x + 27 _________ -28x _________ -28x 2x + 15 = 27 _______ - 15 ____ -15 2x = 12

2x ___ 2 = 12 ___

2

x = 6

y = 30x + 15 y = 30(6) + 15 y = 180 + 15 y = 195 In 6 days, the total cost for each service will be the

same, $195. Fido’s; it will cost less per day after 6 days.


CS10_A1_MESK710372_C05.indd 192 3/30/11 11:04:04 PM

11. 3x - y = 7 ___________ + 2x + y = 3 5x + 0 = 10 5x = 10

5x ___ 5 = 10 ___

5

x = 2

2x + y = 3 2(2) + y = 3 4 + y = 3 ______ -4 ___ -4 y = -1 (2, -1)

12. 4x + y = 0 ___________ -(x + y) = 3

4x + y = 0 _____________ + (-x - y = 3) 3x + 0 = 3 3x = 3

3x ___ 3 = 3 __

3

x = 1

x + y = -3 1 + y = -3 ______ -1 ___ -1 y = -4 (1, -4)

13. 2(2x + y = 3) _____________ + x - 2y = -1

4x + 2y = 6 ____________ + x - 2y = -1 5x + 0 = 5 5x = 5

5x ___ 5 = 5 __

5

x = 1

2x + y = 3 2(1) + y = 3 2 + y = 3 ______ -2 ___ -2 y = 1 (1, 1)

14. y = 6x - 1 → y = 6x - 1 6x - y = 1 → y = 6x - 1 This system is consistent and dependent. It has


15. y = -3x - 3 → y = -3x - 3 3x + y = 3 → y = -3x + 3 This system is inconsistent. It has no solution.

16. 2x - y = 1 → y = 2x - 1 -4x + y = 1 → y = 4x + 1 This system is consistent and independent. It has

one solution.

17. Graph the boundary line y = 2x - 5. Use a dashed line for <. The inequality is <, so shade below the line.

-4

-2

x y

0 3 -3

18. -y ≥ 8 -1(-y) ≤ -1(8) y ≤ -8 Graph the boundary line y = -8. Use a solid line for

≤. The inequality is ≤, so shade below the line.

2

-10

-4

x y

0 2 -2

19. Graph the boundary line y = 1 __ 3 x. Use a dashed line

for >. The inequality is >, so shade above the line.

2

-2

x

y

0 2

20.

-10

x y

0 2 -2

Possible answer: solutions: (2, 0), (4, 0); not solutions: (-2, 0), (-4, 0)

21. 3x - y > 3 _______ -3x ____ -3x -y > -3x + 3 -1(-y) < -1(-3x + 3) y < 3x - 3

2

-2

x

y

0 2 -2


22. y - 2x < 6 _____ + 2x ____ +2x y < 2x + 6

6

-6

x

y

6 -6

Possible answer: solutions: (0, 1), (0, 2); not solutions: (-6, 0), (4, 1)


CS10_A1_MESK710372_C05.indd 193 3/30/11 11:04:08 PM

23. Let x represent the number of coupons Tara sold and let y represent the number of coupons Erza sold.

x + y ≥ 150 y - 2x ≤ 30 The graph should only be in the first quadrant

because sales cannot be negative.

0

40

80

120

60 40 20

Coupon Book Sales

Tara

Ezra

Erza and Tara could have sold any combination represented by an ordered pair of whole numbers in the solution region.

Possible answer: (50, 125), (65, 95)


CS10_A1_MESK710372_C05.indd 194 3/30/11 11:04:09 PM

Documents

CHAPTER System of Equations and Inequalities 5 Solutions Key