Aat Solutions - Ch03

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115Algebra 2

Worked-Out Solution Key

Prerequisite Skills (p. 150)

1. The linear inequality that represents the graph shown at

the right is y < 2 3 } 4 x 1 3.

2. The graph of a linear inequality in two variables is the set of all points in a coordinate plane that are solutions of the inequality.

3. x 1 y 5 4 4. y 5 3x 2 3

1

x

y

21

1

x

y

21

5. 22x 1 3y 5 212 1

x

y

21

6. 2x 2 12 5 16 7. 23x 2 7 5 12

2x 5 28 23x 5 19

x 5 14 x 5 2 19

} 3

8. 22x 1 5 5 2x 2 5 9. y ≥ 2x 1 2

5 5 4x 2 5

1

x

y

21

10 5 4x

5 }

2 5 x

10. x 1 4y < 216 11. 3x 1 5y > 25

y < 2 1 } 4 x 2 4 y > 2

3 } 5 x 2 1

1

x

y

22

1

x

y

22

Lesson 3.1

Investigating Algebra Activity 3.1 (p. 152)

1. y1 5 y2 5 3 when x 5 21. The solution is (21, 3).

2. no solution; There are no values of x where y1 5 y2.


4. no solution; There are no values of x where y1 5 y2.


6. y1 5 y2 for all values of x. There are infi nitely many

solutions.

7. A system of linear equations can have either none, one, or infi nitely many solutions.

3.1 Guided Practice (pp. 153–155)

1. 3x 1 2y 5 24 x 1 3y 5 1

2y 5 23x 2 4 3y 5 1 2 x

y 5 2 3 } 2 x 2 2 y 5

1 } 3 2

1 } 3 x

1x

y

22

From the graph, the lines appear to intersect at (22, 1).

Check:

3x 1 2y 5 24 x 1 3y 5 1

3(22) 1 2(1) 0 24 22 1 3(1) 0 1

26 1 2 0 24 22 1 3 0 1

24 5 24 ✓ 1 5 1 ✓

The solution is (22, 1).

2. 4x 2 5y 5 210 2x 2 7y 5 4

4x 1 10 5 5y 2x 2 4 5 7y

4 } 5 x 1 2 5 y

2 } 7 x 2

4 } 7 5 y

1

x

y

21


Check:

4x 2 5y 5 210 2x 2 7y 5 4

4(25) 2 5(22) 0 210 2(25) 2 7(22) 0 4

220 1 10 0 210 210 1 14 0 4

210 5 210 ✓ 4 5 4 ✓


3. 8x 2 y 5 8 3x 1 2y 5 216

y 5 8x 2 8 2y 5 23x 2 16

y 5 2 3 } 2 x 2 8

4

x

y

22


Check:

8x 2 y 5 8 3x 1 2y 5 216

8(0) 2 (28) 0 8 3(0) 1 2(28) 0 216

0 1 8 0 8 0 2 16 0 216

8 5 8 ✓ 216 5 216 ✓


Chapter 3

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116Algebra 2Worked-Out Solution Key

4. 2x 1 5y 5 6

x

y

1

1

2x 1 5y 5 6

4x 1 10y 5 12

4x 1 10y 5 12

Each point on the line is a solution, and the system has infinitely many solutions. The system is consistent and dependent.

5. 3x 2 2y 5 10

x

y

1

1

3x 2 2y 5 10

3x 2 2y 5 2

3x 2 2y 5 2

The two lines have no point of intersection, so the system has no solution. The system is inconsistent.

6. 22x 1 y 5 5

x

y

1

1

22x 1 y 5 5

y 5 2x 1 2(21, 3)

y 5 2x 1 2

The lines intersect at (21, 3), so (21, 3) is the solution. The system is consistent and independent.

7. y 5 1 p x 1 36 y 5 2.5 p x

y 5 x 1 36 y 5 2.5x

Number of rides

To

tal c

ost

(d

olla

rs)

x

y

100 20 30 40

20

0

40

60

y 5 2.5x

y 5 x 1 36(24, 60)

The lines appear to intersect at (24, 60).

Check:

y 5 x 1 36 y 5 2.5x

60 0 24 1 36 60 0 2.5(24)

60 5 60 ✓ 60 5 60 ✓

The total costs are equal after 24 rides. If the monthly pass is increased to $36, it will take longer for both options to cost the same.

3.1 Exercises (pp. 156–158)

Skill Practice

1. A consistent system that has exactly one solution is called independent.

2. If the two lines intersect at one point, then the solution is the point of intersection. If the two lines are parallel, there is no solution. If the two lines coincide then there are infinitely many solutions.

3.

1

x

y

21


y 5 23x 1 2 y 5 2x 2 3

21 0 23(1) 1 2 21 0 2(1)23

21 0 23 1 2 21 0 2 2 3

21 5 21 ✓ 21 5 21 ✓


4. 1

x

y

21


y 5 5x 1 2 y 5 3x

23 0 5(21) 1 2 23 0 3(21)

23 0 25 1 2 23 5 23 ✓

23 5 23 ✓


5.

1

x

y

21


y 5 2x 1 3 2x 2 3y 5 21

21 0 2(4) 1 3 2(4) 2 3(21) 0 21

21 0 24 1 3 24 1 3 0 21

21 5 21 ✓ 21 5 21 ✓


Chapter 3, continued

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117Algebra 2


6.

2

x

y

21


x 1 2y 5 2 x 2 4y 5 14

6 1 2(22) 0 2 6 2 4(22) 0 14

6 2 4 0 2 6 1 8 0 14

2 5 2 ✓ 14 5 14 ✓


7.

1

x

y

21


y 5 2x 2 10 x 2 4y 5 5

0 0 2(5) 2 10 5 2 4(0) 0 5

0 0 10 2 10 5 2 0 0 5

0 5 0 ✓ 5 5 5 ✓


8.

1

x

y

22


2x 1 6y 5 212 x 1 6y 5 12

2(12) 1 6(0) 0 212 12 1 6(0) 0 12

212 1 0 0 212 12 1 0 0 12

212 5 212 ✓ 12 5 12 ✓


9.

1

y

x21


y 5 23x 2 2 5x 1 2y 5 22

4 0 23(22) 2 2 5(22) 1 2(4) 0 22

4 0 6 2 2 210 1 8 0 22

4 5 4 ✓ 22 5 22 ✓


10.

1

x

y

21


y 5 23x 2 13 2x 2 2y 5 24

5 0 23(26) 2 13 2(26)22(5) 0 24

5 0 18 2 13 6 2 10 0 24

5 5 5 ✓ 24 5 24 ✓


11.

1

x

y

21

x 2 7y 5 6 23x 1 21y 5 218

7y 5 6 2 x 21y 5 3x 2 18

y 5 x } 7 2

6 } 7 y 5

x } 7 2

6 } 7

The graphs of the equations are the same line. Each point on the line is a solution. There are infi nitely many solutions.

12.

3

x

y

22

y 5 4x 1 3 20x 2 5y 5 215

25y 5 215 2 20x

y 5 4x 1 3

The graphs of the equations are the same line. Each point on the line is a solution, so there are infi nitely many solutions.

13.

1

x

y

21

From the graph, the lines appear to intersect at (3.5, 2.2).

4x 2 5y 5 3 3x 1 2y 5 15

4(3.5) 2 5(2.2) 0 3 3(3.5) 1 2(2.2) 0 15

14 2 11 0 3 10.5 1 4.4 0 15

3 5 3 ✓ 14.9 ø 15 ✓

The solution is approximately (3.5, 2.2).


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14.1

x

y

21


7x 1 y 5 217 3x 2 10y 5 24

7(22) 1 (23) 0 217 3(22) 2 10(23) 0 24

214 2 3 0 217 26 1 30 0 24

217 5 217 ✓ 24 5 24 ✓


15. C;

x

y1

1

7x 1 2y 5 25

24x 2 y 5 2

(1, 26)


24x 2 y 5 2 7x 1 2y 5 25

24(1) 2 (26) 0 2 7(1) 1 2(26) 0 25

24 1 6 0 2 7 2 12 0 25

2 5 2 ✓ 25 5 25 ✓


16. The student did not check the solution in the second equation.

3x 2 2y 5 2 x 1 2y 5 6

3(0) 2 2(21) 0 2 0 1 2(21) 0 6

0 1 2 0 2 22 Þ 6

2 5 2 ✓

The solution is not (0, 21).

17.

x

y

1

1

y 5 21 (2, 21)

3x 1 y 5 5


y 5 21 3x 1 y 5 5

21 5 21 ✓ 3(2) 1 (21) 0 5

6 2 1 0 5

5 5 5 ✓

The solution is (2, 21). The system is consistent and independent.

18.

x

y

1

1

(3, 2)

2x 2 y 5 4

x 2 2y 5 21


2x 2 y 5 4 x 2 2y 5 21

2(3) 2 2 0 4 3 2 2(2) 0 21

6 2 2 0 4 3 2 4 0 21

4 5 4 ✓ 21 5 21 ✓


19.

x

y

2

1

y 5 3x 2 2

y 5 3x 1 2

y 5 3x 1 2 y 5 3x 2 2

The graphs of the equations are two parallel lines. The system has no solution. The system is inconsistent.

20.

x

y

1

1

y 5 2x 2 1

26x 1 3y 5 23

y 5 2x 2 1 26x 1 3y 5 23

3y 5 6x 2 3

y 5 2x 2 1

The graphs of the equations are the same line. The system has infi nitely many solutions. The system is consistent and dependent.


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119Algebra 2


21.

x

y

1

2

220x 1 12y 5 224

5x 2 3y 5 6

220x 1 12y 5 224 5x 2 3y 5 6

12y 5 20x 2 24 23y 5 25x 1 6

y 5 5 } 3 x 2 2 y 5

5 } 3 x 2 2


22.

x

y

2

1

4x 2 5y 5 0

3x 2 5y 5 25(5, 4)


4x 2 5y 5 0 3x 2 5y 5 25

4(5) 2 5(4) 0 0 3(5) 2 5(4) 0 25

20 2 20 0 0 15 2 20 0 25

0 5 0 ✓ 25 5 25 ✓


The system is consistent and independent.

23.

x

y

2

1

3x 1 7y 5 62x 1 9y 5 4

(2, 0)


3x 1 7y 5 6 2x 1 9y 5 4

3(2) 1 7(0) 0 6 2(2) 1 9(0) 0 4

6 1 0 0 6 4 1 0 0 4

6 5 6 ✓ 4 5 4 ✓



24.

x

y

2

21

4x 1 5y 5 3

6x 1 9y 5 9

(23, 3)


4x 1 5y 5 3 6x 1 9y 5 9

4(23) 1 5(3) 0 3 6(23) 1 9(3) 0 9

212 1 15 0 3 218 1 27 0 9

3 5 3 ✓ 9 5 9 ✓



25.

x

y

1

1

8x 1 9y 5 15

5x 2 2y 5 17

(3, 21)


8x 1 9y 5 15 5x 2 2y 5 17

8(3) 1 9(21) 0 15 5(3) 2 2(21) 0 17

24 2 9 0 15 15 1 2 0 17

15 5 15 ✓ 17 5 17 ✓



26.

x

y

2

8

(8, 22)

x 2 3y 5 1012

x 1 2y 5 2214


1 }

2 x 2 3y 5 10

1 }

4 x 1 2y 5 22

1 }

2 (8) 2 3(22) 0 10

1 }

4 (8) 1 2(22) 0 22

4 1 6 0 10 2 2 4 0 22

10 5 10 ✓ 22 5 22 ✓




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27.

x

y

2

2

3x 2 2y 5 215

x 2 y 5 2523

3x 2 2y 5 215 x 2 2 } 3 y 5 25

22y 5 23x 2 15 2 2 } 3 y 5 2x 2 5

y 5 3 } 2 x 1

15 } 2 y 5

3 } 2 x 1

15 } 2


28.

x

y

1

1

52 x 2 y 5 24

145x 2 2y 5

5 }

2 x 2 y 5 24 5x 2 2y 5

1 } 4

2y 5 2 5 } 2 x 2 4 22y 5 25x 1

1 } 4

y 5 5 } 2 x 1 4 y 5

5 } 2 x 2

1 } 8

The graphs of the equations are parallel lines. The system has no solution, so it is inconsistent.

29. A;

x

y

1

1

3x 1 4y 5 26

212x 1 16y 5 10

212x 1 16y 5 10 3x 1 4y 5 26

There is exactly one solution of the system. The system is consistent and independent.

30. a. Sample answer: 3x 1 2y 5 9

2x 1 y 5 5

b. Sample answer: 3x 2 y 5 2

26x 1 2y 5 10

c. Sample answer: 2x 2 3y 5 4

24x 1 6y 5 28

31.

1

x

y

22

y 5 x 1 2 y 5 x

There is no solution.

32.

1

x

y

21

From the graph, the lines appear to intersect at (2.5, 1.5).

y 5 x 2 1 y 5 2x 1 4

1.5 0 2.5 2 1 1.5 0 22.5 1 4

1.5 0 1.5 1.5 5 1.5 ✓

1.5 5 1.5 ✓

The solution is (2.5, 1.5).

33.

1

x

y

22

From the graph, the lines appear to intersect at (24, 2) and (4, 2).

y 5 x 22 y 5 2

2 0 24 2 2 2 5 2 ✓

2 0 4 2 2

2 5 2 ✓

2 0 4 2 2

2 0 4 2 2 2 5 2 ✓

The solutions are (24, 2) and (4, 2).

34. a. The system is consistent and independent when a Þ c.

b. The system is consistent and dependent when a 5 c and b 5 d.

c. The system is inconsistent when a 5 c and b Þ d.

Problem Solving

35. Let x 5 hrs as lifeguard.

20 4 6 8 10 12 14

20

468

10121416

(6, 8)

Hours as a lifeguard

Ho

urs

as

a ca

shie

r

x

y

Let y 5 hrs as cashier.

x 1 y 5 14

8x 1 6y 5 96

You worked 6 hours as a lifeguard and 8 hours as acashier last week.


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121Algebra 2


36. Let x 5 warnings.

x

y

x 1 y 5 375

y 1 37 5 x

00

80

120

240

320

80 160 240 320

(206, 169)

Number of warnings

Nu

mb

er o

f ti

cket

s

Let y 5 speeding tickets.

x 1 y 5 375

y 1 37 5 x

The state trooper issued 206 warnings and 169 speeding tickets.

37. Let y 5 total cost (in dollars).

Let x 5 days.

0 4 62 8 10 12 14 160

5025

75100

150125

175

(11, 132)

Number of days

Tota

l co

st (

do

llars

)

x

y y 5 12x

y 5 121 1 x

Option A: y 5 121 1 x

Option B: y 5 12x

The plans are equal when used for 11 days. If the daily cost of Option B increases, the plans will be equal in fewer days. The graph of the new option B equation will be the rotation of y 5 12x counterclockwise and the x-intersection with the graph of the option A equation will be less than 11.

38. a. Let y 5 total cost (in dollars).

Let x 5 years since purchase.

Refrigerator A: y 5 600 1 50x

Refrigerator B: y 5 1200 1 40x

b.

B

A

(60, 3600)

Co

st (

do

llars

)

0200 40 60 80

1000

2000

3000

4000

Number of years

x

y

In 60 years the total costs of owning the refrigerators will be equal.

c. No. It is not likely that the refrigerators would be in use for 60 years. You can conclude that, for the life of the refrigerators, the cost of owning refrigerator A will always be less than the cost of owning refrigerator B.

39. a. m 5 20.09583x 1 50.84 (Men)

b. w 5 20.1241x 1 57.08 (Women)

c.

0 80 160 240 32008

16243240485664

Years since 1972

Win

nin

g t

imes

(se

con

ds)

x

y

men

women

The lines appearto intersect at(220.7, 29.7).

You can predict that the women’s performance will catch up to the men’s performance about 221 years after 1972, or in 2193.

d. No. The lines may be good models for the years contained in the data set, but eventually the swimming times will start leveling off as swimmers approach the maximum swimming speed for a human. So, it is not reasonable to assume that the winning times will continue to decrease indefi nitely at the given rates.

40. a. Distance from park(ft)

5 Total distance(ft)

2 Speed(ft/sec)

p Time(sec)

d 5 5000 2 25 p t

An equation is d 5 5000 2 25t.

b. Let d 5 0 to fi nd when you reach the park.

0 5 5000 2 25t

d 5 5000 2 25t

d 5 3000 2 15tDis

tan

ce f

rom

par

k (f

eet)

0500 100 150 200

1000

2000

4000

5000

3000

Time (seconds)

t

d

(200, 0)

25t 5 5000

t 5 200

Use the verbal model from part (a) to write the equation d 5 3000 2 rt for your friend’s distance d from the park after t seconds, where r represents your friend’s speed. Let d 5 0 and t 5 200 and solve for r.

0 5 3000 2 r (200)

r 5 15

Your friend travels at a speed of 15 feet per second.

c. An equation for your friend’s distance is

d 5 3000 2 15t.

Mixed Review

41. 8x 1 1 5 3x 2 14

8x 5 3x 2 15

5x 5 215

x 5 23

42. 24(x 1 3) 5 5x 1 9

24x 2 12 5 5x 1 9

24x 2 21 5 5x

221 5 9x

2 7 } 3 5 x

43. x 1 2 5 3

} 2 x 2 5 } 4

x 1 13

} 4 5 3 } 2 x

13

} 4 5

1 } 2 x

13

} 2 5 x


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44. x 2 18 5 9

x 2 18 5 9 or x 2 18 5 29

x 5 27 or x 5 9

45. 2x 1 5 5 12

2x 1 5 5 12 or 2x 1 5 5 212

2x 5 7 or 2x 5 217

x 5 7 } 2 or x 5 2

17 } 2

46. 5x 2 18 5 17

5x 2 18 5 17 or 5x 2 18 5 217

5x 5 35 or 5x 5 1

x 5 7 or x 5 1 } 5

47. 3x 2 2y 5 8 48. 25x 1 y 5 212

22y 5 23x 1 8 y 5 5x 2 12

y 5 3 } 2 x 2 4 When x 5 9:

When x 5 22: y 5 5(9) 2 12

y 5 3 } 2 (22) 2 4 y 5 45 2 12

y 5 23 2 4 y 5 33

y 5 27

49. 8x 2 3y 5 10 50. 8x 2 2y 5 7

23y 5 28x 1 10 22y 5 28x 1 7

y 5 8 } 3 x 2

10 } 3 y 5 4x 2

7 } 2

When x 5 8: When x 5 21:

y 5 8 } 3 (8) 2

10 } 3 y 5 4(21) 2

7 } 2

y 5 64

} 3 2 10

} 3 y 5 24 2 7 } 2

y 5 18 y 5 2 15

} 2

51. 16x 1 9y 5 224 52. 212x 1 9y 5 260

9y 5 216x 2 24 9y 5 12x 2 60

y 5 2 16

} 9 x 2 8 } 3 y 5

4 } 3 x 2

20 } 3

When x 5 26: When x 5 27:

y 5 2 16

} 9 (26) 2 8 } 3 y 5

4 } 3 (27) 2

20 } 3

y 5 32

} 3 2 8 } 3 y 5 2

28 } 3 2

20 } 3

y 5 8 y 5 216

53. C 5 5 } 9 (F 2 32)

C 5 5 } 9 (101 2 32)

C 5 5 } 9 (69)

C 5 115

} 3 ø 38.3

Your dog’s temperature is about 38.38C. Your dog does not have a fever.

3.1 Graphing Calculator Activity (p. 159)

1. The solution is about (2.3,20.3).

2. The solution is about (2.71, 9.57).


4. The solution is (24,243).



7. Let x 5 days in San Antonio.

Let y 5 say in Dallas.

x 1 y 5 7

275x 1 400y 5 2300

Using a graphing calculator, the solution is (4, 3). You should spend 4 days in San Antonio and 3 days in Dallas.

8. Let x 5 number of adult tickets sold.

Let y 5 number of child tickets sold.

x 1 y 5 800

7x 1 5y 5 4600

Using a graphing calculator, the solution is (300, 500). The movie theater admitted 300 adults and 500 children that day.

Lesson 3.2


1. 4x 1 3y 5 22

x 1 5y 5 29 → x 5 29 2 5y

When x 5 29 2 5y:

4x 1 3y 5 22

4(29 2 5y) 1 3y 5 22

236 2 20y 1 3y 5 22

217y 5 34

y 5 22

When y 5 22:

x 5 29 2 5y

x 5 29 2 5(22)

x 5 29 1 10

x 5 1


Check: 4x 1 3y 5 22 x 1 5y 5 29

4(1) 1 3(22) 0 22 1 1 5(22) 0 29

4 2 6 0 22 1 2 10 0 29

22 5 22 ✓ 29 5 29 ✓


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123Algebra 2



2. 3x 1 3y 5 215 3 3 9x 1 9y 5 245

5x 2 9y 5 3 5x 2 9y 5 3

14x 5 242

x 5 23

When x 5 23:

3x 1 3y 5 215

3(23) 1 3y 5 215

29 1 3y 5 215

y 5 22


Check: 3x 1 3y 5 215 5x 2 9y 5 3

3(23) 1 3(22) 0 215 5(23) 2 9(22) 0 3

29 2 6 0 215 215 1 18 0 3

215 5 215 ✓ 3 5 3 ✓

3. 3x 2 6y 5 9 → x 5 2y 1 3

24x 1 7y 5 216

When x 5 2y 1 3:

24x 1 7y 5 216

24(2y 1 3) 1 7y 5 216

28y 2 12 1 7y 5 216

2y 5 24

y 5 4

When y 5 4:

3x 2 6y 5 9

3x 2 6(4) 5 9

3x 5 33

x 5 11


Check: 3x 2 6y 5 9 24x 1 7y 5 216

3(11) 2 6(4) 0 9 24(11) 1 7(4) 0 216

33 2 24 0 9 244 1 28 0 216

9 5 9 ✓ 216 5 216 ✓

4.ShortSleeve Cost($/shirt)

p Short Sleeve Shirts(shirts)

1

LongSleeveCost($/shirt)

p LongSleeveShirts(shirts)

5

TotalCost($)

5 p x 1 7 p y 5 3715

ShortSleeve SellingPrice($/shirt)

p

Short Sleeve Shirts(shirts)

1

LongSleeveSellingPrice($/shirt)

p

LongSleeveShirts(shirts)

5

Totalrevenue($)

8 p x 1 12 p y 5 6160

5x 1 7y 5 3715 3 28 240x 2 56y 5 229,720

8x 1 12y 5 6160 3 5 40x 1 60y 5 30,800

4y 5 1080

y 5 270

When y 5 270:

5x 1 7y 5 3715

5x 1 7(270) 5 3715

5x 5 1825

x 5 365

The school sold 365 short sleeve T-shirts and 270 long sleeve T-shirts.

5. 12x 2 3y 5 29

24x 1 y 5 3 → y 5 4x 1 3

When y 5 4x 1 3:

12x 2 3y 5 29

12x 2 3(4x 1 3) 5 29

12x 2 12x 2 9 5 29

29 5 29

There are infi nitely many solutions.

6. 6x 1 15y 5 212 6x 1 15y 5 212

22x 2 5y 5 9 3 3 26x 2 15y 5 227

0 Þ 239


7. 5x 1 3y 5 20 5x 1 3y 5 20

2x 2 3 } 5 y 5 24 3 5 25x 2 3y 5 220

0 5 0


8. 12x 2 2y 5 21 12x 2 2y 5 21

3x 1 12y 5 24 3 (24) 212x 2 48y 5 16

250y 5 37

y 5 2 37

} 50

When y 5 2 37

} 50 :

12x 2 2y 5 21

12x 2 2(2 37

} 50 ) 5 21

12x 1 37

} 25 5 21

12x 5 488

} 25

x 5 122

} 75

The solution is 1 122 } 75 ,2

37 } 50 2 .

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9. 8x 1 9y 5 15 3 5 40x 1 45y 5 75

5x 2 2y 5 17 3 (28) 240x 1 16y 5 2136

61y 5 261

y 5 21

When y 5 21:

8x 1 9y 5 15

8x 1 9(21) 5 15

8x 2 9 5 15

x 5 3


10. 5x 1 5y 5 5 5x 1 5y 5 5

5x 1 3y 5 4.2 3 (21) 25x 2 3y 5 24.2

2y 5 0.8

y 5 0.4

When y 5 0.4:

5x 1 5y 5 5

5x 1 5(0.4) 5 5

5x 1 2 5 5

x 5 0.6

The solution is (0.6, 0.4).


Skill Practice

1. To solve a linear system where one of the coeffi cients is 1 or 21, it is usually easiest to use the substitution method.

2. Multiply one (or both) of the equations by a constant to obtain coeffi cients that differ only in sign for one of the variables. Add the revised equations together and solve for the remaining variable. Substitute this value into either of the original equations and solve for the other variable.

3. 2x 1 5y 5 7

x 1 4y 5 2 → x 5 2 2 4y

When x 5 2 2 4y: When y 5 21:

2x 1 5y 5 7 x 5 2 2 4y

2(2 2 4y) 1 5y 5 7 x 5 2 2 4(21)

4 2 8y 1 5y 5 7 x 5 6

23y 5 3

y 5 21

The solution is (6,21).

4. 3x 1 y 5 16 → y 5 23x 1 16

2x 2 3y 5 24

When y 5 23x 1 16: When x 5 4:

2x 2 3(23x 1 16) 5 24 3x 1 y 5 16

2x 1 9x 2 48 5 24 3(4) 1 y 5 16

11x 5 44 12 1 y 5 16

x 5 4 y 5 4


5. 6x 2 2y 5 5

23x 1 y 5 7 → y 5 3x 1 7

When y 5 3x 1 7:

6x 2 2(3x 1 7) 5 5

6x 2 6x 2 14 5 5

214 Þ 5


6. x 1 4y 5 1 → x 5 1 2 4y

3x 1 2y 5 212

When x 5 1 2 4y: When y 5 3 } 2 :

3(1 2 4y) 1 2y 5 212 x 1 4 1 3 } 2 2 5 1

3 2 12y 1 2y 5 212 x 1 6 5 1

210y 5 215 x 5 25

y 5 3 } 2

The solution is 1 25, 3 }

2 2 .

7. 3x 2 y 5 2 → y 5 3x 2 2

6x 1 3y 5 14

When y 5 3x 2 2: When x 5 4 } 3 :

6x 1 3(3x 2 2) 5 14 3 1 4 }

3 2 2 y 5 2

6x 1 9x 2 6 5 14 4 2 y 5 2

15x 5 20 y 5 2

x 5 4 } 3

The solution is 1 4 }

3 , 2 2 .

8. 3x 2 4y 5 25

2x 1 3y 5 25 → x 5 5 1 3y

When x 5 5 1 3y: When y 5 24:

3(5 1 3y) 2 4y 5 25 2x 1 3(24) 5 25

15 1 9y 2 4y 5 25 2x 2 12 5 25

5y 5 220 x 5 27

y 5 24


9. 3x 1 2y 5 6

x 2 4y 5 212 → x 5 4y 2 12

When x 5 4y 2 12: When y 5 3:

3(4y 2 12) 1 2y 5 6 x 2 4(3) 5 212

12y 2 36 1 2y 5 6 x 2 12 5 212

14y 5 42 x 5 0

y 5 3


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125Algebra 2



10. 6x 2 3y 5 15

22x 1 y 5 25 → y 5 2x 2 5

When y 5 2x 2 5:

6x 2 3(2x 2 5) 5 15

6x 2 6x 1 15 5 15

15 5 15


11. 3x 1 y 5 21 → y 5 23x21

2x 1 3y 5 18

When y 5 23x 2 1: When x 5 23:

2x 1 3(23x 2 1) 5 18 3(23) 1 y 5 21

2x 2 9x 2 3 5 18 y 5 8

27x 5 21

x 5 23


12. 2x 2 y 5 1 → y 5 21 1 2x

8x 1 4y 5 6

When y 5 21 1 2x: When x 5 5 } 8 :

8x 1 4(21 1 2x) 5 6 2 1 5 } 8 2 2 y 5 1

8x 2 4 1 8x 5 6 5 }

4 2 y 5 1

16x 5 10 y 5 1 } 4

x 5 5 } 8

The solution is 1 5 } 8

, 1

} 4 2 .

13. 3x 1 7y 5 13

x 1 3y 5 2 7 → x 5 27 2 3y

When x 5 27 2 3y: When y 5 217:

3(27 2 3y) 1 7y 5 13 x 1 3(217) 5 27

221 2 9y 1 7y 5 13 x 2 51 5 27

22y 5 34 x 5 44

y 5 217


14. 2x 1 5y 5 10

23x 1 y 5 36 → y 5 3x 1 36

When y 5 3x 1 36: When x 5 210:

2x 1 5(3x 1 36) 5 10 23(210) 1 y 5 36

2x 1 15x 1 180 5 10 30 1 y 5 36

17x 5 2170 y 5 6

x 5 210


15. 2x 1 6y 5 17 2x 1 6y 5 17

2x 2 10y 5 9 3 (21) 22x 1 10y 5 29

16y 5 8

When y 5 1 } 2 : y 5

1 } 2

2x 1 6 1 1 } 2 2 5 17

2x 1 3 5 17

x 5 7

The solution is 1 7, 1 }

2 2 .

16. 4x 2 2y 5 216 3 2 8x 2 4y 5 232

23x 1 4y 5 12 23x 1 4y 5 12

5x 5 220

When x 5 24: x 5 24

4(24) 2 2y 5 216

216 2 2y 5 216

y 5 0


17. 3x 2 4y 5 210 3 (22) 26x 1 8y 5 20

6x 1 3y 5 242 6x 1 3y 5 242

11y 5 222

When y 5 22: y 5 22

3x 2 4(22) 5 210

3x 1 8 5 210

x 5 26


18. 4x 2 3y 5 10 3 (22) 28x 1 6y 5 220

8x 2 6y 5 20 8x 2 6y 5 20

0 5 0


19. 5x 2 3y 5 23 3 2 10x 2 6y 5 26

2x 1 6y 5 0 2x 1 6y 5 0

12x 5 26

When x 5 2 1 } 2 : x 5 2

1 } 2

5 1 2 1 } 2 2 2 3y 5 23

2 5 } 2 2 3y 5 23

23y 5 2 1 } 2

y 5 1 } 6

The solution is 1 2 1

} 2 , 1 }

6 2 .

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20. 10x 2 2y 5 16 10x 2 2y 5 16

5x 1 3y 5 212 3 (22) 210x 2 6y 5 24

28y 5 40

When y 5 25: y 5 25

10x 2 2(25) 5 16

10x 1 10 5 16

x 5 3 } 5

The solution is 1 3 } 5 , 25 2 .21. 2x 1 5y 5 14 3 (23) 26x 2 15y 5 242

3x 2 2y 5 236 3 2 6x 2 4y 5 272

219y 5 2114

When y 5 6: y 5 6

2x 1 5(6) 5 14

2x 1 30 5 14

x 5 28


22. 7x 1 2y 5 11 3 (23) 221x 2 6y 5 233

22x 1 3y 5 29 3 2 24x 1 6y 5 58

225x 5 25

When x 5 21: x 5 21

7(21) 1 2y 5 11

27 1 2y 5 11

y 5 9


23. 3x 1 4y 5 18 3 (22) 26x 2 8y 5 236

6x 1 8y 5 18 6x 1 8y 5 18

0 Þ 218


24. 2x 1 5y 5 13 3 (23) 26x 2 15y 5 239

6x 1 2y 5 213 6x 1 2y 5 213

213y 5 252

When y 5 4: y 5 4

2x 1 5(4) 5 13

2x 1 20 5 13

x 5 2 7 } 2

The solution is 1 2 7 } 2 , 4 2 .

25. 4x 2 5y 5 13 3 (23) 212x 1 15y 5 239

6x 1 2y 5 48 3 2 12x 1 4y 5 96

19y 5 57

When y 5 3: y 5 3

4x 2 5(3) 5 13

4x 2 15 5 13

x 5 7


26. 6x 2 4y 5 14 3 2 12x 2 8y 5 28

2x 1 8y 5 21 2x 1 8y 5 21

14x 5 49

When x 5 7

} 2 : x 5 7 } 2

6 1 7 } 2

2 2 4y 5 14

21 2 4y 5 14

y 5 7 } 4

The solution is 1 7 } 2

, 7 }

4 2 .

27. The error was made when multiplying the fi rst equation by the constant 22. Not every term was multiplied by 22.

3x 1 2y 5 7 3 (22) 26x 2 4y 5 214

5x 1 4y 5 15 5x 1 4y 5 15

2x 5 21

x 5 1

28. 3x 1 2y 5 11

4x 1 y 5 22 → y 5 24x 2 2

When y 5 24x 2 2: When x 5 23:

3x 1 2(24x 2 2) 5 11 y 5 24(23) 2 2

3x 2 8x 2 4 5 11 y 5 10

25x 5 15

x 5 23


29. 2x 2 3y 5 8 3 2 4x 2 6y 5 16

24x 1 5y 5 210 24x 1 5y 5 210

2y 5 6

When y 5 26: y 5 26

2x 2 3(26) 5 8

2x 1 18 5 8

x 5 25



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127Algebra 2


30. 3x 1 7y 5 21 3 (22) 26x 2 14y 5 2

2x 1 3y 5 6 3 3 6x 1 9y 5 18

25y 5 20

When y 5 24: y 5 24

3x 1 7(24) 5 21

3x 2 28 5 21

x 5 9


31. 4x 2 10y 5 18 4x 2 10y 5 18

22x 1 5y 5 29 3 2 24x 1 10y 5 218

0 5 0


32. 3x 2 y 5 22 → y 5 3x 1 2

5x 1 2y 5 15

When y 5 3x 1 2: When x 5 1:

5x 1 2(3x 1 2) 5 15 y 5 3(1) 1 2

5x 1 6x 1 4 5 15 y 5 5

11x 5 11

x 5 1


33. x 1 2y 5 28 → x 5 28 2 2y

3x 2 4y 5 224

When x 5 2822y: When y 5 0:

3(28 2 2y) 2 4y 5 224 x 5 28 2 2(0)

224 2 6y 2 4y 5 224 x 5 28

210y 5 0

y 5 0


34. 2x 1 3y 5 26 3 (23) 26x 2 9y 5 18

3x 2 4y 5 25 3 2 6x 2 8y 5 50

217y 5 68

When y = 24: y 5 24

2x 1 3(24) 5 26

2x 2 12 5 26

x 5 3


35. 3x 1 y 5 15 → y 5 23x 1 15

2x 1 2y 5 219

When y 5 23x 1 15: When x 5 7:

2x 1 2(23x 1 15) 5 219 y 5 23(7) 1 15

2x 2 6x 1 30 5 219 y 5 26

27x 5 249

x 5 7


36. 4x 2 3y 5 8 3 2 8x 2 6y 5 16

28x 1 6y 5 16 28x 1 6y 5 16

0 Þ 32


37. 4x 2 y 5 210 → y 5 4x 1 10

6x 1 2y 5 21

When y 5 4x 1 10: When x 5 2 3 } 2 :

6x 1 2(4x 1 10) 5 21 y 5 4 1 2 3 } 2 2 1 10

6x 1 8x 1 20 5 21 y 5 4

14x 5 221

x 5 2 3 } 2

The solution is 1 2 3 } 2 , 4 2 .

38. 7x 1 5y 5 212 3 4 28x 1 20y 5 248

3x 2 4y 5 1 3 5 15x 2 20y 5 5

43x 5 243

When x 5 21: x 5 21

7(21) 1 5y 5 212

5y 5 25

y 5 21


39. 2x 1 y 5 21 → y 5 22x 2 1

24x 1 6y 5 6

When y 5 22x 2 1: When x 5 2 3 } 4 :

24x 1 6(22x 2 1) 5 6 y 5 22 1 2 3 } 4 2 2 1

24x 2 12x 2 6 5 6 y 5 1 } 2

216x 5 12

x 5 2 3 } 4

The solution is 1 2 3 } 4 ,

1 }

2 2 .

40. B; 3x 1 2y 5 4 3 (22) 26x 2 4y 5 28

6x 2 3y 5 227 6x 2 3y 5 227

27y 5 235

When y 5 5: y 5 5

3x 1 2(5) 5 4

3x 5 26

x 5 22



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41. Let (x1, y1) 5 (1, 4) and (x2, y2) 5 (5, 0).

m 5 0 2 4

} 5 2 1 5 21

y 2 y1 5 m(x, 2 x1)

y 2 4 5 21(x 2 1)

y 5 2x 1 5

Let (x1, y1) 5 (0, 2) and (x2, y2) 5 (4, 4).

m 5 4 2 2

} 4 2 0 5

1 } 2

y 2 y1 5 m(x 2 x1)

y 2 2 5 1 } 2 (x 2 0)

y 5 1 } 2 x 1 2

System of equations:

y 5 2x 1 5

y 5 1 } 2 x 1 2

When y 5 2x 1 5: When x 5 2:

2x 1 5 5 1 } 2 x 1 2 y 5 2(2) 1 5

3 5 3 } 2 x y 5 3

2 5 x

The diagonals of the quadrilateral intersect at (2, 3).

42. Let (x1, y1) 5 (6, 1) and (x2, y2) 5 (3, 7).

m 5 7 2 1

} 3 2 6 5 22

y 2 y1 5 m(x 2 x1)

y 2 1 5 22(x2 6)

y 2 1 5 22x 1 12

y 5 22x 1 13

Let (x1, y1) 5 (1, 6) and (x2, y2) 5 (7, 4).

m 5 4 2 6

} 7 2 1 5 2 1 } 3

y 2 y1 5 m(x 2 x1)

y 2 6 5 2 1 } 3 (x 2 1)

y 2 6 5 2 1 } 3 x 1

1 } 3

y 5 2 1 } 3 x 1

19 } 3


y 5 22x 1 13

y 5 2 1 } 3 x 1

19 } 3

When y 5 22x 1 13: When x 5 4:

22x 1 13 5 2 1 } 3 x 1

19 } 3 y 5 22(4) 1 13

2 5 } 3 x 5 2

20 } 3 y 5 5

x 5 4


43. Let (x1, y1) 5 (7, 0) and (x2, y2) 5 (1, 3).

m 5 3 2 0

} 1 2 7 5 2 1 } 2

y 2 y1 5 m(x 2 x1)

y 2 0 5 2 1 } 2 (x 2 7)

y 5 2 1 } 2 x 1

7 } 2

Let (x1, y1) 5 (1, 21) and (x2, y2) 5 (5, 5).

m 5 5 1 1

} 5 2 1 5 3 } 2

y 2 y1 5 m(x 2 x1)

y 1 1 5 3 } 2 (x 2 1)

y 5 3 } 2 x 2

5 } 2


y 5 2 1 } 2 x 1

7 } 2

y 5 3 } 2 x 2

5 } 2

When y 5 2 1 } 2 x 1

7 } 2 : When x 5 3:

2 1 } 2 x 1

7 } 2 5

3 } 2 x 2

5 } 2 y 5 2

1 } 2 (3) 1

7 } 2

6 5 2x y 5 2 3 } 2 1

7 } 2

3 5 x y 5 2


44. 0.02x 2 0.05y 5 20.38 3 (2300) 26x 1 15y 5 114

0.03x 1 0.04y 5 1.04 3 200 6x 1 8y 5 208

23y 5 322

When y 5 14: y 5 14

0.02x 2 0.05(14) 5 20.38

0.02x 2 0.7 5 20.38

0.02x 5 0.32

x 5 16



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129Algebra 2


45. 0.05x 2 0.03y 5 0.21 3 200 10x 2 6y 5 42

0.07x 1 0.02y 5 0.16 3 300 21x 1 6y 5 48

31x 5 90

When x 5 90

} 31 : x 5 90

} 31

10 1 90 }

31 2 2 6y 5 42

900

} 31

2 6y 5 42

26y 5 402

} 31

y 5 2 67

} 31


31 ,2

67 } 31 2 .

46. 2 }

3 x 1 3y 5 234

x 2 1 } 2 y 5 21 → x 5

1 } 2 y 2 1

When x 5 1 } 2 y 2 1: When y 5 210:

2

} 3 1 1 }

2 y 2 1 2 1 3y 5 234 x 5

1 } 2 (210) 2 1

1 }

3 y 2

2 } 3 1 3y 5 234 x 5 26

10

} 3 y 5 2

100 } 3

y 5 210


47. 1 }

2 x 1

2 } 3 y 5

5 } 6 3 (230) 215x 2 20y 5 225

5 }

12 x 1

7 } 12 y 5

3 } 4 3 36 15x 1 21y 5 27

When y 5 2: y 5 2

1 }

2 x 1

2 } 3 (2) 5

5 } 6

1 }

2 x 1

4 } 3 5

5 } 6

x 5 21


48. x 1 3

} 4 1

y 2 1 } 3 5 1

2x 2 y 5 12 → y 5 2x 2 12

When y 5 2x 2 12: When x 5 5:

12 1 x 1 3 }

4 1

(2x 2 12) 2 1 }} 3 2 5 1 p 12 y 5 2(5) 2 12

3(x 1 3) 1 4(2x 2 13) 5 12 y 5 22

3x 1 9 1 8x 2 52 5 12

11x 5 55

x 5 5


49. x 2 1

} 2 1

y 1 2 } 3 5 4

x 2 2y 5 5 → x 5 2y 1 5

When x 5 2y 1 5:

6 1 2y 1 5 2 1 }

2 1

y 1 2 } 3 2 5 4 p 6

3(2y 1 4) 1 2(y 1 2) 5 24

6y 1 12 1 2y 1 4 5 24

8y 5 8

y 5 1

When y 5 1:

x 5 2(1) 1 5

x 5 7


50. Sample answer: Two lines intersect at (21, 4). Choose another point in the plane to create one line. Then choose a point not on that line to create a different line.

x

y

1

1

(21, 4)

(0, 1)

(0, 5)

A B

Line A: (x1, y1) 5 (0, 1), (x2, y2) 5 (21, 4)

m 5 4 2 1

} 21 2 0 5 23

y 2 y1 5 m(x 2 x1)

y 2 1 5 23(x 2 0)

y 5 23x 1 1

Line B: (x1, y1) 5 (0, 5), (x2, y2) 5 (21, 4)

m 5 4 2 5

} 21 2 0 5 1

y 2 y1 5 m(x 2 x1)

y 2 5 5 1(x 2 0)

y 5 x 1 5


y 5 23x 1 1

y 5 x 1 5

Use substitution to check:

When y 5 x 1 5: When x 5 21:

x 1 5 5 23x 1 1 y 5 23(21) 1 1

4x 5 24 y 5 4

x 5 21

The lines intersect at (21, 4).


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51. 7y 1 18xy 5 30

13y 2 18xy 5 90

20y 5 120

y 5 6

When y 5 6:

7(6) 1 18x(6) 5 30

108x 5 212

x 5 2 1 } 9

The solution is 1 2 1 } 9 , 6 2 .

52. xy 2 x 5 14 xy 2 x 5 14

5 2 xy 5 2x 2xy 2 2x 5 25

23x 5 9

When x 5 23: x 5 23

23y 2 (23) 5 14

23y 1 3 5 14

23y 5 11

y 5 2 11

} 3

The solution is 1 23,2 11

} 3 2 .

53. 2xy 1 y 5 44 2xy 1 y 5 44

32 2 xy 5 3y 3 2 64 2 2xy 5 6y

2xy 1 y 5 44

22xy 2 6y 5 264

25y 5 220

y 5 4

When y 5 4:

2x(4) 1 4 5 44

8x 5 40

x 5 5


54. 23x 2 5y 5 9

rx 1 sy 5 t

a. The system will have no solution if the equations represent different lines with the same slope. Sample answer: r 5 23, s 5 25, and t Þ 9.

b. The system will have infi nitely many solutions if the equations represent the same line. Sample answer: r 5 23a, s 5 25a, and t 5 9a, where a is any real number.

c. The system will have a solution of (2,23) for any values or r, s, and t that satisfy the equation 2r 2 3s 5 t. Sample answer: r 5 1, s 5 1, and t 5 21.

Problem Solving

55. Let x 5 acoustic guitars and y 5 electric guitars.

Number of acoustics guitars 1

Number of electric guitars 5

Total number of guitars

x 1 y 5 9

Price peracousticguitar

p

Numberofacousticguitars

1

Price perelectricguitar

p

Numberofelectricguitars

5 Totalrevenue

339 p x 1 479 p y 5 3611

x 1 y 5 9 → y 5 9 2 x

339x 1 479y 5 3611

When y 5 9 2 x:

339x 1 479(9 2 x) 5 3611

339x 1 4311 2 479x 5 3611

2140x 5 2700

x 5 5

When x 5 5:

y 5 9 2 5

y 5 4

The music store sold 5 acoustic guitars and 4 electric guitars.

56. Let x 5 price of adult pass and y 5 price of children’s pass.

Adultprice 5

Childprice

1 2

x 5 y 1 2

Numberof adultpasses

p Adultprice

1

Numberof children’spasses

p Childprice

5 Total

revenue

378 p x 1 214 p y 5 2384

x 5 y 1 2

378x 1 214y 5 2384

When x 5 y 1 2:

378(y 1 2) 1 214y 5 2384

378y 1 756 1 214y 5 2384

592y 5 1628

y 5 2.75

When y 5 2.75:

x 5 2.75 1 2

x 5 4.75

The cost of an adult pass is $4.75.


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131Algebra 2



57. The company can fi ll its orders by operating Factory A for 5 weeks and Factory B for 3 weeks.

Let x 5 weeks of operation at Factory A, and y 5 weeks of operation at Factory B.

Factory A’s gasmowersper wk

p

Wks atFactoryA

1

Factory B’s gasmowersper wk

p

Wks atFactoryB

5

200 p x 1 400 p y 5 2200

Factory A’s electricmowersper wk

p

Wks atFactoryA

1

Factory B’s electricmowersper wk

p

Wks atFactoryB

5

Total #of electricmowers

100 p x 1 300 p y 5 1400

200x 1 400y 5 2200 200x 1 400y 5 2200

100x 1 300y 5 1400 3 (22) 2200x 2 600y 5 22800

2200y 5 2600

When y 5 3: y 5 3

200x 1 400(3) 5 2200

200x 5 1000

x 5 5


58. B;

Let x 5 price of regular gasoline and y 5 price of premium gasoline.

Gallonsofregular

p

Regularprice

1

Gallonsofpremium

p

Premiumprice

5 Total

price

11 p x 1 16 p y 5 58.55

Premiumgas

5

Regularprice

1 0.2

y 5 x 1 0.2

11x 1 16y 5 58.55

y 5 x 1 0.2

When y 5 x 1 0.2:

11x 1 16(x 1 0.2) 5 58.55

11x 1 16x 1 3.2 5 58.55

27x 5 55.35

x 5 2.05

When x 5 2.05:

y 5 2.05 1 0.2

y 5 2.25

A gallon of premium gasoline costs $2.25.

59. Let x 5 doubles games in progress and y 5 singles games in progress.

Number of doublesgames

1

Number of singlesgames

5

Total number of games

x 1 y 5 26

Playerspergame

p

Doublegames 1

Playerspergame

p

Singlesgame

5 Totalnumber ofplayers

4 p x 1 2 p y 5 76

x 1 y 5 26 → y 5 26 2 x

4x 1 2y 5 76

When y 5 26 2 x: When x 5 12:

4x 1 2(26 2 x) 5 76 y 5 26 2 x

4x 1 52 2 2x 5 76 y 5 14

2x 5 24

x 5 12

There were 12 doubles games and 14 singles games in progress.

60. a. Let t 5 hours since 10:00 A.M. and d 5 distance traveled (miles).

Martha’sdistance

5 Martha’srate

p Martha’stime

d 5 4 p t

An equation for the distance Martha travels is d 5 4t.

b. Carol’sdistance

5 Carol’srate

p Carol’stime

d 5 6 p (t 2 2)

An equation for the distance Carol travels is d 5 6(t 2 2).

c. d 5 4t

d 5 6t 2 12

When d 5 4t:

4t 5 6t 2 12

t 5 6

Carol will catch up to Martha 6 hours after 10:00 A.M.,

or at 4:00 P.M.

d. Changing starting time:

Let h 5 the number of hours Carol starts after Martha.

d 5 4t

d 5 6(t 2 h) → d 5 6t 2 6h

When t 5 5: When d 5 20 and t 5 5:

d 5 4(5) 5 20 20 5 6(5) 2 6h

6h 5 10

h 5 5 } 3

Total #of gasmowers

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Carol should start 5 }

3 hours, or 1 hour 40 minutes, after

Martha. It is reasonable that Carol could change her starting time.

Changing speed:

Let r 5 Carol’s speed (miles per hour).

d 5 4t

d 5 r(t 2 2)

When t 5 5: When d 5 20 and t 5 5:

d 5 4(5) 5 20 20 5 r (5 2 2)

r 5 20

} 3 5 6 2

} 3

Carol should run at a speed of 6 2

} 3 miles per hour.

This is a reasonable speed.

61. Let x 5 pounds of peanuts in one pound of mix and y 5 pounds of cashews in one pound mix.

Peanutpriceperpound

p

Amount of peanuts in 1lb of mix

1

Cashewpriceperpound

p

Amount ofcashewsin 1lb ofmix

5

Priceof 1lbof mix

2.80 p x 1 5.30 p y 5 3.30

Amount ofpeanuts in1lb of mix

1

Amount ofcashews in1lb of mix

5

Total weight of 1lb of mix

x 1 y 5 1

2.8x 1 5.3y 5 3.3

x 1 y 5 1 → y 5 1 2 x

When y 5 1 2 x: When x 5 0.8:

2.8x 1 5.3(1 2 x) 5 3.3 y 5 1 2 x

2.8x 1 5.3 2 5.3x 5 3.3 y 5 0.2

22.5x 5 22

x 5 0.8

One pound of mix should contain 0.8 pounds of peanuts and 0.2 pounds of cashews.

(0.8)100 5 80lb peanuts

(0.2)100 5 20lb cashews

The wholesaler should use 80 pounds of peanuts and20 pounds of cashews for 100 pounds of mix.

62. Let x 5 speed of the plane in calm air and y 5 speed of the wind.

Plane’sspeed incalm air

1

Speed oftailwind

5

Actual planespeed

x 1 y 5 1000

} 5 5 200

Plane’sspeed incalm air

2 Speed of

headwind

5

Actual planespeed

x 2 y 5 500

} 5 5 100

x 1 y 5 200 → y 5 200 2 x

x 2 y 5 100

When y 5 200 2 x: When x 5 150:

x 2 (200 2 x) 5 100 y 5 200 2 x

x 2 200 1 x 5 100 y 5 50

2x 5 300

x 5 150

The speed of the plane in calm air was 150 mi/h, and the speed of the wind was 50 mi/h.

63. Let x 5 hours the electrician worked and y 5 hours the apprentice worked.

Apprentice’shours

1 4 5

Electrician’shours

y 1 4 5 x

Electri-cian’spay rate

p

Electri-cian’shours

1

Appren-tice’spay rate

p Appren-

tice’shours

5

Totalearnings

50 p x 1 20 p y 5 550

y 1 4 5 x

50x 1 20y 5 550

When x 5 y 1 4:

50(y 1 4) 1 20y 5 550

50y 1 200 1 20y 5 550

70y 5 350

y 5 5

The apprentice earned 20(5) 5 $100.

When y 5 5:

x 5 y 1 4

x 5 9

The electrician earned 50(9) 5 $450.

Mixed Review

64. 25x 1 4 5 29

25x 5 25

x 5 25

65. 6(2a 2 3) 5 230

12a 2 18 5 230

12a 5 212

a 5 21

66. 1.2m 5 2.3m 2 2.2

21.1m 5 22.2

m 5 2

67. x 1 3 5 4

x 1 3 5 4 or x 1 3 5 24

x 5 1 or x 5 27

68. 2x 1 11 5 3

2x 1 11 5 3 or 2x 1 11 5 23

2x 5 28 or 2x 5 214

x 5 24 or x 5 27


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133Algebra 2


69. 2x 1 7 5 13

2x 1 7 5 13 or 2x 1 7 5 213

2x 5 6 or 2x 5 220

x 5 26 or x 5 20

70. Line 1: m1 5 10 2 5

} 2 2 1 5 5

Line 2: m2 5 27 2 (28)

} 3 2 8 5 2 1 } 5

Because m2 5 2 1 } m1 , the lines are perpendicular.

71. Line 1: m1 5 5 1 2

} 4 2 9 5 2 7 } 5

Line 2: m2 5 26 1 1

} 6 1 2 5 2 5 } 8

Because m1 Þ m2 and m2 Þ 2 1 } m1 , the lines are neither

parallel nor perpendicular.

72. (x1, y1) 5 (2, 4), (x2, y2) 5 (5, 1)

m 5 y2 2 y1

} x2 2 x1 5

1 2 4 } 5 2 2 5 21

y 2 y1 5 m(x 2 x1)

y 2 4 5 21(x 2 2)

y 5 2x 1 6

73. (x1, y1) 5 (3, 1), (x2, y2) 5 (21, 27)

m 5 y2 2 y1

} x2 2 x1 5

27 2 1 }

21 2 3 5 2

y 2 y1 5 m(x 2 x1)

y 2 1 5 2(x 2 3)

y 5 2x 2 5

74. (x1, y1) 5 (2, 4), (x2, y2) 5 (22, 22)

m 5 y2 2 y1

} x2 2 x1 5

22 2 4 }

22 2 2 5 3 } 2

y 2 y1 5 m(x 2 x1)

y 2 4 5 3 } 2 (x 2 2)

y 5 3 } 2 x 1 1

75. x < 23 76. y ≥ 2

1

x

y

21

1

x

y

21

77. 2x 1 y > 1 78. y ≤ 2x 1 4

1

x

y

21

1

x

y

21

79. 4x 2 y ≥ 5 80. y < 23x 1 2

1

x

y

21

1

x

y

21

Quiz 3.1–3.2 (p. 167)

1.

1

x

y

21


The solution appears to be (2, 5).

3x 1 y 5 11 x 2 2y 5 28

3(2) 1 5 0 11 2 2 2(5) 0 28

6 1 5 0 11 2 2 10 0 28

11 5 11 ✓ 28 5 28 ✓

2.

1

x

y

21



2x 1 y 5 25 2x 1 3y 5 6

2(23) 1 1 0 25 2(23) 1 3(1) 0 6

26 1 1 0 25 3 1 3 0 6

25 5 25 ✓ 6 5 6 ✓

3. 2

x

y

21



x 2 2y 5 22 3x 1 y 5 220

26 2 2(22) 0 22 3(26) 1 (22) 0 220

26 1 4 0 22 218 2 2 0 220

22 5 22 ✓ 220 5 220 ✓

4.

x

y

2

1

4x 1 8y 5 8

x 1 2y 5 6

The graphs of the equations are parallel lines. There is no solution. The system is inconsistent.


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5.

x

y

2

2

25x 1 3y 5 25

y 5 x 1 153

The graphs of the equations are parallel lines. There is no solution. The system is inconsistent.

6.

x

y1

21

x 2 2y 5 2

2x 2 y 5 25

(24, 23)


7. 3x 2 y 5 24

x 1 3y 5 228 → x 5 228 2 3y

When x 5 228 2 3y: When y 5 28:

3(228 2 3y) 2 y 5 24 x 5 228 2 3(28)

284 2 9y 2 y 5 24 x 5 24

210y 5 80

y 5 28


8. x 1 5y 5 1 → x 5 1 2 5y

23x 1 4y 5 16

When x 5 1 2 5y: When y 5 1:

23(1 2 5y) 1 4y 5 16 x 5 1 2 5(1)

23 1 15y 1 4y 5 16 x 5 24

19y 5 19

y 5 1


9. 6x 1 y 5 26 → y 5 26x 2 6

4x 1 3y 5 17

When y 5 26x 2 6: When x 5 2 5 } 2 :

4x 1 3(26x 2 6) 5 17 y 5 26 1 2 5 } 2 2 2 6

4x 2 18x 2 18 5 17 y 5 9

214x 5 35

x 5 2 5 } 2

The solution is 1 2 5 } 2 , 9 2 .

10. 2x 2 3y 5 21

2x 1 3y 5 219

4x 5 220

x 5 25

When x 5 25:

2(25) 2 3y 5 21

23y 5 9

y 5 23


11. 3x 2 2y 5 10 3 2 6x 2 4y 5 20

26x 1 4y 5 220 26x 1 4y 5 220

0 5 0


12. 2x 1 3y 5 17 3 (25) 210x 2 15y 5 285

5x 1 8y 5 20 3 2 10x 1 16y 5 40

y 5 245

When y 5 245:

2x 1 3(245) 5 17

2x 5 152

x 5 76


13. Let x 5 cost per foot of cable and y 5 cost per connector.

6 p Cost per

foot of cable

1 2 p

Cost perconnector

5

Cost of 6 footcable with

connectors

6 p x 1 2 p y 5 15.50

3 p Cost per

foot of cable

1 2 p

Cost perconnector

5

Cost of 3 footcable with

connectors

3 p x 1 2 p y 5 10.25

6x 1 2y 5 15.50 6x 1 2y 5 15.50

3x 1 2y 5 10.25 3 (21) 23x 2 2y 5 210.25

3x 5 5.25

When x 5 1.75: x 5 1.75

6(1.75) 1 2y 5 15.50

2y 5 5

y 5 2.5

4 - foot cable 5 4x 1 2y

5 4(1.75) 1 2(2.50)

5 12.00

A 4 - foot cable should cost $12.00.


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135Algebra 2


Lesson 3.3


1. y ≤ 3x 2 2

1

x21

y

y > 2x 1 4

2. 2x 2 1 } 2 y ≥ 4

22

21

y

x

4x 2 y ≤ 5

3. x 1 y > 23 4. y ≤ 4 26x 1 y < 1 y ≥ x 2 5

1

x21

y

1

x

y

1

5. y > 22 6. y ≥ 2x 1 1 y ≤ 2x 1 2 y < x 1 1

1

x1

y

x

y

2

1

7. Range of discounts: 20%–50%

Regular prices: $40– $100

a. Let x 5 regular footwear price and y 5 sale footwear price.

x ≥ 40

00 20 40 60 10080

20

40

80

60

x

y

Regular prices (dollars)

Sal

e p

rice

s (d

olla

rs)

y 5 0.8x

x 5 40

y 5 0.5x

x 5 100 x ≤ 100

y ≥ 0.5x

y ≤ 0.8x

b. When x 5 60:

0.5(60) ≤ y ≤ 0.8(60)

30 ≤ y ≤ 48

Footwear regularly priced at $60 sells for between $30 and $48, inclusive, during the sale.


Skill Practice

1. The ordered pair must be in the area of the graph that is common to all of the inequalities in the system.

2. Graph each inequality in the system. Then identify the region common to all the graphs of the inequalities.

3. D

4. x > 21 5. x ≤ 2 x < 3 y ≤ 5

1

x22

y

1

x

y

21

6. y ≥ 5 7. 2x 1 y < 23

y ≤ 1 2x 1 y > 4

x

y

211

x

y

1

1

8. y < 10 9. 4x 2 4y ≥ 216

y > x 2x 1 2y ≥ 24

1

x21

y

1

x

y

21

10. 2x ≥ y 11. y > x 2 4

2x 1 y ≥ 25 3y < 22x 1 9

1

x21

y

1 x

21

y


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12. x 1 y ≥ 23 13. 2y < 25x 2 10

26x 1 4y < 14 5x 1 2y > 22

1

21

y

x

x

y

1

1

14. 3x 2 y > 12 15. x 2 4y ≤ 210

2x 1 8y > 24 y ≤ 3x 2 1

1

3

y

x

1

y

x21

16. The graph of y ≤ 2x 2 2 should be shaded to the right of the line y 5 2x 2 2.

21

1

y

x

17. x < 6 18. x ≥ 28

y > 21 y ≤ 21

y < x y < 22x 2 4

1

x21

y

1

y

x23

19. 3x 1 2y > 26 20. x 1 y < 5

25x 1 2y > 22 2x 2 y > 0

y < 5 2x 1 5y > 220

1

x

y

23

1

21

y

x

21. x ≥ 2 22. y ≥ x 23x 1 y < 21 x 1 3y < 5

4x 1 3y < 12 2x 1 y ≥ 23

1

21 x

y

1

22 x

y

23. y ≥ 0 24. x 1 y < 5

x > 3 x 1 y > 25

x 1 y ≥ 22 x 2 y < 4

y < 4x x 2 y > 22

2

1 x

y

1

21 x

y

25. x ≤ 10

24 x

y6

x ≥ 22

3x 1 2y < 6

6x 1 4y > 212

26. B; y

x

2

2

Q III Q IV

Q IQ II

y ≤ 2x 2 3 1 2

4x 2 5y ≤ 20

27. Sample answer: x ≥ 0 y ≤ 21

28. y < x 29. y ≤ x 2 2 y > 2x y ≥ x 2 2

1

21 x

y

1

x

y

22


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137Algebra 2


30. y ≤ 2x 2 3 1 2

1

21 x

y

y > x 2 3 2 1

31. y ≤ 3 y ≥ 22

x ≥ 23

x ≤ 432. Line through points (22, 2) and (2, 4):

m 5 4 2 2

} 2 2 (22)

5 1 } 2

y 2 y1 5 m(x 2 x1)

y 2 4 5 1 } 2 (x 2 2)

y 5 1 } 2 x 1 3 (boundary)

Line through points (22, 22) and (2, 24):

m 5 24 2 (22)

} 2 2 (22)

5 2 1 } 2

y 2 y1 5 m(x 2 x1)

y 1 4 5 2 1 } 2 (x 2 2)

y 5 2 1 } 2 x 2 3 (boundary)

System:

y ≤ 1 } 2 x 1 3

y ≥ 2 1 } 2 x 2 3

x ≥ 22

x ≤ 2 33. Line through points (22, 23) and (1, 3):

m 5 3 2 (23)

} 1 2 (22)

5 2

y 2 y1 5 m(x 2 x1)

y 2 3 5 2(x 2 1)

y 5 2x 1 1


m 5 22 2 (23)

} 2 2 (22)

5 1 } 4

y 2 y1 5 m(x 2 x1)

y 1 2 5 1 } 4 (x 2 2)

y 5 1 } 4 x 2

5 } 2


m 5 22 2 3

} 2 2 1 5 25

y 2 y1 5 m(x 2 x1)

y 2 3 5 25(x 2 1)

y 5 25x 1 8

System:

y ≤ 2x 1 1

y ≥ 1 } 4 x 2

5 } 2

y ≤ 25x 1 8

34. Let x 5 the hours dog walking and y 5 the hours car washing.

x 1 y ≤ 20

x ≥ 0 y ≥ 0 7.5x 1 6y ≥ 92

35. Discount: 30% –70%

Regular prices: $20 –$50

Let x 5 the regular price and y 5 the sale price.

x ≥ 20

100 20 30 40 50

10

0

20

30

40

Regular price (dollars)

Sal

e p

rice

(d

olla

rs)

x

y

y 5 0.3x

y 5 0.7x

x 5 50x 5 20 x ≤ 50

y ≥ 0.3x

y ≤ 0.7x

When x 5 20:

0.3(20) ≤ y ≤ 0.7(20)

6 ≤ y ≤ 14

Games regularly priced at $20 sell between $6 and $14, inclusive, during the sale.

36. x > 8.0

00 8.0 8.1 8.2 8.3

75

77

81

79

x

y

pH level

Tem

per

atu

re (

8F) x < 8.3

y > 76

y < 80

In degrees Celsius:

C 5 5 } 9 (F 2 32) C 5

5 } 9 (F 2 32)

C 5 5 } 9 (80 2 32) C 5

5 } 9 (76 2 32)

C 5 5 } 9 (48) C 5

5 } 9 (44)

C 5 26. } 6 C 5 24. } 4

The graphs would look similar; the y axis would be incremented differently to show 24.44 < y < 26.67.

37. a. x ≥ 2 b.

10

23456

x

y

10 2 3 4 5 6 7 8

Number of juniors

Nu

mb

er o

f se

nio

rs

y ≥ 2 x 1 y ≤ 8 x 1 y ≥ 5

c. Sample answer:

3 juniors, 2 seniors

6 juniors, 2 seniors


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38. Let x 5 the horizontal distance (in inches) from the left side of home plate and y 5 the distance above the ground (in inches).

x ≥ 0

00 4 8 12 16

10

20

40

30

x

y

Horizontal distance (in.)

Ver

tica

l dis

tan

ce (

in.) x ≤ 17

y ≥ 20

y ≤ 42

39. a. x ≤ 65 b.

Hea

rt r

ate

(bea

ts p

er m

inu

te)

Age (years)

0100 20 30 40 50 70 80 9060

255075

100125

200225

175150

x

y

x ≥ 20

y ≤ 0.75(220 2 x)

y ≥ 0.5(220 2 x)

c. No; the target zone for a 40-year-old is between 0.5(220 2 40) 5 90 and 0.75(220 2 40) 5 135. A heart rate of 158 is outside of this range.

40.

5000 1000 1500 2000

500

0

1000

1500

2000

Your guess

You

fri

end

’s g

ues

s

x

y

x ≥ 500

y ≥ 500

y 2 1000 > x 2 1000 When y < 1000:

y 2 1000 5 1000 2 y > x 2 1000 2y > x 2 1000 2 1000

y < 1000 2 x 2 1000 When y ≥ 1000:

y 2 1000 5 y 2 1000 > x 2 1000 y > 1000 1 x 2 1000 System of inequalities:

x ≥ 500

y ≥ 500

y 2 1000 > x 2 1000

Mixed Review

41. 6x 2 8y 5 6(4) 2 8(21)

5 24 1 8

5 32

42. 12x 1 3y 5 12(4) 1 3(25)

5 48 2 15

5 33

43. x2 2 2xy 1 3y 5 (22)2 2 2(22)(3) 1 3(3)

5 4 1 12 1 9

5 25

44. 4x2y2 2 xy 5 4(5)2(26)2 2 (5)(26)

5 4(25)(36) 1 30

5 3600 1 30

5 3630

45. x 2 8 ≤ 25

x ≤ 3

21 0 1 2 3 4

46. 5x 2 11 > 2x 1 7

6x > 18

x > 3

21 0 1 2 3 4

47. 9x 1 2 ≥ 23x 2 13

12x ≥ 215

x ≥ 2 5 } 4

0 2 424 22

542

48. 25x 1 y 5 211

4x 2 y 5 7 → y 5 4x 2 7

When y 5 4x 2 7: When x 5 4:

25x 1 4x 2 7 5 211 y 5 4(4) 2 7

2x 5 24 y 5 9

x 5 4


49. 9x 1 4y 5 27 9x 1 4y 5 27

3x 2 5y 5 234 3 (23) 29x 1 15y 5 102

19y 5 95

When y 5 5: y 5 5

9x 1 4(5) 5 27

9x 5 227

x 5 23



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139Algebra 2


50. 4x 1 9y 5 210 3 (22) 28x 2 18y 5 20

8x 1 18y 5 20 8x 1 18y 5 20

0 Þ 40


51. x 2 5y 5 18 → x 5 5y 1 18

2x 1 3y 5 10

When x 5 5y 1 18: When y 5 22:

2(5y 1 18) 1 3y 5 10 x 5 5(22) 1 18

10y 1 36 1 3y 5 10 x 5 8

13y 5 226

y 5 22


52. 16x 2 12y 5 28 16x 2 12y 5 28

8x 2 6y 5 24 3 (22) 216x 1 12y 5 8

0 5 0


53. 16x 1 5y 5 24 16x 1 5y 5 24

8x 2 2y 5 7 3 (22) 216x 1 4y 5 214

9y 5 218

When y 5 22: y 5 22

8x 2 2(22) 5 7

8x 5 3

x 5 3 } 8

The solution is 1 3 } 8 , 22 2 .

3.3 Extension (p. 176)

1. At (0, 7): C 5 0 1 2(7) 5 14

At (1, 0): C 5 1 1 2(0) 5 1

At (8, 0): C 5 8 1 2(0) 5 8

Minimum value: 1; maximum value: 14

2. At (28, 4): C 5 4(28) 2 2(4) 5 240

At (28, 28): C 5 4(28) 2 2(28) 5 216

At (2, 28): C 5 4(2) 2 2(28) 5 24

At (6, 22): C 5 4(6) 2 2(22) 5 28


3. At (20, 60): C 5 3(20) 1 5(60) 5 360

At (40, 10): C 5 3(40) 1 5(10) 5 170

At (80, 0): C 5 3(80) 1 5(0) 5 240

At (100, 40): C 5 3(100) 1 5(40) 5 500

At (60, 80): C 5 3(60) 1 5(80) 5 580


4.

x

y

1

1

(0, 5)

(5, 0)(0, 0)

At (0, 0): C 5 3(0) 1 4(0) 5 0

At (5, 0): C 5 3(5) 1 4(0) 5 15

At (0, 5): C 5 3(0) 1 4(5) 5 20


5.

x

y

2

2

(5, 9)

(25, 3) (5, 3)

At (25, 3): C 5 2(25) 1 5(3) 5 5

At (5, 3): C 5 2(5) 1 5(3) 5 25

At (5, 9): C 5 2(5) 1 5(9) 5 55


6.

x

y

1

1

(0, 4)

(2, 22)(0, 0)

At (0, 4): C 5 3(0) 1 4 5 4

At (0, 0): C 5 3(0) 1 0 5 0

At (2, 22): C 5 3(2) 1 (22) 5 4

Minimun value: 0; no maximum value

7. x 5 the number of mini piñatas

y 5 the number of regular-sized piñatas

2x 1 3y ≤ 30

x

y

3

3

(6, 6)

(15, 0)(12, 0)

x 1 y ≥ 12

x ≥ 0

y ≥ 0

P 5 12x 1 24y

At (12, 0): P 5 12(12) 1 24(0) 5 144

At (15, 0): P 5 12(15) 1 24(0) 5 180

At (6, 6): P 5 12(6) 1 24(6) 5 216

The owner should make 6 mini piñatas and 6 regular-sized piñatas to maximize profi t.


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8. x 5 the number of inkjet printers

y 5 the number of laser printers

x 1 y ≤ 60

x

y

10

10 (60, 0)

(30, 30)(0, 40)

(0, 0)

x 1 3y ≤ 120

x ≥ 0

y ≥ 0

P 5 40x 1 60y

At (0, 0): P 5 40(0) 1 60(0) 5 0

At (60, 0): P 5 40(60) 1 60(0) 5 2400

At (30, 30): P 5 40(30) 1 60(30) 5 3000

At (0, 40): P 5 40(0) 1 60(40) 5 2400

The company should make 30 inkjet printers and30 laser printers.

9. t 5 the number of jars of tomato sauce

s 5 the number of jars of salsa

10t 1 5s ≤ 180

s

t

6

3

(0, 15)

(0, 0)

6013

18013( (,

t 1 0.25s ≤ 15

t ≥ 3s

s ≥ 0

t ≥ 0

P 5 2t 1 1.5s

At (0, 0): P 5 2(0) 1 1.5(0) 5 0

At (0, 15): P 5 2(15) 1 1.5(0) 5 30

At 1 60 }

13 ,

180 }

13 2 : P 5 2 1 180

} 13

2 1 1.5 1 60 }

13 2 5

450 } 13 ø 34.62

The maximum value of P occurs when t 5 180

} 13 ø 13.85

and s 5 60

} 13 ø 4.62.

You should make 13 jars of tomato sauce and 4 jarsof salsa.

10. a. Sample answer: b. Sample answer:

1

x

y

21

(5, 7)

(7, 2)

1

x

y

21

(2, 6)

(6, 1)(0, 0)

Lesson 3.4

Investigating Algebra Activity 3.4 (p. 177)

1. 4x 1 3y 1 2z 5 12 2. 2x 1 2y 1 3z 5 6

(3, 0, 0)

(0, 4, 0)

(0, 0, 6)

y

x

z

(3, 0, 0)

(0, 3, 0)

(0, 0, 2)

y

x

z

3. x 1 5y 1 3z 5 15

(15, 0, 0)

(0, 3, 0)

(0, 0, 5)

y

x

z

4. 5x 2 y 2 2z 5 10

(0, 210, 0)

(0, 0, 25)

(2, 0, 0)

y

x

z

5. 27x 1 7y 1 2z 5 14

(22, 0, 0)

(0, 2, 0)

(0, 0, 7)

y

x

z

6. 2x 1 9y 2 3z 5 218

(29, 0, 0)

(0, 22, 0)

(0, 0, 6)

y

x

z


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141Algebra 2


7. Sample answer:

The planes of three linear equations in three variables can intersect in three different ways. They could intersect in exactly one point, intersect in a line, or never intersectat all.


1. 3x 1 y 2 2z 5 10 3 (24) 212x 2 4y 1 8z 5 240

x 1 4y 1 3z 5 7 x 1 4y 1 3z 5 7

211x 1 11z 5 233

2x 1 z 5 23

6x 2 2y 1 z 5 22 3 2 12x 2 4y 1 2z 5 24

x 1 4y 1 3z 5 7 x 1 4y 1 3z 5 7

13x 1 5z 5 3

2x 1 z 5 23 3 (25) 5x 2 5z 5 15

13x 1 5z 5 3 13x 1 5z 5 3

18x 5 18

x 5 1

21 1 z 5 23 → z 5 22

3(1) 1 y 2 2(22) 5 10 → y 5 3

The solution is (1, 3, 22).

2. x 1 y 2 z 5 2 3 (22) 22x 2 2y 1 2z 5 24

2x 1 2y 2 2z 5 6 2x 1 2y 2 2z 5 6

0 5 2

Because this is a false equation, there is no solution.

3. x 1 y 1 z 5 3 x 1 y 2 z 5 3

x 1 y 2 z 5 3 2x 1 2y 1 z 5 6

2x 1 2y 5 6 3x 1 3y 5 9

2x 1 2y 5 6 3 3 6x 1 6y 5 18

3x 1 3y 5 9 3 (22) 26x 2 6y 5 218

0 5 0

Because you obtain the identity 0 5 0, the system has infi nitely many solutions.

4. x 1 y 1 z 5 60

x 1 (x 1 z) 1 z 5 60

2x 1 2z 5 60

1000x 1 200y 1 500z 5 25,000

1000x 1 200(x 1 z) 1 500z 5 25,000

1200x 1 700z 5 25,000

2x 1 2z 5 60 3 (2600) 21200x 2 1200z 5 236,000

1200x 1 700z 5 25,000 1200x 1 700z 5 25,000

2500z 5 211,000

z 5 22

2x 1 2(22) 5 60 → x 5 8

8 1 y 1 22 5 60 → y 5 30

The department should run 8 TV ads, 30 radio ads, and22 newspaper ads each month.


Skill Practice

1. A linear equation in three variables has the formax 1 by 1 cz 5 d. The graph of this equation will bea plane.

2. Sample answer: If one variable is expressed in terms of the other two, substitute that expression into the remaining two equations, and solve the new linear system in two variables.

3. 2x 2 y 1 z 5 25 x 2 3y 1 z 5 25

2(1) 2 4 1 (23) 0 25 1 2 3(4) 1 (23) 0 25

25 5 25 ✓ 214 Þ 25

5x 1 2y 2 2z 5 19

5(1) 1 2(4) 2 2(23) 0 19

19 5 19 ✓

(1, 4, 23) is not a solution.

4. 4x 2 y 1 3z 5 13

4(21) 2 (22) 1 3(5) 0 13

13 5 13 ✓

x 1 y 1 z 5 2

21 1 (22) 1 5 0 2

2 5 2 ✓

x 1 3y 2 2z 5 217

21 1 3(22) 2 2(5) 0 217

217 5 217 ✓

(21, 22, 5) is a solution.

5. x 1 4y 2 2z 5 12

6 1 4(0) 2 2(23) 0 12

12 5 12 ✓

3x 2 y 1 4z 5 6

3(6) 2 0 1 4(23) 0 6

6 5 6 ✓

2x 1 3y 1 z 5 29

2(6) 1 3(0) 1 (23) 0 29

29 5 29 ✓



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6. 3x 1 4y 2 2z 5 211

3(25) 1 4(1) 2 2(0) 0 211

211 5 211 ✓

2x 1 y 2 z 5 11

2(25) 1 1 2 (0) 0 11

29 Þ 11

x 1 4y 1 3z 5 21

25 1 4(1) 1 3(0) 0 21

21 5 21 ✓


7. 3x 2 y 1 5z 5 34

3(2) 2 (8) 1 5(4) 0 34

18 Þ 34

x 1 3y 2 6z 5 2

2 1 3(8) 2 6(4) 0 2

2 5 2 ✓

23x 1 y 2 2z 5 26

23(2) 1 8 2 2(4) 0 26

26 5 26 ✓


8. 2x 1 4y 2 z 5 223

2(0) 1 4(24) 2 7 0 223

223 5 223 ✓

x 2 5y 2 3z 5 21

0 2 5(24) 2 3(7) 0 21

21 5 21 ✓

2x 1 y 1 4z 5 24

2(0) 1 (24) 1 4(7) 0 24

24 5 24 ✓


9. 3x 1 y 1 z 5 14

5x 2 y 1 5z 5 30

8x 1 6z 5 44

2x 1 2y 2 3z 5 29 2x 1 2y 2 3z 5 29

5x 2 y 1 5z 5 30 3 2 10x 2 2y 1 10z 5 60

9x 1 7z 5 51

8x 1 6z 5 44 3 (27) 256x 2 42z 5 2308

9x 1 7z 5 51 3 6 54x 1 42z 5 306

22x 5 22

x 5 1

9(1) 1 7z 5 51 → z 5 6

3(1) 1 y 1 6 5 14 → y 5 5


10. 2x 2 y 1 2z 5 27 2x 2 y 1 2z 5 27

x 1 4y 2 6z 5 21 3 (22) 22x 2 8y 1 12z 5 2

29y 1 14z 5 25

2x 1 2y 2 4z 5 5

x 1 4y 2 6z 5 21

6y 2 10z 5 4

29y 1 14z 5 25 3 5 245y 1 70z 5 225

6y 2 10z 5 4 3 7 42y 2 70z 5 28

23y 5 3

y 5 21

29(21) 1 14z 5 25 → z 5 21

x 1 4(21) 2 6(21) 5 21 → x 5 23


11. 3x 2 y 1 2z 5 4 3 (22) 26x 1 2y 2 4z 5 28

6x 2 2y 1 4z 5 28 6x 2 2y 1 4z 5 28

0 5 216

No solution.

12. 4x 2 y 1 2z 5 218 3 2 8x 2 2y 1 4z 5 236

3x 1 3y 2 4z 5 44 3x 1 3y 2 4z 5 44

11x 1 y 5 8

2x 1 2y 1 z 5 11 3 4 24x 1 8y 1 4z 5 44

3x 1 3y 2 4z 5 44 3x 1 3y 2 4z 5 44

2x 1 11y 5 88

11x 1 y 5 8 11x 1 y 5 8

2x 1 11y 5 88 3 11 211x 1 121y 5 968

122y 5 976

y 5 8

2x 1 11(8) 5 88 → x 5 0

2(0) 1 2(8) 1 z 5 11 → z 5 25


13. 5x 1 y 2 z 5 6

x 1 y 1 z 5 2

6x 1 2y 5 8

6x 1 2y 5 8 6x 1 2y 5 8

3x 1 y 5 4 3 (22) 26x 2 2y 5 28

0 5 0

Infi nitely many solutions.


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143Algebra 2


14. 2x 1 y 2 z 5 9

5x 1 7y 1 z 5 4

7x 1 8y 5 13

2x 1 6y 1 2z 5 217 2x 1 6y 1 2z 5 217

5x 1 7y 1 z 5 4 3 (22) 210x 2 14y 2 2z 5 28

211x 2 8y 5 225

7x 1 8y 5 13

211x 2 8y 5 225

24x 5 212

x 5 3

7(3) 1 8y 5 13 → y 5 21

5(3) 1 7(21) 1 z 5 4 → z 5 24


15. x 1 y 2 z 5 4

3x 1 2y 1 4z 5 17

2x 1 5y 1 z 5 8 → z 5 x 2 5y 1 8

x 1 y 2 (x 2 5y 1 8) 5 4 → y 5 2

z 5 x 2 5(2) 1 8 → z 5 x 2 2

3x 1 2(2) 1 4(x 2 2) 5 17 → x 5 3

z 5 3 2 2 → z 5 1


16. 2x 2 y 2 z 5 15 → z 5 2x 2 y 2 15

4x 1 5y 1 2z 5 10

2x 2 4y 1 3z 5 220

4x 1 5y 1 2(2x 2 y 2 15) 5 10

8x 1 3y 5 40

2x 2 4y 1 3(2x 2 y 2 15) 5 220

5x 2 7y 5 25

8x 1 3y 5 40 3 7 56x 1 21y 5 280

5x 2 7y 5 25 3 3 15x 2 21y 5 75

71x 5 355

x 5 5

8(5) 1 3y 5 40 → y 5 0

2(5) 2 0 2 z 5 15 → z 5 25


17. 4x 1 y 1 5z 5 240

23x 1 2y 1 4z 5 10

x 2 y 2 2z 5 22 → x 5 y 1 2z 2 2

4(y 1 2z 2 2) 1 y 1 5z 5 240

5y 1 13z 5 232

23(y 1 2z 2 2) 1 2y 1 4z 5 10

2y 2 2z 5 4

5y 1 13z 5 232 5y 1 13z 5 232

2y 2 2z 5 4 3 5 25y 2 10z 5 20

3z 5 212

z 5 24

2y 2 2(24) 5 4 → y 5 4

x 2 4 2 2(24) 5 22 → x 5 26


18. x 1 3y 2 z 5 12

2x 1 4y 2 2z 5 6

2x 2 2y 1 z 5 26 → z 5 x 1 2y 2 6

2x 1 4y 2 2(x 1 2y 2 6) 5 6

12 5 6

No solution.

19. 2x 2 y 1 z 5 22 → z 5 22x 1 y 2 2

6x 1 3y 2 4z 5 8

23x 1 2y 5 3z 5 26

6x 1 3y 2 4(22x 1 y 2 2) 5 8

14x 2 y 5 0

23x 1 2y 1 3(22x 1 y 2 2 5 26

29x 1 5y 5 0

14x 2 y 5 0 3 5 70x 2 5y 5 0

29x 1 5y 5 0 29x 1 5y 5 0

61x 5 0

x 5 0

14(0) 2 y 5 0 → y 5 0

2(0) 2 0 1 z 5 22 → z 5 22



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20. 3x 1 5y 2 z 5 12

x 1 y 1 z 5 0 → x 5 2y 2 z

2x 1 2y 1 2z 5 227

3(2y 2 z) 1 5y 2 z 5 12

2y 2 4z 5 12

2(2y 2 z) 1 2y 1 2z 5 227

3y 1 3z 5 227

2y 2 4z 5 12 3 (23) 26y 1 12z 5 236

3y 1 3z 5 227 3 2 6y 1 6z 5 254

18z 5 290

z 5 25

2y 2 4(25) 5 12 → y 5 24

x 1 (24) 1 (25) 5 0 → x 5 9


21. The error is in line 2. Equation 2 is 3x 1 2y 1 z 5 11. So 2 times Equation 2 is 6x 1 4y 1 2z 5 22, not6x 1 2y 1 2z 5 22.

22. The error is in line 1. Equation 2 is 3x 1 2y 1 z 5 11. After you solve for z, you should get z 5 11 2 3x 2 2y, not z 5 11 1 3x 1 2y.

23. A;

2x 1 5y 1 3z 5 10

2(7) 1 5(1) 1 3(23) 0 10

10 5 10 ✓

3x 2 y 1 4z 5 8

3(7) 2 (1) 1 4(23) 0 8

8 5 8 ✓

5x 2 2y 1 7z 5 12

5(7) 2 2(1) 1 7(23) 0 12

12 5 12 ✓

Because all three equations are true, (7, 1, 23) is the solution of the system.

24. B;

2x 2 2y 2 z 5 6

2(x) 2 2(x 2 3) 2 0 0 6

6 5 6 ✓

2x 1 y 1 3z 5 23

2x 1 (x 2 3) 1 0 0 23

23 5 23

3x 2 3y 1 2z 5 9

3x 2 3(x 2 3) 1 0 0 9

9 5 9 ✓

Because all three equations are true, (x, x 2 3, 0) describes all of the solutions of the system.

25. x 1 5y 2 2z 5 21

2x 2 2y 1 z 5 6

3y 2 z 5 5

2x 2 2y 1 z 5 6 3 (22) 2x 1 4y 2 2z 5 212

22x 2 7y 1 3z 5 7 22x 2 7y 1 3z 5 7

23y 1 z 5 25

3y 2 z 5 5

23y 1 z 5 25

0 5 0


26. 4x 1 5y 1 3z 5 15

x 2 3y 1 2z 5 26 → x 5 3y 2 2z 2 6

2x 1 2y 2 z 5 3

4(3y 2 2z 2 6) 1 5y 1 3z 5 15

17y 2 5z 5 39

2(3y 2 2z 2 6) 1 2y 2 z 5 3

2y 1 z 5 23

17y 2 5z 5 39 17y 2 5z 5 39

2y 1 z 5 23 3 5 25y 1 5z 5 215

12y 5 24

y 5 2

22 1 z 5 23 → z 5 21

2x 1 2(2) 2 (21) 5 3 → x 5 2


27. 6x 1 y 2 z 5 22 6x 1 y 2 z 5 22

2x 1 y 1 2z 5 5 3 6 26x 1 6y 1 12z 5 30

7y 1 11z 5 28

x 1 6y 1 3z 5 23

2x 1 y 1 2z 5 5

7y 1 5z 5 28

7y 1 11z 5 28 7y 1 11z 5 28

7y 1 5z 5 28 3 (21) 27y 2 5z 5 228

6z 5 0

z 5 0

7y 1 5(0) 5 28 → y 5 4

x 1 6(4) 1 3(0) 5 23 → x 5 21



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145Algebra 2


28. x 1 2y 5 21 → x 5 22y 2 1

3x 2 y 1 4z 5 17

24x 1 2y 2 3z 5 230

3(22y 2 1) 2 y 1 4z 5 17

27y 1 4z 5 20

24(22y 2 1) 1 2y 2 3z 5 230

10y 2 3z 5 234

27y 1 4z 5 20 3 3 221y 1 12z 5 60

10y 2 3z 5 234 3 4 40y 2 12z 5 2136

19y 5 276

y 5 24

27(24) 1 4z 5 20 → z 5 22

x 1 2(24) 5 21 → x 5 7


29. 2x 2 y 1 2z 5 221 3 (22) 24x 1 2y 2 4z 5 42

23x 1 2y 1 4z 5 6 23x 1 2y 1 4z 5 6

27x 1 4y 5 48

x 1 5y 2 z 5 25 3 4 4x 1 20y 2 4z 5 100

23x 1 2y 1 4z 5 6 23x 1 2y 1 4z 5 6

x 1 22y 5 106

27x 1 4y 5 48 27x 1 4y 5 48

x 1 22y 5 106 3 7 7x 1 154y 5 742

158y 5 790

y 5 5

27x 1 4(5) 5 48 → x 5 24

2(24) 2 5 1 2z 5 221 → z 5 24


30. 4x 2 8y 1 2z 5 10 4x 2 8y 1 2z 5 10

2x 2 4y 1 z 5 8 3 (22) 24x 1 8y 2 2z 5 216

0 5 26

No solution.

31. 2x 1 5y 2 z 5 216

x 1 y 2 z 5 28

6y 2 2z 5 224

2x 1 3y 1 4z 5 18 2x 1 3y 1 4z 5 18

x 1 y 2 z 5 28 3 (22) 22x 2 2y 1 2z 5 16

y 1 6z 5 34

6y 2 2z 5 224 3 3 18y 2 6z 5 272

y 1 6z 5 34 y 1 6z 5 34

19y 5 238

y 5 22

6(22) 2 2z 5 224 → z 5 6

x 1 (22) 2 6 5 28 → x 5 0


32. 2x 2 y 1 4z 5 19 3 (22) 24x 1 2y 2 8z 5 238

4x 1 2y 1 3z 5 37 4x 1 2y 1 3z 5 37

4y 2 5z 5 21

2x 1 3y 2 2z 5 27 3 4 24x 1 12y 2 8z 5 228

4x 1 2y 1 3z 5 37 4x 1 2y 1 3z 5 37

14y 2 5z 5 9

4y 2 5z 5 21 3 (21) 24y 1 5z 5 1

14y 2 5z 5 9 14y 2 5z 5 9

10y 5 10

y 5 1

4(1) 2 5z 5 21 → z 5 1

2x 1 3(1) 2 2(1) 5 27 → x 5 8


33. x 1 y 1 z 5 3 → x 5 2y 2 z 1 3

3x 2 4y 1 2z 5 228

2x 1 5y 1 z 5 23

3(2y 2 z 1 3) 2 4y 1 2z 5 228

27y 2 z 5 237

2(2y 2 z 1 3) 1 5y 1 z 5 23

6y 1 2z 5 26

27y 2 z 5 237 3 2 214y 2 2z 5 274

6y 1 2z 5 26 6y 1 2z 5 26

28y 5 248

y 5 6

27(6) 2 z 5 237 → z 5 25

x 1 6 1 (25) 5 3 → x 5 2


34. a. Sample answer: b. Sample answer:

x 1 6y 2 2z 5 7 3x 2 2y 1 z 5 10

3x 2 y 1 z 5 4 8x 2 y 1 4z 5 24

x 2 12y 1 10z 5 7 26x 1 4y 2 2z 5 3

Solution is (1, 2, 3). No Solution

c. Sample answer:

4x 2 6y 1 2z 5 72

x 1 y 2 z 5 26

24x 2 4y 1 4z 5 24 Infi nitely many solutions.

35. x 1 1 } 2 y 1

1 } 2 z 5

5 } 2 3 2 2x 1 y 1 z 5 5

1 }

3 x 1

3 } 2 y 1

2 } 3 z 5

13 } 6 3 (26) 22x 2 9y 2 4z 5 213

28y 2 3z 5 28

3 }

4 x 1

1 } 4 y 1

3 } 2 z 5

7 } 4 3 (28) 26x 2 2y 2 12z 5 214

1 }

3 x 1

3 } 2 y 1

2 } 3 z 5

13 } 6 3 18 6x 1 27y 1 12z 5 39

25y 5 25

y 5 1

28(1) 2 3z 5 28 → z 5 0

x 1 1 } 2 (1) 1

1 } 2 (0) 5

5 } 2 → x 5 2



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36. 1 }

3 x 1

5 } 6 y 1

2 } 3 z 5

4 } 3 3 (212) 24x 2 10y 2 8z 5 216

2 }

3 x 1

1 } 6 y 1

3 } 2 z 5

4 } 3 3 6 4x 1 y 1 9z 5 8

29y 1 z 5 28

1 }

6 x 1

2 } 3 y 1

1 } 4 z 5

5 } 6 3 24 4x 1 16y 1 6z 5 20

2 }

3 x 1

1 } 6 y 1

3 } 2 z 5

4 } 3 3 (26) 24x 2 y 2 9z 5 28

15y 2 3z 5 12

29y 1 z 5 28 3 3 227y 1 3z 5 224

15y 2 3z 5 12 15y 2 3z 5 12

212y 5 212

y 5 1

29(1) 1 z 5 28 → z 5 1

1 }

3 x 1

5 } 6 (1) 1

2 } 3 (1) 5

4 } 3 → x 5 2

1 } 2

The solution is 1 2 1 } 2 , 1, 1 2 .

37. Substitute 21 for x, 2 for y, and 23 for z.

21 1 2(2) 2 3(23) 5 a → a 5 12

2(21) 2 2 1 (23) 5 b → b 5 24

2(21) 1 3(2) 2 2(23) 5 c → c 5 10

When a 5 12, b 5 24, and c 5 10, the system has(21, 2, 23) as its solution.

38. w 1 x 1 y 1 z 5 2 Add Equation 1

3w 1 x 1 y 2 z 5 25 to Equation 4.

4w 1 2x 1 y 5 23 New Equation 1

22w 1 x 2 2y 1 z 5 21 Add 21 times Equation 2

3w 1 x 1 y 2 z 5 25 to Equation 4.

w 1 2x 2 y 5 26 New Equation 2

2w 1 2x 2 y 1 2z 5 22 Add Equation 3

6w 1 2x 1 2y 2 2z 5 210 to 2 times Equation 4.

5w 1 4x 1 y 5 212 New Equation 3

4w 1 2x 1 2y 5 23 Add new Equation 1

210w 2 8x 2 2y 5 24 to 22 times new Eq. 3.

26w 2 6x 5 21 Call this Equation 5.

w 1 2x 2 y 5 26 Add new Equation 2

5w 1 4x 1 y 5 212 to new Equation 3.

6w 1 6x 5 218 Call this Equation 6.

26w 2 6x 5 21 Add Equation 5

6w 1 6x 5 218 to Equation 6.

0 5 3

Because 0 5 3 is a false equation, the system has no solution.

39. 2w 1 x 2 3y 1 z 5 4 Add Equation 1

22w 2 2x 1 2y 2 6z 5 228 to 22 times Eq. 4.

2x 2 y 2 5z 5 224 New Equation 1

w 2 3x 1 y 1 z 5 32 Add Equation 2

2w 2 x 1 y 2 3z 5 214 to 21 times Equation 4.

24x 1 2y 2 2z 5 18 New Equation 2

2w 1 2x 1 2y 2 z 5 210 Add Equation 3

w 1 x 2 y 1 3z 5 14 to Equation 4.

3x 1 y 1 2z 5 4 New Equation 3

2x 2 y 2 5z 5 224 Add new Equation 1

3x 1 y 1 2z 5 4 to new Equation 3.

2x 2 3z 5 220 Call this Equation 4.

24x 1 2y 2 2z 5 18 Add new Equation 2

26x 2 2y 2 4z 5 28 to 22 times new Eq. 3.

210x 2 6z 5 10 Call this Equation 5.

24x 1 6z 5 40 Add 22 times Eq. 4

210x 2 6z 5 10 to Equation 5

214x 5 50

x 5 2 25

} 7 Solve for x.

z 5 30

} 7 Substitute into Equation 4 or 5 to fi nd z.

y 5 43

} 7 Substitute into a new equation to fi nd y.

w 5 76

} 7 Substitute into an original equation to fi nd w.

The solution is 1 76 } 7 , 2

25 } 7 ,

43 } 7 ,

30 } 7 2 .

40. 210w 1 5x 1 20y 1 10z 5 235 Add 5 times Eq. 2

22w 2 4x 1 4y 2 10z 5 26 to 22 times Eq. 3.

212w 1 x 1 24y 5 241 New Eq. 2

24w 1 48x 1 120y 5 264 Add 24 times Eq. 1

60w 2 5x 2 120y 5 205 to 25 times new Eq. 2.

84w 1 43x 5 469 New Eq. 1

x 5 3w 2 1 Rewrite Eq 4.

84w 1 43(3w 2 1) 5 469 Substitute 3w 2 1 for x in new Eq. 1.

213w 5 512

w 5 512

} 213 Solve for w.

x 5 441

} 71 Substitute into Eq. 4 to fi nd x.

y 5 2 163

} 213 Substitute into new Eq. 2 to fi nd y.

z 5 2 569

} 213 Substitute into an original equation to fi nd z.


213 ,

441 }

71 , 2

163 } 213 , 2

569 } 273 2 .


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147Algebra 2


41. 2w 1 7x 2 3y 5 41 Add Equation 1

22w 2 2x 1 2y 5 216 to 2 times Equation 4.

5x 2 y 5 25 New Equation 1

2w 2 2x 1 y 5 213 Add Equation 2

w 1 x 2 y 5 8 to 21 times Equation 4.

2x 5 25 New Equation 2

x 5 5 Solve for x.

y 5 0 Substitute into new Equation 1 to fi nd y.

w 5 3 Substitute into an original equation for fi nd w.

z 5 22 Substitute into original Equation 3 to fi nd z.

The solution is (3, 5, 0, 22).

Problem Solving

42. p 5 cost of a small pizza

x 5 cost of a liter of soda

y 5 cost of a salad

Equation 1: 2p 1 x 1 y 5 14

Equation 2: p 1 x 1 3y 5 15

Equation 3: 3p 1 x 5 16 → x 5 16 2 3p

2p 1 (16 2 3p) 1 y 5 14

2p 1 y 5 22

p 1 (16 2 3p) 1 3y 5 15

22p 1 3y 5 21

2p 1 y 5 22 3 (22) 2p 2 2y 5 4

22p 1 3y 5 21 22p 1 3y 5 21

y 5 3

2p 1 y 5 22 → p 5 5

2(5) 1 x 1 3 5 14 → x 5 1

A small pizza costs $5, a liter of soda costs $1, and a salad costs $3.

43. x 5 gallons received in 1st delivery

y 5 gallons received in 2nd delivery

z 5 gallons received in 3rd delivery

Equation 1: 0.7x 1 0.5y 1 0.3z 5 1200

Equation 2: 0.2x 1 0.3y 1 0.3z 5 900

Equation 3: 0.1x 1 0.2y 1 0.4z 5 1000

21.4x 2 y 2 0.6z 5 22400

1.4x 1 2.1y 1 2.1z 5 6300

1.1y 1 1.5z 5 3900

0.7x 1 0.5y 1 0.3z 5 1200

20.7x 2 1.4y 2 2.8z 5 27000

20.9y 2 2.5z 5 25800

1.1y 1 1.5z 5 3900 3 5

20.9y 2 2.5z 5 25800 3 3

5.5y 1 7.5z 5 19,500

22.7y 2 7.5z 5 217,400

2.8y 5 2100

y 5 750

1.1(750) 1 1.5z 5 3900 → z 5 2050

0.7x 1 0.5(750) 1 0.3(2050) 5 1200 → x 5 300

The health club received 300 gallons in the fi rst delivery, 750 gallons in the second delivery, and 2050 gallons in the third delivery.

44. a. c 5 comedy shows

d 5 dramas

r 5 reality shows

30c 1 60d 1 60r 5 360

c 1 d 1 r 5 7

2c 5 d

b. 30c 1 60d 1 60r 5 360 Add Eq. 1

260c 2 60d 2 60r 5 2420 to 260 times Eq. 2.

230c 5 260

c 5 2 Solve for c.

d 5 4 Substitute into Eq. 3 to fi nd d.

r 5 1 Substitute to fi nd r.

There are two comedies, four dramas, and one reality show on the tape.

c. New system of equations:

30c 1 60d 1 60r 5 360

c 1 d 1 r 5 5

2c 5 d

30c 1 60d 1 60r 5 360 Add Equation 1

260c 2 60d 2 60r 5 2300 to 260 times Eq. 2.

230c 5 60

c 5 22 Solve for c.

Because a negative number of shows cannot be recorded, this situation is impossible.

45. a. x 5 athletes who fi nished in 1st place

y 5 athletes who fi nished in 2nd place

z 5 athletes who fi nished in 3rd place

Equation 1: x 1 y 1 z 5 20

Equation 2: 5x 1 3y 1 z 5 68

Equation 3: y 5 x 1 z

x 1 (x 1 z) 1 z 5 20 5x 1 3(x 1 z) 1 z 5 68

2x 1 2z 5 20 8x 1 4z 5 68

2x 1 2z 5 20 3 (24) 28x 2 8z 5 280

8x 1 4z 5 68 8x 1 4z 5 68

24z 5 212

z 5 3

2x 1 2(3) 5 20 → x 5 7

7 1 y 1 3 5 20 → y 5 10

Seven athletes fi nished in fi rst place, ten fi nished second, and three fi nished third.


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b. The new Equation 2 would be 5x 1 3y 1 z 5 70.

x 1 (x 1 z) 1 z 5 20 5x 1 3(x 1 z) 1 z 5 70

2x 1 2z 5 20 8x 1 4z 5 70

2x 1 2z 5 20 3 (24) 28x 2 8z 5 280

8x 1 4z 5 70 8x 1 4z 5 70

24z 5 210

z 5 2.5

This claim must be false because you cannot have 2.5 athletes.

46. m 5 cost of mixed nuts

g 5 cost of granola

d 5 cost of dried fruit

Equation 1: m 1 0.5g 1 d 5 8

Equation 2: 2m 1 0.5g 1 0.5d 5 9

Equation 3: m 1 2g 1 0.5d 5 9

m 1 0.5g 1 d 5 8 m 1 0.5g 1 d 5 8

m 1 2g 1 0.5d 5 9 3 (22) 22m 2 4g 2 d 5 218

2 m 2 3.5g 5 210

2m 1 0.5g 1 0.5d 5 9 2m 1 0.5g 1 0.5d 5 9

m 1 2g 1 0.5d 5 9 3 (21) 2m 2 2g 2 0.5d 5 29

m 2 1.5g 5 0

2m 2 3.5g 5 210

m 2 1.5g 5 0

25g 5 210

g 5 2

m 2 1.5(2) 5 0 → m 5 3

3 1 0.5(2) 1 d 5 8 → d 5 4

Mixed nuts cost $3 per pound, granola costs $2 per pound, and dried fruit costs $4 per pound.

47. a. r 5 number of roses

l 5 number of lilies

i 5 number of irises

Equation 1: r 1 l 1 i 5 12

Equation 2: 2.5r 1 4l 1 2i 5 32

Equation 3: r 5 2(l 1 i) or r 5 2l 1 2i

b. (2l 1 2i) 1 l 1 i 5 12

3l 1 3i 5 12

2.5(2l 1 2i) 1 4l 1 2i 5 32

9l 1 7i 5 32

3l 1 3i 5 12 3 (23) 29l 2 9i 5 236

9l 1 7i 5 32 9l 1 7i 5 32

22i 5 24

i 5 2

3l 1 3(2) 5 12 → l 5 2

r 1 2 1 2 5 12 → r 5 8

Each bouquet should contain 8 roses, 2 lilies, and 2 irises.

c. r 1 l 1 i 5 12 r 1 l 1 i 5 12

(2l 1 2i) 1 l 1 i 5 12 r 1 4 5 12

3l 1 3i 5 12 r 5 8

3(l 1 i) 5 12

l 1 i 5 4

No; Three possible solutions are:(8 roses, 2 lilies, 2 irises), (8 roses, 3 lilies, 1 iris), (8 roses, 1 lily, 3 irises).

48. t 5 weight of one tangerine

b 5 weight of one banana

g 5 weight of one grapefruit

t 1 a 5 g Equation 1

t 1 b 5 a Equation 2

2g 5 3b Equation 3

t 1 (t 1 b) 5 g Substitute t 1 b for a in Eq. 1.

2t 1 b 5 g New Equation 1

g 5 3 } 2 b Rewrite Equation 3.

2t 1 b 5 3 } 2 b Substitute

3 }

2 b for g in new Eq. 1.

2 1 } 2 b 5 22t

b 5 4t Solve for b.

t 1 4t 5 a Substitute 4t for b in Eq. 2

5t 5 a

Because 5t 5 a, fi ve tangerines will balance the applein the fi nal picture.

Mixed Review

49. 15 1 (28) 5 7

50. 24 2 (213) 5 24 1 13 5 9

51. 15 p 7 5 105 52. 24(28) 5 4 p 8 5 32

53. (1, 24), (2, 6)

m 5 6 2 (24)

} 2 2 1 5 10

The slope is positive, so the line rises.

54. (4, 2), (218, 1)

m 5 1 2 2

} 218 2 4 5

1 } 22

The slope is positive, so the line rises.

55. (6, 26), (26, 6)

m 5 6 2 (26)

} 26 2 6 5 21

The slope is negative, so the line falls.

56. (25, 2), (25, 10)

m 5 10 2 2

} 25 2 (25)

5 8 } 0

The slope is undefi ned, so the line is vertical.


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149Algebra 2


57. (22, 4), (26, 8)

m 5 8 2 4 }

26 2 (22) 5 21

The slope is negative, so the line falls.

58. (27, 3), (5, 3)

m 5 3 2 3

} 5 2 (27)

5 0

The slope is 0, so the line is horizontal.

59. 3x 2 y 5 27 3 3 9x 2 3y 5 221

2x 1 3y 5 21 2x 1 3y 5 21

11x 5 0

x 5 0

3(0) 2 y 5 27 → y 5 7


60. 3x 1 2y 5 23 3 3 9x 1 6y 5 29

4x 2 3y 5 238 3 2 8x 2 6y 5 276

17x 5 285

x 5 25

3(25) 1 2y 5 23 → y 5 6


61. 5x 1 y 5 11 → y 5 11 2 5x

2x 1 3y 5 219

2x 1 3(11 2 5x) 5 219

213x 5 252

x 5 4

5(4) 1 y 5 11 → y 5 29


Mixed Review of Problem Solving (p. 186)

1. a. n 5 the number of necklaces

b 5 the number of bracelets

3.5n 1 2.5b 5 121 Equation 1

9.0n 1 7.5b 5 324 Equation 2

b. 210.5n 2 7.5b 5 2363 Add 23 times Eq. 1

9.0n 1 7.5b 5 324 to Equation 2.

21.5n 5 239

n 5 26 Solve for n.

b 5 12 Substitute to fi nd b.

At the craft fair, 26 necklaces and 12 bracelets were sold.

2. a. t 5 number of tapers

p 5 number of pillars

j 5 number of jar candles

t 1 4p 1 6j 5 24 Equation 1

t 1 p 1 j 5 8 Equation 2

t 5 p 1 j Equation 3

b. (p 1 j) 1 4p 1 6j 5 24 Substitute p 1 j for t in Equation1.

5p 1 7j 5 24 New Equation 1

(p 1 j) 1 p 1 j 5 8 Substitute p 1 j for t in Equation 2.

2p 1 2j 5 8 New Equation 2

10p 1 14j 5 48 Add 2 times new Eq. 1.

210p 2 10j 5 240 to 25 times new Eq. 2.

4j 5 8

j 5 2 Solve for j.

p 5 2 Substitute to fi nd p.

t 5 4 Substitute to fi nd t.

There are four tapers, two pillars, and two jar candles in each basket.

c. t 1 p 1 j 5 8 t 1 p 1 j 5 8

(p 1 j) 1 p 1 j 5 8 t 1 4 5 8

2p 1 2j 5 8 t 5 4

2(p 1 j) 5 8

p 1 j 5 4

No; three possible solutions are: (4 tapers, 2 pillars, 2 jar candles), (4 tapers, 3 pillars, 1 jar candle), (4 tapers, 1 pillar, 3 jar candles).

3. Sample answer: y > 1, x > 2, y < 4 2 x

4. a. s 5 the number of small tables

l 5 the number of large tables

s 1 l 5 20 Equation 1

4s 1 6l 5 90 Equation 2

24s 2 4l 5 280 Add 24 times Equation. 1

4s 1 6l 5 90 to Equation 2.

2l 5 10

l 5 5 Solve for l.

s 5 15 Substitute to solve for s.

The restaurant has fi fteen 4-seat tables and fi ve 6-seat tables.

b. Sample answer:

1. The restaurant could buy fi ve more 6-seat tables and fi ve more 4-seat tables.

2. The restaurant could buy one additional 6-seat table and eleven more 4-seat tables.

3. The restaurant could buy three more 6-seat tables and 8 more 4-seat tables.


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5. s 5 the price of a soda

p 5 the price of a pretzel

h 5 the price of a hot dog

s 1 p 1 2h 5 7 Equation 1

2s 1 p 1 2h 5 8 Equation 2

s 1 4h 5 10 Equation 3

s 1 p 1 2h 5 7 Add Equation 1

22s 2 p 2 2h 5 28 to 21 times Equation 2.

2s 5 21

s 5 1 Solve for s.

h 5 2.25 Substitute into original Equation. 3 to fi nd h.

The price of one hotdog is $2.25.

6. h 5 the number of hours skating

b 5 the number of hours bicycling

6h 1 8b 5 34 Equation 1

b 1 h 5 5 Equation 2

b 5 5 2 h Rewrite Equation 2.

6h 1 8(5 2 h) 5 34 Substitute 5 2 h for b in Eq. 1.

22h 5 26

h 5 3 Solve for h.

The student spends three hours skating.

7. a. y 5 0.29x 1 73.14

b. y 5 0.14x 1 79.12

c.

50 10 15 20 25 30 35 40 45

720

7476788082848688

Years since 1996

Life

sp

an (

year

s)

x

y

(40, 84.7)

The lines intersect at about (40, 84.7). This means that in the year 2036, men and women will both have a life span of about 84.7 years.

8. a. s 5 the number of small chairs

l 5 the number of large chairs

45s 1 70l ≤ 2000

80s 1 110l ≥ 2750

Number of small chairs

Nu

mb

er o

f la

rge

chai

rs

0 10 20 30 40 s

l

0

10

20

30

b. Sample answer:

30 small chairs and 5 large chairs



Lesson 3.5


1. F 22 5 11 4 26 8G1 F 23 1 25

22 28 4G 5 F 22 1 (23) 5 1 1 11 1 (25)

4 1 (22) 26 1 (28) 8 1 4G 5 F 25 6 6

2 214 12G 2. F 24 0

7 22

23 1G 2 F 2 2

23 0

5 214G

5 F 24 2 2 0 2 2

7 2 (23) 22 2 0

23 2 5 1 2 (214)G

5 F 26 22

10 22

28 15G

3. 24 F 2 21 23

27 6 1

22 0 25G

5 F 24(2) 24(21) 24(23)

24(27) 24(6) 24(1)

24(22) 24(0) 24(25)G

5 F28 4 12

28 224 24

8 0 20G

4. 3 F 4 21 23 25G 1 F 22 22

0 6G 5 F 3(4) 1 (22) 3(21) 1 (22)

3(23) 1 0 3(25) 1 6G 5 F 10 25 29 29G

5. B 2 A 5 F 95 114

316 215

205 300G 2 F 125 100

278 251

225 270G

5 F 95 2 125 114 2 100

316 2 278 215 2 251

205 2 225 300 2 270G 5 F 230 14

38 236

220 30G

This matrix represents the change in the number of DVD racks sold from last month to this month.


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151Algebra 2


6. 22 (F 23x 21 4 yG 1 F 9 24

25 3G) 5 F 12 10

2 218G 22 F23x 1 9 25

21 y 1 3G 5 F 12 10

2 218G F22(23x 1 9) 22(25)

22(21) 22(y 1 3)G 5 F 12 10

2 218G F6x 2 18 10

2 22y 2 6G 5 F 12 10

2 218G 6x 2 18 5 12 22y 2 6 5 218

x 5 5 y 5 6

The solution is x 5 5 and y 5 6.


Skill Practice

1. The dimensions of a matrix with 3 rows and 4 columns are 3 3 4.

2. To determine if two matrices are equal, fi rst compare the dimensions and then compare the corresponding elements. If they are the same, then the matrices are equal.

3. The fi nal matrix has the wrong dimensions.

It should be F 13.1

21.2 G.

4. F 5 2

21 8G 1 F 28 10

26 3 G 5 F 5 1 (28) 2 1 10 21 1 (26) 8 1 3G

5 F 23 12 27 11G

5. F 10 28 5 23G 2 F 12 23

3 24G 5 F10 2 12 28 2 (23)

5 2 3 23 2 (24)G 5 F 22 25 2 1G

6. This operation is not possible. You cannot subtract a 2 3 1 matrix from a 2 3 2 matrix.

7. F1.2 5.3

0.1 4.4

6.2 0.7G 1 F2.4 20.6

6.1 3.1

8.1 21.9G

5 F 1.2 1 2.4 5.3 1 (20.6)

0.1 1 6.1 4.4 1 3.1

6.2 1 8.1 0.7 1 (21.9)G 5 F 3.6 4.7

6.2 7.5

14.3 21.2G

8. This operation is not possible. You cannot add a 3 3 3 matrix to a 3 3 2 matrix.

9. F 7 23

12 5

24 11G 2 F 9 2

22 6

6 5G

5 F 7 2 9 23 2 2

12 2 (22) 5 2 6

24 2 6 11 2 5G 5 F 22 25

14 21

210 6G

10. 2 F 21 4 3 26 G 5 F 2(21) 2(4)

2(3) 2(26)G

5 F 22 8 6 212G

11. 23 F 2 0 25 4 7 23G 5 F 23(2) 23(0) 23(25)

23(4) 23(7) 23(23)G 5 F 26 0 15

212 221 9G 12. 24 F 2 23 22

2 5 } 8 11 }

2 7 }

4 G

5 F 24(2) 24(23) 24(22)

24 1 2 5 } 8 2 24 1 11 } 2 2 24 1 7 }

4 2 G

5 F 28 12 8

5 } 2 222 27G 13. 1.5 F 22 3.4 1.6

5.4 0 23G 5 F 1.5(22) 1.5(3.4) 1.5(1.6)

1.5(5.4) 1.5(0) 1.5(23)G 5 F 23 5.1 2.4

8.1 0 24.5G 14.

1 }

2 F 22 8 12

20 21 0

28 10 2G

5 F 1 } 2 (22) 1 }

2 (8) 1 } 2 (12)

1 } 2 (20) 1 }

2 (21) 1 } 2 (0)

1 } 2 (28) 1 }

2 (10) 1 } 2 (2)

G 5 F 21 4 6

10 2 1 } 2 0

24 5 1G


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15. 22.2 F 6 3.1 4.5

21 0 2.5

5.5 21.8 6.4G

5 F 22.2(6) 22.2(3.1) 22.2(4.5)

22.2(21) 22.2(0) 22.2(2.5)

22.2(5.5) 22.2(21.8) 22.2(6.4)G

5 F 213.2 26.82 29.9

2.2 0 25.5

212.1 3.96 214.08G

16. A 1 B 5 F 5 24

3 21G 1 F 18 212

26 0G 5 F 5 1 18 24 1 (212)

3 1 (26) 21 1 0G 5 F 23 216

23 21G17. B 2 A 5 F 18 212

26 0G 2 F 5 24

3 21G 5 F 18 2 5 212 2 (24)

26 2 3 0 2 (21)G 5 F 13 28

29 1G18. 4A 2 B 5 4F 5 24

3 21G 2 F 18 212

26 0G 5 F 4(5) 2 18 4(24) 2 (212)

4(3) 2 (26) 4(21) 2 0G 5 F 2 24

18 24G19.

2 }

3 B 5

2 } 3 F 18 212

26 0G 5 F 2 } 3 (18) 2 }

3 (212)

2 } 3 (26) 2 }

3 (0)G

5 F 12 28

24 0G20. C 1 D 5 F 1.8 21.5 10.6

28.8 3.4 0G 1 F 7.2 0 25.4

2.1 21.9 3.3G 5 F 1.8 1 7.2 21.5 1 0 10.6 1 (25.4)

28.8 1 2.1 3.4 1 (21.9) 0 1 3.3G 5 F 9 21.5 5.2

26.7 1.5 3.3G21. C 1 3D 5 F 1.8 21.5 10.6

28.8 3.4 0G 1 3F 7.2 0 25.4

2.1 21.9 3.3G 5 F 1.8 1 3(7.2) 21.5 1 3(0) 10.6 1 3(25.4)

28.8 1 3(2.1) 3.4 1 3(21.9) 0 1 3(3.3)G 5 F 23.4 21.5 25.6

22.5 22.3 9.9G

22. D 2 2C 5 F 7.2 0 25.4

2.1 21.9 3.3G 2 2F 1.8 21.5 10.6

28.8 3.4 0G 5 F 7.2 2 2(1.8) 0 2 2(21.5) 25.4 2 2(10.6)

2.1 2 2(28.8) 21.9 2 2(3.4) 3.3 2 2(0)G 5 F 3.6 3 226.6

19.7 28.7 3.3G 23. 0.5C 2 D 5 0.5F 1.8 21.5 10.6

28.8 3.4 0G 2 F 7.2 0 25.4

2.1 21.9 3.3G5 F 0.5(1.8) 2 7.2 0.5(21.5) 2 0 0.5(10.6) 2 (25.4)

0.5(28.8) 2 2.1 0.5(3.4) 2 (21.9) 0.5(0) 2 3.3G5 F 26.3 20.75 10.7

26.5 3.6 23.3G 24. F 21 3x

24 5G 5 F 21 218

2y 5G 3x 5 218 2y 5 24

x 5 26 y 5 22


25. F 22x 6

1 28G 1 2 F 5 21

27 6G 5 F 29 4

213 yG F 22x 1 2(5) 6 1 2(21)

1 1 2(27) 28 1 2(6)G 5 F 29 4

213 yG F 22x 1 10 4

213 4G 5 F 29 4

213 yG 22x 1 10 5 29 y 5 4

x 5 19

} 2

The solution is x 5 19

} 2 and y 5 4.

26. 2F 8 2x

5 6G 2 F 3 29

10 24yG 5 F 13 4

0 16G F 2(8) 2 3 2(2x) 2 (29)

2(5) 2 10 2(6) 2 (24y)G 5 F 13 4

0 16G F 13 22x 1 9

0 12 1 4yG 5 F 13 4

0 16G 22x 1 9 5 4 12 1 4y 5 16

x 5 5 } 2 y 5 1

The solution is x 5 5 } 2 and y 5 1.


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153Algebra 2


27. 4xF 21 2

3 6G 5 F 8 216

224 3yG F 4x(21) 4x(2)

4x(3) 4x(6)G 5 F 8 216

224 3y G F 24x 8x

12x 24xG 5 F 8 216

224 3yG 24x 5 8 3y 5 24x

x 5 22 3y 5 24(22)

y 5 216


28. C; 2x 5 6.4 3y 5 20.75

x 5 3.2 y 5 20.25

3x 2 2y 5 3(3.2) 2 2(20.25)

5 9.6 1 0.50

5 10.1

29. Sample answer:

A 5 F 10 3

25 4 G, B 5 F 5 2

23 2 G30. a. X 1 F 25 0

4 23G 5 F 7 28

23 5 G F a 1 (25) b 1 0

c 1 4 d 1 (23)G 5 F 7 28

23 5 G a 5 12, b 5 28, c 5 27, and d 5 8.

X 5 F 12 28

27 8 G b. X 2 F 2 3

5 0 G 5 F 8 6

21 3 G F a 2 2 b 2 3

c 2 5 d 2 0G 5 F 8 6

21 3G a 5 10, b 5 9, c 5 4, and d 5 3.

X 5 F 10 9

4 3 G c. 2X 1 F 23 1

4 7 G 5 F 8 29

0 10 G F 2a 1 (23) 2b 1 1

2c 1 4 2d 1 7G 5 F 8 29

0 10 G a 5 211, b 5 10, c 5 4, and d 5 23.

X 5 F 211 10

4 23 G

d. 3X 2 F 11 26

2 1 G 5 F 213 15

219 2 G F 3a 2 11 3b 2 (26)

3c 2 2 3d 2 1G 5 F 213 15

219 2 G a 5 2

2 } 3 , b 5 3, c 5 2

17 } 3 , and d 5 1.

X 5 F 2 2 } 3 3

2 17

} 3 1G

Problem Solving

31. Change in sales 5 Sales for 2004 2 Sales for 2003

5 F 32 47 30 19

5 16 20 14

29 39 36 31G

2 F 32 42 29 20

12 17 25 16

28 40 32 21G

5 F 32 2 32 47 2 42 30 2 29 19 2 20

5 2 12 16 2 17 20 2 25 14 2 16

29 2 28 39 2 40 36 2 32 31 2 21G

5 F 0 5 1 21

27 21 25 22

1 21 4 10G

32. City mpg Highway mpg

Economy

Mid-size

Mini-van

Suv

F 32 40

24 34

18 25

19 22

G After an 8% increase: 1.08F 32 40

24 34

18 25

19 22

G 5 F 34.56 43.2

25.92 36.72

19.44 27

20.52 23.76

G 33. a. May(M) June (J)

A B C A B C

Downtown

Mall

F 31 42 18

22 25 11 GF 25 36 12

38 32 15 G b. M 1 J 5 F 31 42 18

22 25 11 G 1 F 25 36 12

38 32 15 G 5 F 31 1 25 42 1 36 18 1 12

22 1 38 25 1 32 11 1 15 G 5 F 56 78 30

60 57 26 G This matrix represents the total sales for May and June.


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c. 1 }

2 (M 1 J) 5

1 } 2 F 56 78 30

60 57 26G 5 F 28 39 15

30 28.5 13 G34. No, the matrix A 1 B does not give meaningful

information. The team size in each matrix is an average, but the sum of the two averages is not an average.

35. A 5 F 1 1 5 5

1 4 1 4G

2

x

y

2

3A 5 F 3 3 15 15

3 12 3 12 G The height of the large rectangle is three times the

height of the small rectangle, and the width of the large rectangle is three times the width of the small rectangle. Therefore, the large rectangle is nine times the size of the small rectangle.

Mixed Review

36. 5 1 (28) 5 23 37. 27 1 6 5 21

38. 8(27) 5 256

39. 26 2 (215) 5 26 1 15 5 9

40. 25(29) 5 45

41. 6 2 (218) 5 6 1 18 5 24

42. y 5 x 1 1 when x ≤ 0.

y 5 2x 1 1 when x ≥ 0.

y 5 x 1 1

43. vertex: (1, 1) → y 5 ax 2 1 1 1

passes through (3, 22) → 22 5 a(3 2 1) 1 1

a 5 2 3 } 2

y 5 2 3 } 2 x 2 1 1 1

44. vertex: (3, 21) → y 5 ax 2 3 2 1

passes through (2, 2) → 2 5 a2 2 3 2 1

a 5 3

y 5 3x 2 3 2 1

45. 0 1 2(3) ≤? 23 25 1 2(1) ≤? 23

6 µ 23 23 ≤ 23 ✓

(25, 1) is a solution of the inequality, but (0, 3) is not.

46. 5(25) 2 0 >?2 5(5) 2 23 >?2

225 ò 2 2 ò 2

Neither ordered pair is a solution.

47. 28(21) 2 3(1) <? 5 28(3) 2 3(29) <? 5

5ñ 5 3 < 5 ✓


48. 21(2) 2 10(3) >?4 21(21) 2 10(0) >?4

12 > 4 ✓ 221ò 4


Quiz 3.3–3.5 (p. 193)

1.

1

x

y

22

2.

1

x

y

1

3.

x

y

2

1

4. 1

x

y

21

5.

8

x

y

22

6.

1

x

y

21

7. 2x 2 y 2 3z 5 5

x 1 2y 2 5z 5 211

2x 2 3y 5 10 → x 5 210 2 3y

2(210 2 3y) 2 y 2 3z 5 5

27y 2 3z 5 25

(210 2 3y) 1 2y 2 5z 5 211

2y 2 5z 5 21

27y 2 3z 5 25 27y 2 3z 5 25

2y 2 5z 5 21 3 (27) 7y 1 35z 5 7

32z 5 32

z 5 1

27y 2 3(1) 5 25 → y 5 24

x 1 2(24) 2 5(1) 5 211 → x 5 2


8. x 1 y 1 z 5 23 3 (22) 22x 2 2y 2 2z 5 6

4x 2 5y 1 2z 5 16 4x 2 5y 1 2z 5 16

2x 2 7y 5 22

2x 2 3y 1 z 5 9 3 (22) 24x 1 6y 2 2z 5 218

4x 2 5y 1 2z 5 16 4x 2 5y 1 2z 5 16

y 5 22

2x 2 7(22) 5 22 → x 5 4

4 1 (22) 1 z 5 23 → z 5 25



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155Algebra 2


9. 2x 2 4y 1 3z 5 1

22x 1 5y 2 2z 5 2

y 1 z 5 3

6x 1 2y 1 10z 5 19 6x 1 2y 1 10z 5 19

22x 1 5y 2 2z 5 2 3 3 26x 1 15y 2 6z 5 6

17y 1 4z 5 25

y 1 z 5 3 3 (24) 24y 2 4z 5 212

17y 1 4z 5 25 17y 1 4z 5 25

13y 5 13

y 5 1

1 1 z 5 3 → z 5 2

2x 2 4(1) 1 3(2) 5 1 → x 5 2 1 } 2

The solution is 1 2 1 } 2 , 1, 2 2 .

10. A 1 B 5 F 2 25

3 21 G 1 F 24 3

8 10 G 5 F 2 1 (24) 25 1 3

3 1 8 21 1 10 G 5 F 22 22

11 9 G 11. B 2 2A 5 F 24 3

8 10 G 2 2 F 2 25

3 21 G 5 F 24 2 2(2) 3 2 2(25)

8 2 2(3) 10 2 2(21) G 5 F 28 13

2 12G 12. The sum 3A 1 C is not possible. You cannot add a 2 3 2

matrix to a 2 3 3 matrix.

13. 2 } 3 C 5 F 2 }

3 (26) 2 }

3 (22) 2 }

3 (9)

2 } 3 (1) 2 }

3 (24) 2 }

3 (21)G

5 F24 2 4 } 3 6

2 } 3 2 8 } 3 2 2 } 3 G

14. e 5 pounds of Empire apples

r 5 pounds of Red Delicious apples

g 5 pounds of Golden Delicious apples

Equation 1: e 1 r 1 g 5 21

Equation 2: 1.4e 1 1.1r 1 1.3g 5 25

Equation 3: r 5 2(g 1 e), or 22e 1 r 2 2g 5 0

e 1 r 1 g 5 21 3 (21.4)

21.4e 2 1.4r 2 1.4g 5 229.4

1.4e 1 1.1r 1 1.3g 5 25

1.4e 1 1.1r 1 1.3g 5 25

20.3r 2 0.1g 5 24.4

e 1 r 1 g 5 21 3 2 2e 1 2r 1 2g 5 42

22e 1 r 2 2g 5 0 22e 1 r 2 2g 5 0

3r 5 42

r 5 14

20.3(14) 2 0.1g 5 24.4 → g 5 2

e 1 14 1 2 5 21 → e 5 5

You should buy 5 pounds of Empire, 14 pounds of Red Delicious, and 2 pounds of Golden Delicious apples.

Graphing Calculator Activity 3.5 (p. 194)

1. F 19 25

8 8G 2. F 32.24 17.68 23.12

22.08 14.56 28.32G 3. F 26 11 25

21 11 7

18 20 28G 4. F 22 225

38 239

211 12G

5. R M S C Hardcover

Paperback

F 44 36 38 21

76 44 22 50 GLesson 3.6


1. AB is defi ned and has dimensions 5 3 2.

2. AB is not defi ned because the number of columns in A does not equal the number of rows in B.

3. AB 5 F 23 3

1 22 GF 1 5

23 22 G 5 F 23(1) 1 3(23) 23(5) 1 3(22)

1(1) 1 (22)(23) 1(5) 1 (22)(22)G

5 F 212 221

7 9 G 4. A(B 2 C) 5 F 21 2

23 0

4 1G (F 3 2

22 21G 2 F 24 5

1 0G) 5 F 21 2

23 0

4 1G F 7 23

23 21G 5 F 21(7) 1 2(23) 21(23) 1 2(21)

23(7) 1 0(23) 23(23) 1 0(21)

4(7) 1 1(23) 4(23) 1 1(21)G

5 F 213 1

221 9

25 213G


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5. AB 2 AC 5 (F 21 2

23 0

4 1G F 3 2

22 21 G) 2 (F 21 2

23 0

4 1G F 24 5

1 0 G) 5 F 21(3) 1 2(22) 21(2) 1 2(21)

23(3) 1 0(22) 23(2) 1 0(21)

4(3) 1 1(22) 14(2) 1 1(21)G

2 F 21(24) 1 2(1) 21(5) 1 2(0)

23(24) 1 0(1) 23(5) 1 0(0)

4(24) 1 1(1) 4(5) 1 1(0)G

5 F 27 24

29 26

10 7G 2 F 6 25

12 215

215 20G

5 F 213 1

221 9

25 213G

6. 2 1 } 2 (AB) 5 2

1 } 2 F 27 24

29 26

10 7G

5 F 2 1 } 2 (27) 2 1 } 2 (24)

2 1 } 2 (29) 2 1 } 2 (26)

2 1 } 2 (10) 2 1 } 2 (7)G 5 F 7

} 2 2

9 } 2 3

25 2 7 } 2 G

7. F 14 30 18

16 25 20 G F 75

1

45G

5 F 14(75) 1 30(1) 1 18(45)

16(75) 1 25(1) 1 20(45)G 5 F 1890

2125 G The total cost of equipment for the women’s team is

$1890 and the total cost of equipment for the men’s team is $2125.


Skill Practice

1. The product of matrices A and B is defi ned provided the number of columns in A is equal to the number of rowsin B.

2. Sample answer: To fi nd the element in the fi rst row and fi rst column of AB, multiply each element in the fi rst row of A by the corresponding element in the fi rst column of B, then add the products.



5. AB is not defi ned. The number of columns in A does not equal the number of rows in B.


7. AB is not defi ned. The number of columns in A does not equal the number of rows in B.


9. A; Matrix A has 2 rows and matrix B has 2 columns, so AB has dimensions 2 3 2.

10. F 3 21G F 5

7 G 5 F 3(5) 1 (21)(7) G 5 F 8 G

11. F 1

4 G F 22 1 G 5 F 1(22) 1(1)

4(22) 4(1) G 5 F 22 1

28 4 G 12. Not defi ned; The number of columns in the fi rst matrix

does not equal the number of rows in the second matrix.

13.

F 9 23

0 2 G F 0 1

4 22 G 5 F 9(0) 1 (23)(4) 9(1) 1 (23)(22)

0(0) 1 2(4) 0(1) 1 2(22)G

5 F 212 15

8 24 G 14. F 5 0

24 1 G F 23 2

6 2 G 5 F 5(23) 1 0(6) 5(2) 1 0(2)

24(23) 1 1(6) 24(2) 1 1(2)G

5 F 215 10

18 26 G 15. F 5 2

0 24

1 6G F 3 7

22 0 G 5 F 5(3) 1 2(22) 5(7) 1 2(0)

0(3) 1 (24)(22) 0(7) 1 (24)(0)

1(3) 1 6(22) 1(7) 1 6(0)G

5 F 11 35

8 0

29 7G

16. Not defi ned; The number of columns in the fi rst matrix does not equal the number of rows in the second matrix.

17. F 1 3 0

2 12 24 G F 9 1

4 23

22 4G

5 F 1(9) 1 3(4) 1 0(22) 1(1) 1 3(23) 1 0(4)

2(9) 1 12(4) 1 (24)(22) 2(1) 1 12(23) 1 (24)(4)G

5F 21 28

74 250G


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157Algebra 2


18. F 2 5

21 4

3 27G F 0 1 5

23 10 24 G 5 F 2(0) 1 5(23) 2(1) 1 5(10) 2(5) 1 5(24)

21(0) 1 4(23) 21(1) 1 4(10) 21(5) 1 4(24)

3(0) 1 (27)(23) 3(1) 1 (27)(10) 3(5) 1 (27)(24)G

5 F 215 52 210

212 39 221

21 267 43G

19. The multiplication should be row 1 of the left matrix by column 1 of the right matrix.

3(7) 1 (21)(1) 5 20

20. The multiplication should be row 1 of the left matrix by column 1 of the right matrix.

2(4) 1 5(3) 5 23

21. B;

F 1 24

3 22 G F 4 21

0 23 G 5 F 1(4) 1 (24)(0) 1(21) 1 (24)(23)

3(4) 1 (22)(0) 3(21) 1 (22)(23)G

5 F 4 11

12 3G22. 3AB 5 3F 5 23

22 4GF 0 1

4 22G

5 F 15 29

26 12GF 0 1

4 22G

5 F 15(0) 1 (29)(4) 15(1) 1 (29)(22)

26(0) 1 12(4) 26(1) 1 12(22)G

5 F 236 33

48 230G

23. 2 1 } 2 AC 5 2

1 } 2 F 5 23

22 4G F 26 3

4 1G 5 2

1 } 2 F 5(26) 1 (23)(4) 5(3) 1 (23)(1)

22(26) 1 4(4) 22(3) 1 4(1)G

5 F 2 1 } 2 (242) 2 1 } 2 (12)

2 1 } 2 (28) 2

1 } 2 (22)

G 5 F 21 26

214 1G

24. AB 1 AC 5 F 5 23

22 4G F 0 1

4 22 G 1 F 5 23

22 4G F 26 3

4 1G 5 F 5(0) 1 (23)(4) 5(1) 1 (23)(22)

22(0) 1 4(4) 22(1) 1 4(22)G

1 F 5(26) 1 (23)(4) 5(3) 1 (23)(1)

22(26) 1 4(4) 22(3) 1 4(1)G

5 F 212 11

16 210 G 1 F 242 12

28 22 G 5 F 254 23

44 212G 25. AB 2 BA 5 F 5 23

22 4G F 0 1

4 22 G 2 F 0 1

4 22 G F 5 23

22 4G 5 F 5(0) 1 (23)(4) 5(1) 1 (23)(22)

22(0) 1 4(4) 22(1) 1 4(22)G

2 F 0(5) 1 1(22) 0(23) 1 1(4)

4(5) 1 (22)(22) 4(23) 1 (22)(4)G

5 F 212 11

16 210G 2 F 22 4

24 220G 5 F 210 7

28 10G 26. E(D 1 E)

5 F 23 1 4

7 0 22

3 4 21G (F 1 3 2

23 1 4

2 1 22G 1 F 23 1 4

7 0 22

3 4 21G)

5 F 23 1 4

7 0 22

3 4 21GF 22 4 6

4 1 2

5 5 23G

5 F 23(22) 1 1(4) 1 4(5) 23(4) 1 1(1) 1 4(5)

7(22) 1 0(4) 1 (22)(5) 7(4) 1 0(1) 1 (22)(5)

3(22) 1 4(4) 1 (21)(5) 3(4) 1 4(1) 1 (21)(5)

23(6) 1 1(2) 1 4(23)

7(6) 1 0(2) 1 (22)(23)

3(6) 1 4(2) 1 (21)(23)G

5 F 30 9 228

224 18 48

5 11 29 G


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27. (D 1 E)D

5 (F 1 3 2

23 1 4

2 1 22G 1 F 23 1 4

7 0 22

3 4 21G) F 1 3 2

23 1 4

2 1 22G

5 F 22 4 6

4 1 2

5 5 23GF 1 3 2

23 1 4

2 1 22G

5 F 22(1) 1 4(23) 1 6(2) 22(3) 1 4(1) 1 6(1)

4(1) 1 1(23) 1 2(2) 4(3) 1 1(1) 1 2(1)

5(1) 1 5(23) 1 (23)(2) 5(3) 1 5(1) 1 (23)(1)

22(2) 1 4(4) 1 6(22)

4(2) 1 1(4) 1 2(22)

5(2) 1 5(4) 1 (23)(22)G

5 F 22 4 0

5 15 8

216 17 36 G

28. 22(BC) 5 22 F 0 1

4 22G F 26 3

4 1 G 5 22 F 0(26) 1 1(4) 0(3) 1 1(1)

4(26) 1 (22)(4) 4(3) 1 (22)(1)G

5 F 22(4) 22(1)

22(232) 22(10)G 5 F 28 22

64 220G29. 4AC 1 3AB 5 4F 5 23

22 4G F 26 3

4 1 G 1 3F 5 23

22 4G F 0 1

4 22G 5 4F 5(26) 1 (23)(4) 5(3) 1 (23)(1)

(22)(26) 1 4(4) (22)(3) 1 4(1)G

1 3F 5(0) 1 (23)(4) 5(1) 1 (23)(22)

(22)(0) 1 4(4) (22)(1) 1 4(22)G

5 F 4(242) 4(12)

4(28) 4(22)G

1 F 3(212) 3(11)

3(16) 3(210)G

5 F 2168 2 36 48 1 33

112 1 48 28 1 (230)G

5 F 2204 81

160 238G

30. 3(1) 1 2(x) 1 4(3) 5 19

2x 5 4

x 5 2

0(1) 1 (22)(x) 1 4(3) 5 y

0(1) 1 (22)(2) 1 4(3) 5 y

8 5 y


31. 22(9) 1 x(2) 1 1(21) 5 213

2x 5 6

x 5 3

4(9) 1 1(2) 1 3(21) 5 y

35 5 y


32. A2 5 F 1 21

0 2G F 1 21

0 2G 5 F 1(1) 1 (21)(0) 1(21) 1 (21)(2)

0(1) 1 2(0) 0(21) 1 2(2)G

5 F 1 23

0 4G A3 5 F 1 21

0 2G F 1 21

0 2G F 1 21

0 2G 5 F 1 23

0 4G F 1 21

0 2G 5 F 1(1) 1 (23)(0) 1(21) 1 (23)(2)

0(1) 1 4(0) 0(21) 1 4(2)G

5 F 1 27

0 8G 33. A2 5 F 24 1

2 21G F 24 1

2 21G 5 F (24)(24) 1 1(2) (24)(1) 1 1(21)

2(24) 1 (21)(2) 2(1) 1 (21)(21)G

5 F 18 25

210 3G A3 5 F 24 1

2 21G F 24 1

2 21G F 24 1

2 21G 5 F 18 25

210 3G F 24 1

2 21G 5 F 18(24) 1 (25)(2) 18(1) 1 (25)(21)

210(24) 1 3(2) 210(1) 1 3(21)G

5 F 282 23

46 213G


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159Algebra 2


34. A2 5 F 2 0 21

1 3 2

22 21 0G F 2 0 21

1 3 2

22 21 0G

5 F 6 1 22

1 7 5

25 23 0G

A3 5 F 2 0 21

1 3 2

22 21 0G F 2 0 21

1 3 2

22 21 0G F 2 0 21

1 3 2

22 21 0G

5 F 6 1 22

1 7 5

25 23 0GF 2 0 21

1 3 2

22 21 0G

5 F 17 5 24

21 16 13

213 29 21G

35. Sample answer:

A 5 F 2 6

4 9G, B 5 F 1 0

0 1G36. k(AB) 5 k(F a b

c dGF e f

g hG)

5 kF ae 1 bg af 1 bh

ce 1 dg cf 1 dhG 5 F kae 1 kbg kaf 1 kbh

kce 1 kdg kcf 1 kdhG (kA)B 5 (kF a b

c dG) F e f

g hG

5 F ka kb

kc kdG F e f g hG

5 F kae 1 kbg kaf 1 kbh

kce 1 kdg kcf 1 kdhG A(kB) 5 F a b

c dG (kF e f g hG)

5 F a b c dG F ke kf

kg khG 5 F kae 1 kbg kaf 1 kbh

kce 1 kdg kcf 1 kdhG All of the matrices are equal, so k(AB) 5 (kA)B 5 A(kB).

Problem Solving

37. Equipment Cost (dollars)

Bats Balls Uniforms Bats

Balls

Uniforms

F 21

4

30G F 12 45 15 G

Total cost 5 F 12 45 15 G F 21

4

30G

5 F 12(21) 1 45(4) 1 15(30) G 5 F 882 G The total cost of equipment for the softball team is $882.

38. Art Supplies

Paint Brushes Canvases

Class 1

Class 2

F 24 12 17

20 14 15G Cost (dollars)

Paint

Brushes

Canvases

F 3.35

1.75

4.50G

Total cost 5 F 24 12 17

20 14 15G F 3.35

1.75

4.50G

5 F 24(3.35) 1 12(1.75) 1 17(4.50) 20(3.35) 1 14(1.75) 1 15(4.50)G 5 F 177.9

159.00G The supplies for class 1 cost $177.90 and the supplies for

class 2 cost $159.00.

39. Attendance

Students Adults Seniors

Friday

Saturday

F 120 150 40

192 215 54G Ticket cost (dollars)

Students

Adults

Seniors

F 2

5

4G

F 120 150 40

192 215 54G F 2

5

4G

5 F 120(2) 1 150(5) 1 40(4) 192(2) 1 215(5) 1 54(4)G 5 F 1150

1675G The income from Friday night’s play was $1150 and the

income from Saturday night’s play was $1675.

Chapter 3 continued

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40. Medals Won

Gold Silver Bronze

USA

China

Russia

F 35 39 29

32 17 14

27 27 38G

Points

Gold

Silver

Bronze

F 3

2

1G

F 35 39 29

32 17 14

27 27 38G F 3

2

1G

5 F 35(3) 1 39(2) 1 29(1)

32(3) 1 17(2) 1 14(1)

27(3) 1 27(2) 1 38(1)G 5 F 212

144

173G

USA scored 212 points, China scored 144 points, and Russia scored 173 points.

41. SP: (3 3 2)(1 3 3) PS: (1 3 3)(3 3 2) not equal equal

The matrix PS is defi ned.

PS 5 F 650 825 1050G F 21 16

40 33

15 19G 5

F 650(21)1825(40)11050(15) 650(16)1825(33)11050(19)G 5 F 62,400 57,575 G This matrix shows that dealer A made a profi t of $62,400

and dealer B made a profi t of $57,575.

42. Grades (G)

Homework Quizzes Tests Jean

Ted

Pat

Al

Matt

F 82 88 86

92 88 90

82 73 81

74 75 78

88 92 90

G Weights (W)

Homework

Quizzes

Tests

F 0.2

0.3

0.5G

GW 5 F 82(0.2) 1 88(0.3) 1 86(0.5)

92(0.2) 1 88(0.3) 1 90(0.5)

82(0.2) 1 73(0.3) 1 81(0.5)

74(0.2) 1 75(0.3) 1 78(0.5)

88(0.2) 1 92(0.3) 1 90(0.5)

G 5 F 85.8

89.8

78.8

76.3

90.2

G The students received the following overall grades:

Jean, 85.8; Ted, 89.8; Pat, 78.8; Al, 76.3; and Matt, 90.2.

43. a. T 5 F 1 2 p q

p 1 2 qG 5 F 1 2 0.2 0.05

0.2 1 2 0.05G 5 F 0.8 0.05

0.2 0.95G b. M1 5 F 0.8 0.05

0.2 0.95G F 5000 8000G

5 F 0.8(5000) 1 0.05(8000) 0.2(5000) 1 0.95(8000)G 5 F 4400

8600G This matrix represents the number of commuters who

will drive and use public transportation after one year.

c. M2 5 F 0.8 0.05

0.2 0.95G F 4400 8600G

5 F 0.8(4400) 1 0.05(8600) 0.2(4400) 1 0.95(8600)G 5 F 3950

9050G M3 5 F 0.8 0.05

0.2 0.95G F 3950 9050G

5 F 0.8(3950) 1 0.05(9050) 0.2(3950) 1 0.95(9050)G 5 F 3612.5

9387.5G M4 5 F 0.8 0.05

0.2 0.95G F 3612.5 9387.5G

5 F 0.8(3612.5) 1 0.05(9387.5) 0.2(3612.5) 1 0.95(9387.5)G 5 F 3359.375

9640.625G M2, M3, and M4 represent the number of commuters

after 2, 3, and 4 years, respectively.

44. a. Cost (C) Price (P)

Plain

Class year

School name

Mascot

F 10

15

20

20

G Plain

Class year

School name

Mascot

F 15

20

25

30

G b. Sales (S)

Plain Class year School name Mascot

Year 1

Year 2

Year 3

F 0 20 100 0

10 100 50 30

20 300 100 50G


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161Algebra 2


c. SC 5 F 0 20 100 0

10 100 50 30

20 300 100 50G F 10

15

20

20

G 5 F 0(10) 1 20(15) 1 100(20) 1 0(20)

10(10) 1 100(15) 1 50(20) 1 30(20)

20(10) 1 300(15) 1 100(20) 1 50(20)G

5 F 2300

3200

7700G

SC represents the cost of making the scarves each year.

SP 5 F 0 20 100 0

10 100 50 30

20 300 100 50G F 15

20

25

30

G 5 F 0(15) 1 20(20) 1 100(25) 1 0(30)

10(15) 1 100(20) 1 50(25) 1 30(30)

20(15) 1 300(20) 1 100(25) 1 50(30)G

5 F 2900

4300

10,300G

SP represents the total price received for the scarveseach year.

d. SP 2 SC 5 F 2900

4300

10,300G 2 F 2300

3200

7700G

5 F 2900 2 2300

4300 2 3200

10,300 2 7700G 5 F 600

1100

2600G

This matrix represents the profi t that is made each year.

45. a. AB 5 F 0 21

1 0G F 27 24 24

4 8 2G 5 F 24 28 22

27 24 24G

1

x

y

21

(24, 27)

(22, 24)(28, 24)

b. For the triangle rotated 1808:

F 0 21

1 0G F 0 21

1 0G F 27 24 24

4 8 2G 5 F 21 0

0 21G F 27 24 24

4 8 2G 5 F 7 4 4

24 28 22G The vertices of this triangle are (7, 24), (4, 28),

and (4, 22).

For the triangle rotated 2708:

F 0 21

1 0G F 0 21

1 0G F 0 21

1 0G F 27 24 24

4 8 2G 5 F 21 0

0 21G F 0 21

1 0GF 27 24 24

4 8 2G 5 F 0 21

1 0GF 27 24 24

4 8 2G 5 F 4 8 2

7 4 4G The vertices of this triangle are (4, 7), (8, 4), and (2, 4).

Mixed Review

46.

1

x

y

21

47. 1

x

y

21

48.

1

x

y

21

49. 1

x

y

21

50. 1

x

y

21

51.

1

x

y

21

52. y 2 (24) 5 2(x 2 0) 53. y 2 2 5 23(x 2 5)

y 1 4 5 2x y 2 2 5 23x 1 15

y 5 2x 2 4 y 5 23x 1 17


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54. y 2 (21) 5 2 2 } 3 (x 2 0) 55. y 2 3 5

3 } 4 (x 2 0)

y 1 1 5 2 2 } 3 x y 2 3 5

3 } 4 x

y 5 2 2 } 3 x 2 1 y 5

3 } 4 x 1 3

56. m 5 2 2 8

} 1 2 4 5 2

y 2 2 5 2(x 2 1)

y 2 2 5 2x 2 2

y 5 2x

57. m 5 1 2 8

} 0 2 (28)

5 2 7 } 8

y 2 1 5 2 7 } 8 (x 2 0)

y 2 1 5 2 7 } 8 x

y 5 2 7 } 8 x 1 1

58. 3x 1 2y 5 5

2x 1 3y 5 13 → x 5 3y 2 13

3(3y 2 13) 1 2y 5 5 → y 5 4

2x 1 3(4) 5 13 → x 5 21


59. 3x 2 5y 5 11

2x 1 5y 5 24

5x 5 35

x 5 7

3(7) 2 5y 5 11 → y 5 2


60. 3x 2 y 5 4 3 3 9x 2 3y 5 12

22x 1 3y 5 226 22x 1 3y 5 226

7x 5 214

x 5 22

3(22) 2 y 5 4 → y 5 210


61. 4x 2 3y 5 17 4x 2 3y 5 17

2x 1 5y 5 15 3 (22) 24x 2 10y 5 230

213y 5 213

y 5 1

4x 2 3(1) 5 17 → x 5 5


62. 4x 2 2y 5 14 4x 2 2y 5 14

22x 1 y 5 27 3 2 24x 1 2y 5 214

0 5 0


63. x 1 4y 5 4 x 1 4y 5 4

3x 2 2y 5 19 3 2 6x 2 4y 5 38

7x 5 42

x 5 6

6 1 4y 5 4 → y 5 2 1 } 2

The solution is 1 6, 2 1 } 2 2 .

Lesson 3.7


1. 3 2 6 1 5 3(1) 2 6(22) 5 15

2. 4 21 2 23 22 21

0 5 1 4 21

23 22

0 5

5 (28 1 0 2 30) 2 (0 2 20 1 3)

5 238 2 (217) 5 221

3. 10 22 3 2 212 4

0 27 22 10 22

2 212

0 27

5 (240 1 0 2 42) 2 (0 2 280 1 8)

5 198 1 272 5 470

4. Area 5 6 1 } 2

5 11 1 9 2 1

1 3 1 5 11

9 2

1 3

5 6 1 } 2 [(10 1 11 1 27) 2 (2 1 15 1 99)]

5 6 1 } 2 (268) 5 34

The area of the triangle is 34 square units.

5. 3 24 2 5 5 15 2 (28) 5 23

x 5

215 24 13 5

} 23

5 275 2 (252)

}} 23 5 21

y 5

3 215 2 13

} 23

5 39 2 (230)

} 23 5 3


6. 4 7 23 22 5 28 2 (221) 5 13

x 5

2 7 28 22

} 13

5 24 2 (256)

} 13 5 4

y 5

4 2 23 28

} 13

5 232 2 (26)

} 13 5 22



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163Algebra 2


7. 3 24 2 4 1 25

2 23 1 3 24

4 1

2 23

5 (3 1 40 1 (224)) 2 (4 1 45 1 (216)) 5 214

x 5

18 24 2

213 1 25

11 23 1 18 24

213 1

11 23

}}} 214

5 (18 1 220 1 78) 2 (22 1 270 1 52)

}}} 214 5 2

y 5

3 18 2 4 213 25

2 11 1 3 18

4 213

2 11

}} 214

5 (239 2 180 1 88) 2 (252 2 165 1 72)

}}}} 214 5 21

z 5

3 24 18 4 1 213

2 23 11 3 24

4 1

2 23

}} 214

5 (33 1 104 2 216) 2 (36 1 117 2 176)

}}} 214 5 4



Skill Practice

1. The determinant of a 2 3 2 matrix is the difference of the products of the elements of the diagonals.

2. Cramer’s rule is used to solve a system of linear equations. The numerators for x and y are the determinants of the matrices formed by replacing the coeffi cients of x and y, respectively, with the column of constants. The denominator of x and y is the determinant of the coeffi cient matrix.

3. 2 21 4 25 5 2(25) 2 4(21) 5 26

4. 7 1 0 3 5 7(3) 2 0(1) 5 21

5. 24 3 1 27 5 24(27) 2 1(3) 5 25

6. 1 23 2 6 5 1(6) 2 2(23) 5 12

7. 10 26 27 5 5 10(5) 2 (26)(27) 5 8

8. 0 3 5 23 5 0(23) 2 5(3) 5 215

9. 9 23 7 2 5 9(2) 2 7(23) 5 39

10. 25 12 4 6 5 25(6) 2 4(12) 5 278

11. 21 12 4 0 2 25

3 0 15 21 12 4 0 2 25

3 0 1 21 12

0 2

3 0

5 (22 2 180 1 0) 2 (24 1 0 1 0)

5 2206

12. 1 2 3 5 28 1

2 4 35 1 2 3 5 28 1

2 4 3 1 2

5 28

2 4

5 (224 1 4 1 60) 2 (248 1 4 1 30)

5 54

13. 5 0 2 23 9 22

1 24 0 5 5 0 2 23 9 22

1 24 0 5 0

23 9

1 24

5 (0 1 0 1 24) 2 (18 1 40 1 0)

5 234

14. 27 4 5 1 2 24

210 1 65 27 4 5 1 2 24

210 1 6 27 4

1 2

210 1

5 (284 1 160 1 5) 2 (2100 1 28 1 24)

5 129

15. 12 5 8 0 6 28

1 10 45 12 5 8 0 6 28

1 10 4 12 5

0 6

1 10

5 (288 2 40 1 0) 2 (48 2 960 1 0)

5 1160

16. 24 3 29 12 6 0

8 212 0 5 24 3 29 12 6 0

8 212 0 24 3

12 6

8 212

5 (0 1 0 1 1296) 2 (2432 1 0 1 0)

5 1728

17. 22 6 0 8 15 3

4 21 7 5 22 6 0 8 15 3

4 21 7 22 6

8 15

4 21

5 (2210 1 72 1 0) 2 (0 1 6 1 336)

5 2480

18. 5 7 6

24 0 8

1 8 7 5 5 7 6

24 0 8

1 8 7 5 7

24 0

1 8

5 (0 1 56 2 192) 2 (0 1 320 2 196)

5 2260


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19. The values in the upward diagonals should be subtracted from the values in the downward diagonals.

2 0 21 4 1 6

23 2 5 2 0

4 1

23 2

5 (10 1 0 2 8) 2 (3 1 24 1 0)

5 2 2 27 5 225

The determinant is 225.

20. The column extensions should be on the right sideof the matrix.

3 0 1 2 2 23

23 5 0 3 0

2 2

25 5

5 (0 1 0 1 10) 2 (26 2 45 1 0)

5 10 2 (251) 5 61

21. D;

det A 5 212 2 6 5 218

det B 5 8 2 18 5 210

det C 5 25 2 (221) 5 16

det D 5 25 2 (22) 5 27

The greatest determinant is det D.

22. Area 5 6 1 } 2

1 5 1 4 6 1

7 3 1 1 5

4 6

7 3

5 6 1 } 2 [(6 1 35 1 12) 2 (42 1 3 1 20)]

5 6


23. Area 5 6 1 } 2

4 2 1 4 8 1

8 5 1 4 2

4 8

8 5

5 6 1 } 2 [(32 1 16 1 20) 2 (64 1 20 1 8)]

5 12


24. Area 5 6 1 } 2

24 6 1 0 3 1

6 6 1 24 6

0 3

6 6

5 6 1 } 2 [(212 1 36 1 0) 2 (18 2 24 1 0)]

5 15


25. Area 5 6 1 } 2

24 24 1 21 2 1

2 26 1 24 24

21 2

2 26

5 6 1 } 2 [(28 2 8 1 6) 2 (4 1 24 1 4)]

5 21


26. Area 5 6 1 } 2

5 24 1 6 3 1

8 21 1 5 24

6 3

8 21

5 6 1 } 2 [(15 2 32 2 6) 2 (24 2 5 2 24)]

5 9


27. Area 5 6 1 } 2

26 1 1 22 26 1

0 3 1 26 1

22 26

0 3

5 6 1 } 2 [(36 1 0 2 6) 2 (0 2 18 2 2)]

5 25


28. A;

Area 5 6 1 } 2

23 4 1 6 3 1

2 21 1 23 4

6 3

2 21

5 6 1 } 2 [(29 1 8 2 6) 2 (6 1 3 1 24)]

5 20


29. 3 5

21 2 5 6 2 (25) 5 11

x 5

3 5

10 2 }

11 5

6 2 50 } 11 5 24

y 5

3 3 21 10

} 11

5 30 2 (23)

} 11 5 3


30. 2 21

1 2 5 4 2 (21) 5 5

x 5

22 21

14 2 }}

5 5

24 2 (214) } 5 5 2

y 5

2 22

1 14 }

5 5

28 2 (22) } 5 5 6


31. 5 1

2 25 5 225 2 2 5 227

x 5

240 1

11 25 }}

227 5

200 2 11 }

227 5 27

y 5

5 240

2 11 }}

227 5

55 2 (280) }

227 5 25



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165Algebra 2


32. 21 1 1 4 21 4

1 2 21 21 1

4 21

1 2

5 (21 1 4 1 8) 2 (21 2 8 2 4) 5 24

x 5

23 1 1 214 21 4

9 2 21 23 1

214 21

9 2

}}} 24

5 (23 1 36 2 28) 2 (29 2 24 1 14)

}}} 24 5 1

y 5

21 23 1 4 214 4

1 9 21 21 23

24 214

1 9

}}} 24

5 (214 2 12 1 36) 2 (214 2 36 1 12)

}}} 24 5 2

z 5

21 1 23 4 21 214

1 2 9 21 1

24 21

1 2

}}} 24

5 (9 2 14 2 24) 2 (3 1 28 1 36)

}}} 24 5 24


33. 21 22 4 1 1 2

2 1 23 21 22

1 1

2 1

5 (3 2 8 1 4) 2 (8 2 2 1 6) 5 213

x 5

228 22 4

211 1 2

30 1 23 228 22

211 1

30 1 }}} 213

5 (84 2 120 2 44) 2 (120 2 56 2 66)

}}} 213 5 6

y 5

21 228 4

1 211 2

2 30 23 21 228

1 211

2 30 }}} 13

5 (233 2 112 1 120) 2 (288 2 60 1 84)

}}}} 213 5 23

z 5

21 22 228 1 1 211

2 1 30 21 22

1 1

2 1 }}} 213

5 (230 1 44 2 28) 2 (256 1 11 2 60)

}}} 213 5 27


34. 4 1 3 2 25 4

1 21 2 4 1

2 25

1 21

5 (240 1 4 2 6) 2 (215 2 16 1 4) 5 215

x 5

7 1 3

219 25 4

22 21 2 7 1

219 25

22 21

}}} 215

5 (270 2 8 1 57) 2 (30 2 28 2 38)

}}} 215 5 21

y 5

4 7 3

2 219 4

1 22 2 4 7

2 219

1 22

}}} 215

5 (2152 1 28 2 12) 2 (257 2 32 1 28)

}}} 215 5 5

z 5

4 1 7

2 25 219

1 21 22 4 1

2 25

1 21

}} 215

5 (40 2 19 2 14) 2 (235 1 76 2 4)

}}} 215 5 2


35. 5 21 22

1 3 4

2 24 1 5 21

1 3

2 24

5 (15 2 8 1 8) 2 (212 2 80 2 1) 5 108

x 5

26 21 22

16 3 4

215 24 1 26 21

16 3

215 24

}}} 108

5 (218 1 60 1 128) 2 (90 1 96 2 16)

}}} 108 5 0

y 5

5 26 22

1 16 4

2 215 1 5 26

1 16

2 215

}}} 108

5 (80 2 48 1 30) 2 (264 2 300 2 6)

}}} 108 5 4

z 5

5 21 26

1 3 16

2 24 215 5 21

1 3

2 24

}}} 108

5 (2225 2 32 1 24) 2 (236 2 320 1 15)

}}}} 108 5 1



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36. 1 1 1

3 23 2

21 2 22 1 1

3 23

21 2

5 (6 2 2 1 6) 2 (3 1 4 2 6) 5 9

x 5

28 1 1

221 23 2

11 2 22 28 1

221 23

11 2 }} 9

5 (248 1 22 2 42) 2 (233 2 32 1 42)

}}} 9 5 25

y 5

1 28 1

3 221 2

21 11 22 1 28

3 221

21 11 }}} 9

5 (42 1 16 1 33) 2 (21 1 22 1 48)

}}} 9 5 0

z 5

1 1 28

3 23 221

21 2 11 1 1

3 23

21 2 }}} 9

5 (233 1 21 2 48) 2 (224 2 42 1 33)

}}} 9 5 23


37. 3 21 1

21 2 23

1 1 1 3 21

21 2

1 1

5 (6 1 3 2 1) 2 (2 2 9 1 1) 5 14

x 5

25 21 1

217 2 23

21 1 1 25 21

217 2

21 1 }} 14

5 (50 1 63 2 17) 2 (42 2 75 1 17)

}}} 14 5 8

y 5

3 25 1

21 217 23

1 21 1 3 25

21 217

1 21 }}} 14

5 (251 2 75 2 21) 2 (217 2 189 2 25)

}}} 14 5 6

z 5

3 21 25

21 2 217

1 1 21 3 21

21 2

1 1 }}} 14

5 (126 1 17 2 25) 2 (50 2 51 1 21)

}}} 14 5 7


38. Sample answer:

F 10 17 5 9G

39. a. AB 5 F 2 21 1 2GF 3 5

22 24G 5 F 2(3) 1 (21)(22) 2(5) 1 (21)(24)

1(3) 1 2(22) 1(5) 1 2(24)G 5 F 8 14

21 23G det AB 5 224 2 (214) 5 210

det A 5 4 2 (21) 5 5

det B 5 212 2 (210) 5 22

The determinant of AB is equal to the product of det A and det B.

b. det kA 5 2k 2k k 2k 5 4k2 2 (2k2) 5 5k2

The determinant of kA is equal to k2 times the determinant of A.

Problem Solving

40. Area 5 6 1 } 2

938 454 1 900 2518 1

0 0 1 938 454

900 2518

0 0

5 6 1 } 2 F (2485,884 1 0 1 0) 2 (0 1 0 1 408,600) G

5 447,242

The area of the Bermuda Triangle is 447,242 square miles.

41. Area 5 6 1 } 2

0 0 1 5 2 1

3 6 1 0 0

5 2

3 6

5 6 1 } 2 [(0 1 0 1 30) 2 (6 1 0 1 0)]

5 12

The area of the garden will be 12 square feet.


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167Algebra 2


42. f 5 number of fl oor seats

n 5 number of other seats

Equation 1: 40f 1 25n 5 185,500

Equation 2: f 1 n 5 6700

Method 1: Cramer’s rule

40 25 1 15 40 2 25 5 15

f 5

185,500 25 6700 1

}} 15

5 185,500(1) 2 6700(25)

}} 15 5 1200

n 5

40 185,500 1 6700

}} 15

5 40(6700) 2 1(185,500)

}} 15 5 5500

Method 2: Substitution

n 5 6700 2 f

40f 1 25(6700 2 f ) 5 185,500

15f 5 18,000

f 5 1200

n 5 6700 2 1200 5 5500

Method 3: Elimination

40f 1 25n 5 185,500 40f 1 25n 5 185,500

f 1 n 5 6700 3 (240) 240f 2 40n 5 2268,000

215n 5 282,500

n 5 5500

f 1 5500 5 6700 → f 5 1200

Each method used above produces the same outcome. 1200 fl oor seats and 5500 other seats were sold.

43. a. s 5 number of single scoop cones

d 5 number of double scoop cones

t 5 number of triple scoop cones

Equation 1: 0.90s 1 1.20d 1 1.60t 5 134

Equation 2: s 1 d 1 t 5 120

Equation 3: s 5 d 1 t, or s 2 d 2 t 5 0

0.90 1.20 0.60 1 1 1

1 21 21 0.90 1.20

1 1

1 21

5 (20.90 1 1.20 2 1.60) 2 (1.60 2 0.90 2 1.20)

5 20.80

s 5

134 1.20 1.60 120 1 1

0 21 21 134 1.20

120 1

0 21 }}} 20.80

5 2(134 1 0 2 192) 2 (0 2 134 2 144)

}}} 20.80 5 60

d 5

0.90 134 1.60 1 120 1

1 0 21 0.90 134

1 1

1 0

}}} 20.80

5 (2108 1 134 1 0) 2 (192 1 0 2 134)

}}} 20.80 5 40

t 5

0.90 1.20 134 1 1 120

1 21 0 0.90 1.20

1 1

1 21 }}} 20.80

5 (0 1 144 2 134) 2 (134 2 108 1 0)

}}} 20.80 5 20

The ice cream shop sells 60 single scoop, 40 double scoop, and 20 triple scoop cones.

b. New prices:

Single scoop: 0.90(1.10) 5 0.99

Double scoop: 1.20(1.10) 5 1.32

Triple scoop: 1.60(1.10) 5 1.76

Number of cones sold:

Single scoop: 60(0.95) 5 57

Double scoop: 40(0.95) 5 38

Triple scoop 20(0.95) 5 19

Total revenue 5 0.99(57) 1 1.32(38) 1 1.76(19)

5 140.03

The total revenue from the ice cream cones is $140.03.

44. F 1 Na 5 42

Na 1 Cl 5 58.5

5F 1 Cl 5 130.5

1 1 0 0 1 1

5 0 1 1 1

0 1

5 0

5 (1 1 5 1 0) 2 (0 1 0 1 0) 5 6

F 5

42 1 0 58.5 1 1

130.5 0 1 42 1

58.5 1

130.5 0 }} 6

5 (42 1 130.5 1 0) 2 (0 1 0 1 58.5)

}}} 6 5 19

Na 5

1 42 0 0 58.5 1

5 130.5 1 1 42

0 58.5

5 130.5 }} 6

5 (58.5 1 210 1 0) 2 (0 1 130.5 1 0)

}}} 6 5 23


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Cl 5

1 1 42 0 1 58.5

5 0 130.5 1 1

0 1

5 0 }}} 6

5 (130.5 1 292.5 1 0) 2 (210 1 0 1 0)

}}} 6 5 35.5

The atomic weights of fl ourine, sodium, and chlorine are 19, 23, and 35.5, respectively.

45. a. Area 5 6 1 } 2

70 128 1 0 70 1

124 36 1 70 128

0 70

124 36

5 6 1 } 2 [(4900 1 15872 1 0) 2 (8680 1 2520 1 0)]

5 4786

The area of the top triangular region is 4786 mi2.

b. Area 5 6 1 } 2

0 70 1 67 0 1

124 36 1 0 70

67 0

124 36

5 6 1 } 2 [(0 1 8680 1 2412) 2 (0 1 0 1 4690)]

5 3201

The area of the bottom triangular region is 3201 mi2.

c. Total area 5 4786 1 3201 5 7987

The total of the Dinosaur Diamond is 7987 mi2.

d. You could connect Vernal, UT, to Moab, UT, to create a left and right triangle.

46. 6 1 } 2

x 120 1 100 50 1

0 0 1 x 120

100 50

0 0

5 5000

6 1 } 2 F (50x 1 0 1 0) 2 (0 1 0 1 12,000) G 5 5000

6 1 } 2 (50x 2 12,000) 5 5000

Two possibilities:

1. 1 }

2 (50x 2 12,000) 5 5000

25x 5 11,000

x 5 440

2. 2 1 } 2 (50x 2 12,000) 5 5000

225x 5 21000

x 5 40

The farmer could put the fi nal post at (40, 120) or(440, 120).

Mixed Review

47. f (8) 5 8 2 12 5 24 48. f (7) 5 4(7) 1 8 5 36

49. f (25) 5 (25)2 2 10 5 15

50. f (3) 5 2(3)2 1 2(3) 5 23

51. f (4) 5 2(4)2 2 4 1 5 5 215

52. f (22) 5 (22)2 2 2(22) 1 4 5 12

53.

1

x

y

21

54. 1

x

y

21

55.

8

x

y

4

56. 1

x

y

21

57. F 2 24 6 1GF 23 0

1 7G 5 F 2(23) 1 (24)(1) 2(0) 1 (24)(7)

6(23) 1 1(1) 6(0) 1 1(7)G 5 F 210 228

217 7G 58. F 26 28

2 24GF 0 5 7 1G

5 F 26(0) 1 (28)(7) 26(5) 1 (28)(1) 2(0) 1 (24)(7) 2(5) 1 (24)(1)G

5 F 256 238 228 6G

59. F 1 0 3 22GF 25 10

2 0 G 5 F 1(25) 1 0(2) 1(10) 1 0(0)

3(25) 1 (22)(2) 3(10) 1 (22)(0)G 5 F 25 10

219 30GLesson 3.8


1. 1 }

24 2 2 F 4 21

22 6G 5 1 } 22

F 4 21

22 6G 5 F 2

} 11

2 1 } 22

2 1 } 11 3 } 11 G

2. 1 }

28 2 (220) F 8 25

4 21G 5 1 } 12

F 8 25

4 21G 5 F 2 } 3 2 5 }

12

1 } 3 2 1 } 12

G


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169Algebra 2


3. 1 }

6 2 4 F 22 4

1 23G 5 1 } 2 F 22 4

1 23G 5 F 21 2

1 } 2 2 3 } 2 G4. 2

1 } 24 F 6 21

0 24G 5 F 2 1 } 4 1 } 24

0 1 } 6 G

F 2 1 } 4 1 } 24

0 1 } 6 GF 24 1

0 6GX 5 F 2 1 } 4 1 } 24

0 1 } 6 GF 8 9

24 6G F 1 0

0 1GX 5 F 22 1 1 2 9 } 4 1 1 } 4

0 1 4 0 1 1G

X 5 F 21 22

4 1G5. A 5 F 2 22 0

2 0 22

12 24 26G; A

21 5 F 21 21.5 0.5

21.5 21.5 0.5

21 22 0.5G

Check:

AA21 5 F 2 22 0

2 0 22

12 24 26GF 21 21.5 0.5

21.5 21.5 0.5

21 22 0.5G

5 F 22 1 3 1 0 23 1 3 1 0 1 2 1 1 0

22 1 0 1 2 23 1 0 1 4 1 1 0 2 1

212 1 6 1 6 218 1 6 1 12 6 2 2 2 3G

5 F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 21 21.5 0.5

21.5 21.5 0.5

21 22 0.5GF 2 22 0

2 0 22

12 24 26G

5 F 22 2 3 1 6 2 1 0 2 2 0 1 3 2 3

23 2 3 1 6 3 1 0 2 2 0 1 3 2 3

22 2 4 1 6 2 1 0 2 2 0 1 4 2 3G

5 F 1 0 0

0 1 0

0 0 1G 5 I

6. A 5 F 23 4 5

1 5 0

5 2 2G

A21 ø F 20.0654 20.0131 0.1634

0.01307 0.2026 20.0327

0.1503 20.1699 0.1242G

Check:

AA21 ø F 23 4 5

1 5 0

5 2 2GF 20.0654 20.0131 0.1634

0.01307 0.2026 20.0327

0.1503 20.1699 0.1242G

5 F 0.1962 1 0.05228 1 0.7515

20.0654 1 0.06535 1 0

20.327 1 0.02614 1 0.3006

0.0393 1 0.8104 2 0.8495

20.0131 1 1.0130 1 0

20.0655 1 0.4052 2 0.3398

20.4902 2 0.1308 1 0.621

0.1634 2 0.1635 1 0

0.817 2 0.0654 1 0.2482G

5 F 0.99998 2 3 1024 0

25 3 1025 0.9999 21 3 1024

22.6 3 1024 21 3 1024 0.9998G

ø F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 20.0654 20.0131 0.1634

0.01307 0.2026 20.0327

0.1503 20.1699 0.1242GF 23 4 5

1 5 0

5 2 2G

5 F 0.1962 2 0.0131 1 0.817

20.03921 1 0.2026 2 0.1635

20.4509 2 0.1699 1 0.621

20.2616 2 0.0655 1 0.3268

0.05228 1 1.013 2 0.0654

0.6012 2 0.8495 1 0.2484

20.327 1 0 1 0.3268

0.06535 1 0 2 0.0654

0.7515 1 0 1 0.2484G

5 F 1.0001 23 3 1024 22 3 1024

21.1 3 1024 0.99988 25 3 1025

2 3 1024 1 3 1024 0.9999G

ø F 1 0 0

0 1 0

0 0 1G 5 I


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7. A 5 F 2 1 22

5 3 0

4 3 8G; A21 5 F 12 27 3

220 12 25

1.5 21 0.5G

Check:

AA21 5 F 2 1 22

5 3 0

4 3 8GF 12 27 3

220 12 25

1.5 21 0.5G

5F 24 2 20 2 3 214 1 12 1 2 6 2 5 2 1

60 2 60 1 0 235 1 36 1 0 15 2 15 1 0

48 2 60 1 12 228 1 36 2 8 12 2 15 1 4G

5 F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 12 27 3

220 12 25

1.5 21 0.5GF 2 1 22

5 3 0

4 3 8G

5 F 24 2 35 1 12 12 2 21 1 9

240 1 60 2 20 220 1 36 2 15

3 2 5 1 2 1.5 2 3 1 1.5

224 1 0 1 24

40 1 0 2 40

23 1 0 1 4G

5 F 1 0 0

0 1 0

0 0 1G 5 I

8. F 4 1 3 5GF x

y G 5 F 10 21G

A21 5 1 } 17 F 5 21

23 4G 5 F 5 } 17

2 1 } 17

2 3 } 17 4 } 17

G X 5 A21B 5 F 5 }

17 2 1 } 17

2 3 } 17 4 } 17

GF 10 21G 5 F 3

22G The solution of the system is (3, 22).

9. F 2 21 6 23GF x

y G 5 F 26 218G

A21 5 1 } 0 F 23 1

26 2G Because A 5 0,

1 }

A is undefi ned and A does not

have an inverse. The system has infi nitely many solutions.

10. F 3 21 24 2GF x

y G 5 F 25 8 G

A21 5 1 } 2 F 2 1

4 3G 5 F 1 1 } 2

2 3 } 2 G X 5 A21B 5 F 1 1 } 2

2 3 } 2 GF 25 8 G 5 F 21

2G The solution of the system is (21, 2).

11. F 2 1 0

2 2 1

4 3 2GF m

p

dG 5 F 17

35

69G

A21B 5 F 8

1

17G

A movie pass costs $8, a package of popcorn costs $1, and a DVD costs $17.


Skill Practice

1. Matrix of variables: F x y G

Matrix of constants:F 4 22G

2. First, you fi nd the determinant of A. Then, switch the elements on the downward diagonal and negate the elements on the upward diagonal. Finally, you

multiply 1 }

det A and the new matrix.

3. 1 }

4 2 5 F 4 5

1 1G 5 21F 4 5 1 1G 5 F 24 25

21 21G 4. 1 }

28 2 (29) F 4 23

3 22G 5 1F 4 23

3 22G 5 F 4 23

3 22G 5.

1 }

12 2 10 F 2 22

25 6G 5 1 } 2 F 2 22

25 6G 5 F 1 21

2 5 } 2 3 G 6.

1 }}

221 2 (218) F 3 9

22 27G 5 2 1 } 3 F 3 9

22 27G 5 F 21 23

2 } 3 7 }

3 G


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171Algebra 2


7. 1 }}

228 2 (224) F 7 6

24 24G 5 2 1 } 4 F 7 6

24 24G 5 F 2 7 } 4 2 3 } 2

1 1G

8.1 }

120 2 264 F 20 22

12 6G 5 2 1 } 144

F 20 22 12 6G

5 F 2 5 } 36 2 11 } 72

2 1 } 12 2 1 } 24 G

9. 1 }}

2720 2 (2360) F 30 260

6 224 G 5 2 1 } 360

F 30 260 6 224 G

5 F 2 1 } 12 1 } 6

2 1 } 60 1 } 15

G10.

1 }

2 4 } 3 2 1 2

10 } 3 2 F 21 2 5 } 6

4 4 } 3 G 5

1 } 2 F 21 2 5 } 6

4 4 } 3 G

5 F 2 1 } 2 2 5 } 12

2 2 } 3 G

11. The new matrix should have been multiplied by 1 }

det A ,

not by det A.

F 2 4 1 5G21

5 1 } 6 F 5 24

21 2G 5 F 5 }

6 2 2 } 3

2 1 } 6 1 } 3 G

12. C;

1 }

210 2 (29) F 21 3

23 10G 5 F 1 23 3 210G

13. 1 }

1 F 5 21

24 1G 5 F 5 21 24 1G

F 5 21 24 1GF 1 1

4 5GX 5 F 5 21 24 1GF 2 3

21 6G F 1 0

0 1GX 5 F 10 1 1 15 2 6 28 2 1 212 1 6G

X 5 F 11 9 29 26G

14. 1 }

2 F 3 28

22 6G 5 F 3 } 2 24

21 3G

F 3 } 2 24

21 3GF 6 8

2 3GX 5 F 3 } 2 24

21 3GF 4 3

0 22G F 1 0

0 1GX 5 F 6 1 0 9 } 2 1 8

24 1 0 23 2 6G

X 5 F 6 25 }

2

24 29G

15. 2 1 } 4 F 4 0

26 21G 5 F 21 0

3 } 2 1 }

4 G

F 21 0

3 } 2 1 }

4 GF 21 0

6 4GX 5 F 21 0

3 } 2 1 }

4 GF 3 21

4 5G F 1 0

0 1GX 5 F 23 1 0 1 1 0

9

} 2 1 1 2 3 } 2 1 5 } 4 G

X 5 F 23 1

11 }

2 2 1 }

4 G

16. 2 1 } 12

F 2 26 21 23G 5 F 2 1 } 6 1 } 2

1 } 12

1 } 4 G F 2 1 } 6 1 } 2

1 } 12

1 } 4 GF 23 6 1 2GX 5 F 2 1 } 6 1 } 2

1 } 12

1 } 4 GF 5 21 8 2G

F 1 0 0 1GX 5 F 2

5 } 6 1 4

1 }

6 1 1

5 } 12

1 2 2 1 } 12 1 1 } 2 G X 5 F 19

} 6 7 }

6

29 }

12 5 } 12 G

17. 2 1 } 2 F 22 25

0 1G 5 F 1 5 } 2

0 2 1 } 2 G F 1 5 }

2

0 2 1 } 2 GF 1 5 0 22GX 5 F 1 5 }

2

0 2 1 } 2 GF 3 21 0 6 8 4G

F 1 0 0 1GX 5 F 3 1 15 21 1 20 0 1 10

0 2 3 0 2 4 0 2 2G X 5 F 18 19 10

23 24 22G


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18.1 } 3 F 3 22

9 25G 5 F 1 2 2 } 3

3 2 5 } 3 G F 1 2 2 } 3

3 2 5 } 3 GF 25 2 29 3GX 5 F 1 2 2 } 3

3 2 5 } 3 GF 4 5 0 3 1 6G

F 1 0 0 1GX 5 F 4 2 2 5 2 2 } 3 0 2 4

12 2 5 15 2 5 } 3 0 2 10G

X 5 F 2 13 }

3 24

7 40 } 3 210

G19. A21 5 F 20.3 20.2 0.3

0.9 0.6 0.1

20.2 0.2 0.2G

Check:

AA21 5 F 1 1 22

22 0 3

3 1 0GF 20.3 20.2 0.3

0.9 0.6 0.1

20.2 0.2 0.2G

5 F 20.3 1 0.9 1 0.4 20.2 1 0.6 2 0.4

0.6 1 0 2 0.6 0.4 1 0 1 0.6

20.9 1 0.9 1 0 20.6 1 0.6 1 0

0.3 1 0.1 2 0.4

20.6 1 0 1 0.6

0.9 1 0.1 1 0G

5 F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 20.3 20.2 0.3

0.9 0.6 0.1

20.2 0.2 0.2GF 1 1 22

22 0 3

3 1 0G

5 F 20.3 1 0.4 1 0.9 20.3 1 0 1 0.3

0.9 2 1.2 1 0.3 0.9 1 0 1 0.1

20.2 2 0.4 1 0.6 20.2 1 0 1 0.2

0.6 2 0.6 1 0

21.8 1 1.8 1 0

0.4 1 0.6 1 0G

5 F 1 0 0

0 1 0

0 0 1G 5 I

20. A21 5 F 21. } 3 1. } 3 20. } 3 20.8 } 3 0. } 3 0.1 } 6 1.1 } 6 20. } 6 0.1 } 6

G Check:

AA21 5 F 1 0 2

2 1 3

1 4 4GF 21. } 3 1. } 3 20. } 3

20.8 } 3 0. } 3 0.1 } 6 1.1 } 6 20. } 6 0.1 } 6

G 5 F 1 0 2

2 1 3

1 4 4GF 2 4 } 3 4 }

3 2 1 } 3

2 5 } 6 1 } 3 1 }

6

7 } 6 2 2 } 3 1 } 6 G

5 F 2 4 } 3 1 0 1 7 } 3 4 }

3 1 0 2 4 }

3 2 1 } 3 1 0 1 1 }

3

2 8 } 3 2 5 } 6 1 7 }

2 8 }

3 1 1 }

3 2 2 2 2 } 3 1 1 }

6 1 1 }

2

2 4 } 3 2 10 }

3 1 14

} 3 4 }

3 1 4 }

3 2 8 }

3 2 1 } 3 1 2 }

3 1 2 }

3 G

5 F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 21. } 3 1. } 3 20. } 3 20.8 } 3 0. } 3 0.1 } 6 21.1 } 6 20. } 6 0.1 } 6

GF 1 0 2

2 1 3

1 4 4G

5F 2 4 } 3 4 } 3 2 1 } 3

2 5 } 6 1 } 3 1 }

6

7 } 6 2 2 } 3 1 } 6 G F 1 0 2

2 1 3

1 4 4G

5 F 2 4 } 3 1 8 } 3 2 1 }

3 0 1 4 }

3 2 4 }

3 2 8 } 3 1 4 2 4 }

3

2 5 } 6 1 2 } 3 1 1 }

6 0 1 1 }

3 1 2 }

3 2 5 } 3 1 1 1 2 }

3

7 } 6 2 4 } 3 1 1 }

6 0 2 2 }

3 1 2 }

3 7 }

3 2 2 1 2 }

3 G

5 F 1 0 0

0 1 0

0 0 1G 5 I


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173Algebra 2


21. A21 5 F 20.5 0 0.5

21.0625 0.125 0.4375

0.21875 0.0625 20.03125G

Check:

AA21 5 F 1 21 2

22 3 10

3 21 2GF 20.5 0 0.5

21.0625 0.125 0.4375

0.21875 0.0625 20.03125G

5 F 20.5 1 1.0625 1 0.4375 0 2 0.125 1 0.125

1 2 3.1875 1 2.1875 0 1 0.375 1 0.625

21.5 1 1.0625 1 0.4375 0 2 0.125 1 0.125

0.5 2 0.4375 2 0.0625

21 1 1.3125 2 0.3125

1.5 2 0.4375 2 0.0625G

5 F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 20.5 0 0.5

21.0625 0.125 0.4375

0.21875 0.0625 20.03125GF 1 21 2

22 3 10

3 21 2G

5 F 20.5 1 0 1 1.5

21.0625 2 0.25 1 1.3125

0.21875 2 0.125 2 0.09375

0.5 1 0 2 0.5

1.0625 1 0.375 2 0.4375

20.21875 1 0.1875 1 0.03125

21 1 0 1 1

22.125 1 1.25 1 0.875

0.4375 1 0.625 2 0.0625G

5 F 1 0 0

0 1 0

0 0 1G 5 I

22. A21 ø F 0.03425 0.03425 0.08904

0.08219 0.08219 0.01370

20.65753 0.34247 20.10960G

Check:

AA21 5 F 22 5 21

0 8 1

12 25 0G

F 0.03425 0.03425 0.08904

0.08219 0.08219 0.01370

20.65753 0.34247 20.10960G

5 F 20.0685 1 0.41095 1 0.65753

0 1 0.65752 2 0.65753

0.411 2 0.41095 1 0

20.0685 1 0.41095 2 0.34247

0 1 0.65752 1 0.34247

0.411 2 0.41095 1 0

20.17808 1 0.0685 1 0.10960

0 1 0.1096 2 0.10960

1.06848 2 0.0685 1 0G

5 F 0.99998 22.0 3 1025 2.0 3 1025

21.0 3 1025 0.99999 0

5 3 1025 5 3 1025 0.99998G

ø F 1 0 0

0 1 0

0 0 1G 5 I

A21A ø F 0.03425 0.03425 0.08904

0.08219 0.08219 0.01370

20.65753 0.34247 20.10960G

F 22 5 21

0 8 1

12 25 0G

5 F 20.0685 1 0 1 1.06848

20.16438 1 0 1 0.1644

1.31506 1 0 2 1.3152

0.17125 1 0.274 2 0.4452

0.41095 1 0.65752 2 0.0685

23.28765 1 2.73976 1 0.548

20.03425 1 0.03425 1 0

20.08219 1 0.08219 1 0

0.65753 1 0.34247 1 0G

5 F 0.99998 5 3 1025 0

2 3 1025 0.99997 0

21.4 3 1024 1.1 3 1024 1G

ø F 1 0 0

0 1 0

0 0 1G 5 I


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23. A21 5 F 0.15 0.3 0.05

20.06875 0.1125 0.01875

20.025 20.05 20.175G

Check:

AA21 5 F 3 28 0

2 4 1

21 0 26GF 0.15 0.3 0.05

20.06875 0.1125 0.01875

20.025 20.05 20.175G

5 F 0.45 1 0.55 1 0 0.9 2 0.9 1 0

0.3 2 0.275 2 0.025 0.6 1 0.45 2 0.05

20.15 1 0 1 0.15 20.3 1 0 1 0.3

0.15 2 0.15 1 0

0.1 1 0.075 2 0.175

20.05 1 0 1 1.05G

5 F 1 0 0

0 1 0

0 0 1G 5 I

A21A 5 F 0.15 0.3 0.05

20.06875 0.1125 0.01875

20.025 20.05 20.175GF 3 28 0

2 4 1

21 0 26G

5 F 0.45 1 0.6 2 0.05

20.20625 1 0.225 2 0.01875

20.075 2 0.1 1 0.175

21.2 1 1.2 1 0 0 1 0.3 2 0.3

0.55 1 0.45 1 0 0 1 0.1125 2 0.1125

0.2 2 0.2 1 0 0 2 0.05 1 1.05G

5 F 1 0 0

0 1 0

0 0 1G 5 I

24. A21 ø F 0.2766 20.2340 20.1915

0.3191 0.1915 20.2979

20.0851 0.1489 0.2128G

AA21 5 F 4 1 5

22 2 1

3 21 6GF 0.2766 20.2340 20.1915

0.3191 0.1915 20.2979

20.0851 0.1489 0.2128G

5 F 1.1064 1 0.3191 2 0.4255

20.5532 1 0.6382 2 0.0851

0.8298 2 0.3191 2 0.5106

20.936 1 0.1915 1 0.7445

0.468 1 0.383 1 0.1489

20.702 2 0.1915 1 0.8934

20.766 2 0.2979 1 1.064

0.383 2 0.5958 1 0.2128

20.5745 1 0.2979 1 1.2768G

5 F 1 0 1 3 1024

21 3 1024 0.9999 0

1 3 1024 21 3 1024 1.0002G

ø F 1 0 0

0 1 0

0 0 1G 5 I

A21A ø F 0.2766 20.2340 20.1915

0.3191 0.1915 20.2979

20.0851 0.1489 0.2128GF 4 1 5

22 2 1

3 21 6G

5 F 1.1064 1 0.468 2 0.5745

1.2764 2 0.383 2 0.8937

20.3404 2 0.2978 1 0.6384

0.2766 2 0.468 1 0.1915

0.3191 1 0.383 1 0.2979

20.0851 1 0.2978 2 0.2128

1.383 2 0.234 2 1.149

1.5955 1 0.1915 2 1.7874

20.4255 1 0.1489 1 1.2768G

5 F 0.9999 1 3 1024 0

23 3 1024 1 24 3 1024

2 3 1024 21 3 1024 1.0002G

ø F 1 0 0

0 1 0

0 0 1G 5 I

25. F 4 21 27 22GF x

yG 5 F 10 225G

A21 5 2 1 } 15 F 22 1

7 4G 5 F 2 } 15

2 1 } 15

2 7 } 15 2 4 } 15 G X 5 A21B 5 F 2 }

15 2 1 } 15

2 7 } 15 2 4 } 15 GF 10 225G 5 F 3

2G The solution is (3, 2).

26. F 4 7 2 3GF x

yG 5 F 216 24G

A21 5 2 1 } 2 F 3 27

22 4G 5 F 2 3 } 2 7 } 2

1 22G

X 5 A21B 5 F 2 3 } 2 7 } 2

1 22GF 216

24G 5 F 10

28G The solution (10, 28).


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175Algebra 2



27. F 3 22 6 25GF x

yG 5 F 5 14G

A21 5 2 1 } 3 F 25 2

26 3G 5 F 5 } 3 2 2 } 3

2 21G

X 5 A21B 5 F 5 } 3 2 2 } 3

2 21GF 5

14G 5 F 21 24G


28. F 1 21 9 210GF x

yG 5 F 4 45G

A21 5 21F 210 1 29 1G 5 F 10 21

9 21G X 5 A21B 5 F 10 21

9 21GF 4 45G 5 F 25


29. F 22 29 4 16GF x

yG 5 F 22 8G

A21 5 1 } 4 F 16 9

24 22G 5 F 4 9 } 4

21 2 1 } 2 G

X 5 A21B 5 F 4 9 } 4

21 2 1 } 2 GF 22

8G 5 F 10 22G


30. F 2 27 21 5GF x

yG 5 F 26 3G

A21 5 1 } 3 F 5 7

1 2G 5 F 5 } 3 7 }

3

1 } 3

2 } 3 G

X 5 A21B 5 F 5 } 3 7 }

3

1 } 3

2 } 3 GF 26

3G 5 F 23 0G


31. F 6 1 21 3GF x

yG 5 F 22 225G

A21 5 1 } 19 F 3 21

1 6G 5 F 3 } 19

2 1 } 19

1 } 19

6 } 19 G

X 5 A21B 5 F 3 } 19

2 1 } 19

1 } 19

6 } 19 GF 22

225 G 5 F 1 28G


32. F 2 1 2 5GF x

yG 5 F 22

38G

A21 5 1 } 8 F 5 21

22 2G 5 F 5 } 8 2 1 } 8

2 1 } 4

1 } 4 G

X 5 A21B 5 F 5 } 8 2 1 } 8

2 1 } 4

1 } 4 GF 22

38G 5 F 26


33. F 5 7 3 5GF x

yG 5 F 20

16G

A21 5 1 } 4 F 5 27

23 5G 5 F 5 } 4 2 7 } 4

2 3 } 4

5 } 4 G

X 5 A21B 5 F 5 } 4 2 7 } 4

2 3 } 4

5 } 4 GF 20

16G 5 F 23


34. C;F 3 25 21 2GF x

yG 5 F 226

10G

A21 5 1F 2 5 1 3G 5 F 2 5

1 3G X 5 A21B 5 F 2 5

1 3GF 226

10G 5 F 22


35. F 1 21 23

5 2 1

23 21 0GF x

y

zG 5 F 2

217

8G

A21 5 F 1 3 5

23 29 216

1 4 7G

X 5 A21B 5 F 1 3 5

23 29 216

1 4 7GF 2

217

8G 5 F 29

19

210G


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36. F 23 1 28

1 22 1

2 22 5GF x

y

zG 5 F 18

211

217G

A21 5 F 21.6 2.2 23

20.6 0.2 21

0.4 20.8 1G

X 5 A21B 5 F 21.6 2.2 23

20.6 0.2 21

0.4 20.8 1GF 18

211

217G 5 F 22

4

21G


37. F 2 4 5

1 2 3

5 24 22GF x

y

zG 5 F 5

4

23G

A21 ø F 0.5714 20.8571 0.1429

1.2143 22.0714 20.0714

21 2 0G

X 5 A21B 5 F 21

22

3G


38. F 4 21 21

6 0 21

21 4 5GF x

y

zG 5 F 220

227

23G

A21 ø F 0.1905 0.0476 0.0476

21.3810 0.9048 20.0952

1.1429 20.7143 0.2857G

X 5 A21B 5 F 24

1

3G


39. F 3 2 21

21 25 4

4 1 1GF x

y

zG 5 F 14

248

2G

A21 5 F 0.75 0.25 20.25

21.41 } 6 20.58 } 3 0.916

21.58 } 3 21.41 } 6 1.08 } 3 G

X 5 A21B 5 F 0.75 0.25 20.25

21.41 } 6 20.58 } 3 0.91 } 6 21.58 } 3 21.41 } 6 1.08 } 3

GF 14

248

2G

5 F 22

10

0G


40. F 6 1 2 1 21 1

21 4 21GF x

y

zG 5 F 11

25

14G

A21 5 F 0.25 20.75 20.25

0 0. } 3 0. } 3 0.25 2.08 } 3 0.58 } 3

GX 5 A21B 5 F 0.25 20.75 20.25

0 0. } 3 0. } 3 0.25 2.08 } 3 0.58 } 3

GF 11

25

14G 5 F 3

3

25G


41. Sample answer: F 4 10 2 5G

42. F 2 5 24 6

0 2 1 27

4 8 27 14

3 6 25 10

G F w

x

y

z

G 5 F 0

52

225

216

G A21 ø F 210 4 27 229

5 22 216 18

4 22 217 20

2 21 27 8

G

X 5 A21B 5 F 23

8

1

25

G The solution is (23, 8, 1, 25).

Problem Solving

43. s 5 hours in a single-engine plane

t 5 hours in a twin-engine plane

s 1 t 5 200

60s 1 240t 5 21,000

F 1 1 60 240GF s

t G 5 F 200 21,000G

A21 5 1 } 180 F 240 21

260 1G 5 F 4 }

3 2 1

} 180

2 1 } 3 1 }

180 G

X 5 A21B 5 F 4 }

3 2 1

} 180

2 1 } 3 1 }

180 GF 200

21,000G 5 F 150 50G

The pilot spent 150 hours fl ying a single-engine plane and 50 hours fl ying a twin-engine plane.


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177Algebra 2



44. x 5 three-point fi eld goals

y 5 two-point fi eld goals

z 5 free throws

x 1 y 1 z 5 976

3x 1 2y 1 z 5 1680

y 2 z 5 135

F 1 1 1

3 2 1

0 1 21G

F x

y

zG 5 F 976

1680

135G

A21 5 F 21 2 } 3 2 1 } 3

1 2 1 } 3 2 } 3

1 2 1 } 3 2 1 } 3 G

X 5 A21B 5 F 21 2 } 3 2 1 } 3

1 2 1 } 3 2 } 3

1 2 1 } 3 2 1 } 3 GF 976

1680

135G 5 F 99

506

371G

Dirk made 99 three-point fi eld goals, 506 two-point fi eld goals, and 371 free throws.

45. a. m 5 number of batches of muffi ns

r 5 number of batches of rolls

m 1 2r 5 8 (cups of buttermilk)

m 1 3r 5 11 (number of eggs)

b. F 1 2 1 3GF m

r G 5 F 8 11G

c. A21 5 1 } 1 F 3 22

21 1G 5 F 3 22 21 1G

X 5 A21B 5 F 3 22 21 1GF 8

11G 5 F 2 3G

The class should make two batches of muffi ns and three batches of rolls.

46 a. c 5 cost of cheese

m 5 cost of meat

2c 1 3m 5 18

3c 1 5m 5 28

F 2 3 3 5GF c

mG 5 F 18 28G

A21 5 1F 5 23 23 2

G 5 F 5 23 23 2

G X 5 A21B 5 F 5 23

23 2GF 18

28G 5 F 6

2G Each cheese costs $6.00 and each meat costs $2.00.

b. 3c 1 5m 5 28

7c 1 10m 5 60

F 3 5 7 10GF c

mG 5 F 28 60G

A21 5 2 1 } 5 F 10 25

27 3G 5 F 22 1

7 } 5 2

3 } 5 G

X 5 A21B 5 F 22 1

7 } 5 2

3 } 5 GF 28

60G 5 F 4 3.2G

Each cheese costs $4.00 and each meat costs $3.20.

c. Sample answer: The results of parts (a) and (b) are not the same. The larger volume of cheese reduces the unit price. The amount of meat is larger and raises the unit price.

47. b 5 ounces of Bran Crunchies

t 5 ounces of Toasted Oats

w 5 ounces of Whole Wheat Flakes

78b 1 104t 1 198w 5 500

b 1 0.6w 5 5

22b 1 25.5t 1 23.8w 5 100

F 78 104 198

1 0 0.6

22 25.5 23.8GF b

t

wG 5 F 500

5 100

G A21 5 F 20.0056 0.9348 0.0223

20.0039 20.9079 0.0549

0.0093 0.1086 20.0378G

X 5 A21B ø F2.3

0.8

1.2G

About 2.3 ounces of Bran Crunchies, 0.8 ounces of Toasted Oats, and 1.2 ounces of Whole Wheat Flakes should be combined.

48. a. Equation 1: Area per

sheet p ( Sheets

of red 1 Sheets of

yellow 1 Sheets

of blue )

5 Total area

Equation 2: Area of red

5 Area of yellow

1 Area of blue

Equation 3:

Cost p Sheets of red

1 Cost p Sheets of yellow

1 Cost p Sheets

of blue 5

Total cost

Equation 1: 0.75r 1 0.75y 1 0.75b 5 9

Equation 2: r 2 y 2 b 5 0

Equation 3: 6.5r 1 4.5y1 8.5b 5 80

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b. F 0.75 0.75 0.75

1 21 21

6.5 4.5 8.5GF r

y

bG 5 F 9

0

80G

c. X 5 A21B 5 F 6

2.5

3.5G

You should buy 6 sheets of red, 2.5 sheets of yellow, and 3.5 sheets of blue tiles.

49. a. AT 5 F 0 1 21 0GF 1 3 5

1 4 2G 5 F 0 1 1 0 1 4 0 1 2

21 1 0 23 1 0 25 1 0G 5 F 1 4 2

21 23 25G AAT 5 A(AT ) 5 F 0 1

21 0GF 1 4 2

21 23 25G 5 F 0 2 1 0 2 3 0 2 5

21 1 0 24 1 0 22 1 0G 5 F 21 23 25

21 24 22G Matrix A rotates the triangle 908 clockwise about

the origin.

x

y

1

4(1, 21)(21, 21)

(23, 24)

(25, 22)

(1, 1)

(5, 2)

(3, 4)

(4, 23)

(2, 25)

ATAAT

A

b. To get back to the original triangle, you must fi nd AAAAT, or A2(AAT). This means the triangle will be rotated 908 clockwise four times from T, or rotated 908 clockwise twice from AAT, so it will be back to its original position.

50. A 5 F a b

c dG, B 5 F d }

ad 2 cb 2b

} ad 2 cb

2c }

ad 2 cb

a }

ad 2 cb G

AB 5 F ad 2 bc }

ad 2 cb 2ab 1 ba

} ad 2 cb

cd 2 dc

} ad 2 cb

2cb 1 da

} ad 2 cb

G 5 F 1 0

0 1G 5 I

BA 5 F ad 2 bc }

ad 2 bc db 2 bd

} ad 2 bc

2ca 1 ac

} ad 2 bc

2bc 1 ad

} ad 2 bc

G 5 F 1 0

0 1G 5 I

Because AB 5 BA 5 I, B is the inverse of A.

Mixed Review

51.

1

x

y

21

52.

1

x

y

21

53.

1

x

y

21

54.

2

x

y

21

55.

1

x

y

21

56. 1

x

y

1

57. The points have approximately no correlation because they show no linear pattern.

58. The points have a positive correlation because y tends to increase as x increases.

59. The points have a negative correlation because y tends to decrease as x increases.

60. F 5 4x 18 6

G 5 F 5 220 3y 6G

4x 5 220 3y 5 18

x 5 25 y 5 6

61. F 23x 29 13 25G 1 F 4 12

25y 16G 5 F 220 3 18 11G

F 23x 1 4 3 13 2 5y 11G 5 F 220 3

18 11G 23x 1 4 5 220 13 2 5y 5 18

23x 5 224 25y 5 5

x 5 8 y 5 21

Quiz 3.6–3.8 (p. 217)

1. 2AB 5 2F 1 24 5 2

GF 2 23 0 1

G 5 F 2 28 10 4

GF 2 23 0 1

G 5 F 4 1 0 26 2 8

20 1 0 230 1 4G 5 F 4 214

20 226G


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179Algebra 2


2. AB 1 AC 5 (F 1 24 5 2

GF 2 23 0 1

G) 1 (F 1 24

5 2GF 26 21

2 4G)

5 F 2 1 0 23 2 4 10 1 0 215 1 2

G 1 F 26 2 8 21 2 16

230 1 4 25 1 8G

5 F 2 2 14 27 2 17 10 2 26 213 1 3

G 5 F 212 224 216 210

G 3. A(B 1 C) 5 F 1 24

5 2G(F 2 23

0 1G 1 F 26 21

2 4G)

5 F 1 24 5 2

GF 24 24 2 5

G 5 F 24 2 8 24 2 20

220 1 4 220 1 10G 5 F 212 224

216 210G

4. (B 2 A)C 5 (F 2 23 0 1

G 2 F 1 24 5 2

G)F 26 21 2 4

G 5 F 1 1

25 21GF 26 21

2 4G

5 F 26 1 2 21 1 4 30 2 2 5 2 4

G 5 F 24 3 28 1

G 5. 5 4

22 23 5 5(23) 2 (22)(4) 5 27

6. 1 0 22 23 1 4

2 3 21 1 0

23 1

2 3

5 (21 1 0 1 18) 2 (24 1 12 1 0) 5 9

7. 2 21 5 23 6 9

22 3 1 2 21

23 6

22 3

5 (12 1 18 2 45) 2 (260 1 54 1 3) 5 212

8. F 1 3

2 7GF x

y G 5 F 22

26G A21 5

1 } 1 F 7 23

22 1G 5 F 7 23

22 1G X 5 A21B 5 F 7 23

22 1GF 22

26G 5 F 4


9. F 3 24

2 23GF x

y G 5 F 5

3G A21 5 21F 23 4

22 3G 5 F 3 24

2 23G X 5 A21B 5 F 3 24

2 23GF 5

3G 5 F 3


10. F 23 2

6 25GF x

y G 5 F 213

24G A21 5

1 } 3 F 25 22

26 23G 5 F 2 5 } 3 2 2 } 3

22 21G

X 5 A21B 5 F 2 5 } 3 2 2 } 3

22 21GF 213

24G 5 F 17 }

3

2G


3 , 2 2 .

11. F 3 21

2 22GF x

y G 5 F 24

28G A21 5 2

1 } 4 F 22 1

22 3G 5 F 1 } 2 2 1 } 4

1 } 2 2 3 } 4

G X 5 A21B 5 F 1 }

2 2 1 } 4

1 } 2 2 3 } 4

GF 24

28G 5 F 0


12. F 7 4

5 3GF x

y G 5 F 6

225G A21 5 1F 3 24

25 7G 5 F 3 24

25 7G X 5 A21B 5 F 3 24

25 7GF 6

225G 5 F 118


13. F 4 1

26 1GF x

y G 5 F 22

18G A21 5

1 } 10 F 1 21

6 4G 5 F 1 } 10

2 1 } 10

3 } 5 2 } 5 G X 5 A21B 5 F 1 }

10 2 1 } 10

3 } 5 2 } 5 GF 22

18G 5 F 22


Chapter 3 continued

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14. Area 5 6 1 } 2

0 2 1 12 2 1

12 26 1 0 2 12 2

12 26

5 6 1 } 2 [(0 1 24 1 312) 2 (24 1 0 1 24)] 5 144

The area of the sail is 144 square feet.

Problem Solving Workshop 3.8 (p. 219)

1. 2m 1 p 5 17.75

2m 1 2p 1 d 5 34.50

4m 1 3p 1 2d 5 67.25 F 2 1 0 � 17.75

2 2 1 � 34.50

4 3 2 � 67.25G

(22)R1 1 R3 F 2 1 0 � 17.75

2 2 1 � 34.50

0 1 2 � 31.75G

(21)R1 1 R2

F 2 1 0 � 17.75

0 1 1 � 16.75

0 1 2 � 31.75G

(21)R2 1 R3 F 2 1 0 � 17.75

0 1 1 � 16.75

0 0 1 � 15G

From the third row, d 5 15. From the second row, p 1 d 5 16.75, so p 1 15 5 16.75, or p 5 1.75. From the fi rst row, 2m 1 p 5 17.75, so 2m 1 1.75 5 17.75, or m 5 8.

A movie pass costs $8, a package of popcorn costs $1.75, and a DVD costs $15.

2. s 5 amount in stocks; b 5 amount in bonds

m 5 amount in money market funds

s 1 b 1 m 5 18000

0.1s 1 0.07b 1 0.05m 5 1440

s 2 b 2 m 5 0

F 1 1 1 � 18000

0.1 0.07 0.05 � 1440

1 21 21 � 0G

100R2

R1 1 (21)R3 F 1 1 1 � 18,000

10 7 5 � 144,000

0 2 2 � 18,000G

(210)R1 1 R2

F 1 1 1 � 18,000

0 23 25 � 236,000

0 2 2 � 18,000G

2R2 1 3R3 F 1 1 1 � 18,000

0 23 25 � 236,000

0 0 24 � 218,000G

(21)R2

1 2 1 } 4 2 R3

F 1 1 1 � 18,000

0 3 5 � 36,000

0 0 1 � 4500G

From row 3, m 5 4500. From row 2, 3b 1 5m 5 36,000, so 3b 1 5(4500) 5 36,000, or b 5 4500. From row 1, s 1 b 1 m 5 18,000, so s 1 4500 1 4500 5 18,000, or s 5 9000. You should invest $9000 in stocks, $4500 in bonds, and $4500 in money market funds.

3. s 5 pounds of sunfl ower seed

t 5 pounds of thistle seed

s 1 t 5 20

0.34s 1 0.79t 5 10.85

F 1 1 � 20 0.34 0.79 � 10.85G

(20.34)R1 1 R2

F 1 1 � 20 0 0.45 � 4.05G

R2

} 0.45

F 1 1 � 20 0 1 � 9G

From row 2, t 5 9. From row 1, s 1 t 5 20. So, s 1 9 5 20, or s 5 11.

The mixture contains 11 pounds of sunfl ower seed and 9 pounds of thistle seed.

4. x 2 2y 1 4z 5 210

5x 1 y 2 z 5 24

3x 2 6y 1 12z 5 230 F 1 22 4 � 210

5 1 21 � 24

3 26 12 � 230G

(23)R1 1 R3 F 1 22 4 � 210

5 1 21 � 24

0 0 0 � 0G

Because row 3 produces the equation 0 5 0, the system has infi nitely many solutions.

Mixed Review of Problem Solving (p. 220)

1. a. A 5 F 4.5 6 2.5 5.5 8 2.5G B 5 F 4 6.5 3.25

5 8.5 3.25G b. B 2 A 5 F 4 6.5 3.25

5 8.5 3.25G 2 F 4.5 6 2.5 5.5 8 2.5G

5 F 20.5 0.5 0.75 20.5 0.5 0.75G

Each element in each row of B 2 A represents the difference in cost (in thousands of dollars) of a daytime, primetime, and late-night commercial on a cable TV network between the two cities.

c. A 5 1.10F 4.5 6 2.5 5.5 8 2.5G 5 F 4.95 6.6 2.75

6.05 8.8 2.75G B 5 1.10F 4 6.5 3.25

5 8.5 3.25G 5 F 4.4 7.15 3.575 5.5 9.35 3.575G

2. a. n 5 number of nickels

d 5 number of dimes

q 5 number of quarters

n 1 d 1 q 5 85

0.05n 1 0.1d 1 0.25q 5 13.25

22d 1 q 5 0

b. F 1 1 1

0.05 0.1 0.25

0 22 1GF n

d

qG 5 F 85

13.25

0G


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181Algebra 2


c. A21 5 F 1. } 3 26. } 6 0. } 3 20. } 1 2. } 2 20. } 4 20. } 2 4. } 4 0. } 1

G X 5 A21B 5 F 1. } 3 26. } 6 0. } 3

20. } 1 2. } 2 20. } 4 20. } 2 4. } 4 0. } 1

GF 85

13.25

0G

5 F 25

20

40G

The person has 25 nickels, 20 dimes, and40 quarters.

3. a. Sample answer:

A 5 F 3 2 8 4G C 5 F 2 5

6 13G det A 5 det C 5 24

b. Sample answer:

B 5 F 7 0 5

1 4 3

2 4 6G

D 5 F 1 2 4

5 0 4

1 6 4G

det B 5 det D 5 64

4. Matrix BA is defi ned because B has three columns andA has three rows.

BA 5 [ 0.03 0.05 0.08]F 175 270

370 225

200 255G

5 [0.03(175) 1 0.05(370) 1 0.08(200)

0.03(270) 1 0.05(225) 1 0.08(255)]

5 [39.75 39.75]

This matrix represents the total commission made by each salesperson. Mary and Mark each made a total of $39.75 in commission.

5. a. H 1 N 1 3O 5 63

2N 1 O 5 44

2H 1 O 5 18

b. A 5 F 1 1 3

0 2 1

2 0 1G

det A 5 1 1 3 0 2 1

2 0 1 1 1

0 2

2 0

5 (2 1 2 1 0) 2 (12 1 0 1 0) 5 28

c. You can use Cramer’s rule to solve the system.

H 5

63 1 3 44 2 1

18 0 1 63 1

44 2

18 0 }} 28

5 (126 1 18 1 0) 2 (108 1 0 1 44)

}}} 28 5 1

N 5

1 63 3 0 44 1

2 18 1 1 63

0 44

2 18 }} 28

5 (44 1 126 1 0) 2 (264 1 18 1 0)

}}} 28 5 14

O 5

1 1 63 0 2 44

2 0 18 1 1

0 2

2 0 }} 28

5 (36 1 88 1 0) 2 (252 1 0 1 0)

}}} 28

5 16

The atomic weights of hydrogen, nitrogen, and oxygen are 1, 14, and 16, respectively.

6. c 5 bushels of corn

s 5 bushels of soybeans

w 5 bushels of wheat

2.35c 1 5.40s 1 3.60w 5 4837

c 1 s 1 w 5 1700

c 2 3.25s 2 3.25w 5 0

F 2.35 5.40 3.60

1 1 1

1 23.25 23.25GF c

s

wG 5 F 4837

1700

0G

A21 5 F 0 0.7647 0.2353

0.5556 21.4690 0.1634

20.5556 1.7042 20.3987G

X 5 A21B 5 F1300

190

210G

The farmer harvested 210 bushels of wheat.

Chapter 3 Review (pp. 222–226)

1. A system of linear equations with at least one solution is consistent, while a system with no solution is inconsistent.

2. A solution (x, y, z) of a system of linear equations in three variables is called an ordered triple.

3. The product of two matrices is defi ned when the number of columns in the fi rst matrix is the same as the number of rows in the second matrix.

4.

2x

y

22

(5, 1)


Check: 2(5) 2 1 0 9 5 1 3(1) 0 8

9 5 9 ✓ 8 5 8 ✓


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5.2

x

y

21

(24, 22)


Check: 2(24) 2 3(22) 0 22 24 1 (22) 0 26

22 5 22 ✓ 26 5 26 ✓

6.

1

x

y

22

(0, 6)


Check: 3(0) 1 6 0 6 20 1 2(6) 0 12

6 5 6 ✓ 12 5 12 ✓

7. 3x 1 2y 5 5 3 2 6x 1 4y 5 10

22x 1 3y 5 27 3 3 26x 1 9y 5 81

13y 5 91

y 5 7

3x 1 2(7) 5 5 → x 5 23


8. 3x 1 5y 5 5 3 3 9x 1 15y 5 15

2x 2 3y 5 16 3 5 10x 2 15y 5 80

19x 5 95

x 5 5

3(5) 1 5y 5 5 → y 5 22


9. 2x 1 3y 5 9 2x 1 3y 5 9

23x 1 y 5 25 3 (23) 9x 2 3y 5 275

11x 5 266

x 5 26

2(26) 1 3y 5 9 → y 5 7


10. r 5 price of regular gasoline

p 5 price of premium gasoline

14r 1 10p 5 46.68 14r 1 10p 5 46.68

2r 1 p 5 0.30 3 14 214r 1 14p 5 4.2

24p 5 50.88

p 5 2.12

2r 1 2.12 5 0.30 → r 5 1.82

Regular gas costs $1.82 per gallon and premium gas costs $2.12 per gallon.

11. 4x 1 y < 1 12. 2x 1 3y > 6

2x 1 2y ≤ 5 2x 2 y ≤ 8

1

x

y

21

1

x

y

21

13. x 1 3y ≥ 5 2x 1 2y < 4

1x

y

21

14. x 2 y 1 z 5 10

23x 1 5y 2 z 5 218

22x 1 4y 5 28

4x 1 y 2 2z 5 15 4x 1 y 2 2z 5 15

23x 1 5y 2 z 5 218 3 (22) 6x 2 10y 1 2z 5 36

10x 2 9y 5 51

22x 1 4y 5 28 3 5 210x 1 20y 5 240

10x 2 9y 5 51 10x 2 9y 5 51

11y 5 11

y 5 1

22x 1 4(1) 5 28 → x 5 6

6 2 1 1 z 5 10 → z 5 5


15. 6x 2 y 1 4z 5 6 6x 2 y 1 4z 5 6

2x 1 2y 2 5z 5 242 3 (23) 26x 2 6y 1 15z 5 126

27y 1 19z 5 132

2x 2 3y 1 z 5 31 3 2 22x 2 6y 1 2z 5 62

2x 1 2y 2 5z 5 242 2x 1 2y 2 5z 5 242

24y 2 3z 5 20

27y 1 19z 5 132 3 4 228y 1 76z 5 528

24y 2 3z 5 20 3 (27) 28y 1 21z 5 2140

97z 5 388

z 5 4

27y 1 19(4) 5 132 → y 5 28

6x 2 (28) 1 4(4) 5 6 → x 5 23



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183Algebra 2


16. 5x 1 y 2 z 5 40 5x 1 y 2 z 5 40

2x 1 3y 1 z 5 16 3 5 25x 1 15y 1 5z 5 80

16y 1 4z 5 120

x 1 7y 1 4z 5 44

2x 1 3y 1 z 5 16

10y 1 5z 5 60

16y 1 4z 5 120 3 5 280y 2 20z 5 2600

10y 1 5z 5 60 3 4 40y 1 20z 5 240

240y 5 2360

y 5 9

16(9) 1 4z 5 120 → z 5 26

5x 1 9 2 (26) 5 40 → x 5 5


17. w 5 number of wind instruments

s 5 number of string instruments

p 5 number of percussion instruments

Equation 1: w 1 s 1 p 5 15

Equation 2: w 5 2(s 1 p)

Equation 3: s 5 3

w 5 2(3 1 p) → w 5 6 1 2p

(6 1 2p) 1 3 1 p 5 15 → p 5 2

w 5 6 1 2(2) → w 5 10

Ten students play wind instruments, three play string instruments, and two play percussion.

18. F 4 25 2 3G 1 F 21 3

27 4G 5 F 4 1 (21) 25 1 3 2 1 (27) 3 1 4G

5 F 3 22 25 7G

19. F 21 8 2 23G 1 F 7 24

6 21G 5 F 21 1 7 8 1 (24) 2 1 6 23 1 (21)G

5 F 6 4 8 24G

20. F 10 24 5 1G 2 F 0 9

2 7G 5 F 10 2 0 24 2 9 5 2 2 1 2 7G

5 F 10 213 3 26G

21. F 22 3 5

21 6 22G 2 F 24 7 5

28 0 29G 5 F 22 2 (24) 3 2 7 5 2 5

21 2 (28) 6 2 0 22 2 (29)G 5 F 2 24 0

7 6 7G

22. 23F 5 22 3 6G 5 F 23(5) 23(22)

23(3) 23(6)G 5 F 215 6 29 218G

23. 8F 8 4 5

21 6 22G 5 F 8(8) 8(4) 8(5)

8(21) 8(6) 8(22)G 5 F 64 32 40

28 48 216G 24. [21 21]F 8 2

26 29G 5 [21(8) 1 (21)(26) 21(2) 1 (21)(29)]

5 [22 7]

25. F 11 7 1 25GF 0 25

4 23G 5 F 11(0) 1 7(4) 11(25) 1 7(23)

1(0) 1 (25)(4) 1(25) 1 (25)(23)G 5 F 28 276

220 10G 26. F 4 21

1 7GF 5 22 4 3 12 6G

5 F 4(5) 1 (21)(3) 4(22) 1 (21)(12) 4(4) 1 (21)(6) 1(5) 1 7(3) 1(22) 1 7(12) 1(4) 1 7(6)G

5 F 17 220 10 26 82 46G

27. F 22 5 0 3GF 6 23 5

2 0 1G 5 F 22(6) 1 5(2) 22(23) 1 5(0) 22(5) 1 5(21)

0(6) 1 3(2) 0(23) 1 3(0) 0(5) 1 3(21)G 5 F 22 6 215

6 0 23G 28. C 5 BI 5 F 109.99 0 0

0 319.99 0

0 0 549.99G

AC 5 F 5000 6000 8000 4000 10,000 5000G

F 109.99 0 0

0 319.99 0

0 0 549.99G

5 F 549,950 1,919,940 4,399,920 439,960 3,199,900 2,749,950G


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The labels for the matrix are shown below. Total Value (Dollars)

19 in. 27 in. 32 in.

Warehouse 1

Warehouse 2 F 549,950 1,919,940 4,399,920 439,960 3,199,900 2,749,950G

In warehouse 1, the total value of the 19 in. TVs was $549,950, the total value of the 27 in. TVs was $1,919,940, and the total value of the 32 in. TVs was $4,399,920. In warehouse 2, the total value of the 19 in. TVs was $439,960, the total value of the 27 in. TVs was $3,199,900, and the total value of the 32 in. TVs was $2,749,950.

29. 24 2 5 8 5 24(8) 2 5(2) 5 242

30. 3 25 2 6 5 3(6) 2 2(25) 5 28

31. 3 0 1 6 5 3(6) 2 1(0) 5 18

32. Area 5 6 1 } 2

0 0 1 0 50 1

70 20 1 0 0

0 50

70 20

5 6 1 } 2 [(0 1 0 1 0) 2 (3500 1 0 1 0)]

5 1750 square inches

(1750 in.2) 1 1 ft2 }

144 in.2 2 ø 12.2 ft2

You will need approximately 12.2 square feet of material.

33. F 1 4 2 25GF x

yG 5 F 11 9G

A21 5 1 }

213 F 25 24

22 1G 5 F 5 }

13 4 }

13

2 } 13

2 1 } 13 G X 5 A21B 5 F

5 }

13 4 }

13

2 } 13

2 1 } 13 GF 11 9G 5 F 7


34. F 3 1

21 2GF x yG 5 F 21

12G A21 5

1 } 7 F 2 21

1 3G 5 F 2 } 7 2 1 } 7

1 } 7 3 } 7 G X 5 A21B 5 F 2 } 7 2 1 } 7

1 } 7 3 } 7 GF 21 12G 5 F 22


35. F 3 2

4 23GF x yG 5 F 211

8G A21 5 2

1 } 17 F 23 22

24 3G 5 F 3 } 17

2 } 17

4 } 17

2 3 } 17 G X 5 A21B 5 F 3 }

17 2 }

17

4 } 17

2 3 } 17 GF 211 8G 5 F 21


Chapter 3 Test (p. 227)

1.

x

y

1

1

4x 1 y 5 5

3x 2 y 5 2

(1, 1)


Check: 3(1) 2 1 0 2 4(1) 1 1 0 5

2 5 2 ✓ 5 5 5 ✓

2.

x

y1

1

x 1 2y 5 26

26x 2 2y 5 214

(4, 25)


Check: 4 1 2(25) 0 26 26(4) 2 2(25) 0 214

26 5 26 ✓ 214 5 214 ✓

3.

x

y

1

1

x 2 y 5 2332

2x 2 3y 5 15

The lines do not intersect, so there is no solution.


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185Algebra 2


4.

x

y

2

2

2x 1 8y 5 24

3x 2 y 5 12

(4, 0)


Check: 3(4) 2 0 0 12 24 1 8(0) 0 24

12 5 12 ✓ 24 5 24 ✓

5. 2x 1 y < 6 6. x 2 3y ≥ 9 y > 22

1 }

3 x 2 y ≤ 3

1

x

y

21

1

x

y

22

The solution is a line.

7. x 2 2y ≤ 214 8. 23x 1 4y > 212

y ≥ x y < 22x 1 5

2

x

y

22

1

x

y

21

9. 3x 1 y 5 29 3 2 6x 1 2y 5 218

x 2 2y 5 210 x 2 2y 5 210

7x 5 228

x 5 24

3(24) 1 y 5 29 → y 5 3


10. 2x 1 3y 5 22 3 (22) 24x 2 6y 5 4

4x 1 7y 5 26 4x 1 7y 5 26

y 5 22

2x 1 3(22) 5 22 → x 5 2


11. x 1 4y 5 226 → x 5 226 2 4y

25x 2 2y 5 214

25(226 2 4y) 2 2y 5 214

18y 5 2144

y 5 28

x 1 4(28) 5 226 → x 5 6


12. x 2 y 1 z 5 23 3 2 2x 2 2y 1 2z 5 26

4x 1 2y 2 z 5 2 4x 1 2y 2 z 5 2

6x 1 z 5 24

2x 2 y 1 5z 5 4 3 2 4x 2 2y 1 10z 5 8

4x 1 2y 2 z 5 2 4x 1 2y 2 z 5 2

8x 1 9z 5 10

6x 1 z 5 24 3 (29) 254x 2 9z 5 36

8x 1 9z 5 10 8x 1 9z 5 10

246x 5 46

x 5 21

6(21) 1 z 5 24 → z 5 2

21 2 y 1 2 5 23 → y 5 4


13. x 1 y 1 z 5 3

2x 1 3y 1 2z 5 28

4y 1 3z 5 25

5y 1 z 5 2 3 (23) 215y 2 3z 5 26

4y 1 3z 5 25 4y 1 3z 5 25

211y 5 211

y 5 1

5(1) 1 z 5 2 → z 5 23

x 1 1 1 (23) 5 3 → x 5 5


14. 2x 2 5y 2 z 5 17 3 2 4x 2 10y 2 2z 5 34

24x 1 6y 1 z 5 220 24x 1 6y 1 z 5 220

24y 2 z 5 14

x 1 y 1 3z 5 19 3 4 4x 1 4y 1 12z 5 76

24x 1 6y 1 z 5 220 24x 1 6y 1 z 5 220

10y 1 13z 5 56

24y 2 z 5 14 3 13 252y 2 13z 5 182

10y 1 13z 5 56 10y 1 13z 5 56

242y 5 238

y 5 2 17

} 3

24 1 2 17

} 3 2 2 z 5 14 → z 5 26

} 3

2x 2 5 1 2 17

} 3 2 2 26

} 3 5 19 → x 5 2 4 } 3

The solution is 1 2 4 } 3 , 2

17 } 3 ,

26 }

3 2 .

15. 2A 1 B 5 2F 1 22 4 23

G 1 F 3 5

21 0G

5 F 2(1) 1 3 2(22) 1 5

2(4) 1 (21) 2(23) 1 0G 5 F 5 1

7 26G


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16. C 2 3B 5 F 26 8

10 15G 2 3F 3 5

21 0G

5 F 26 2 3(3) 8 2 3(5)

10 2 3(21) 15 2 3(0)G 5 F 215 27

13 15G

17. Not possible; A and D do not have the same dimensions.

18. 4D 1 E 5 4F 21 3 22

2 0 21G 1 F 4 21 3

6 22 1G

5 F 4(21) 1 4 4(3) 1 (21) 4(22) 1 3

4(2) 1 6 4(0) 1 (22) 4(21) 1 1G

5 F 0 11 25

14 22 23G

19. AC 5 F 1 22

4 23GF 26 8

10 15G

5 F 1(26) 1 (22)(10) 1(8) 1 (22)(15)

4(26) 1 (23)(10) 4(8) 1 (23)(15)G

5 F 226 222

254 213G

20. Not possible; the number of columns in D does not equal the number of rows in E.

21. (A 1 B)D 5 (F 1 22 4 23

G 1 F 3 5

21 0G)F 21 3 22

2 0 21G

5 F 4 3 3 23

GF 21 3 22

2 0 21G

5 F 4(21) 1 3(2) 4(3) 1 3(0) 4(22) 1 3(21)

3(21) 1 (23)(2) 3(3) 1 (23)(0) 3(22) 1 (23)(21)G

5 F 2 12 211

29 9 23G

22. A(C 2 B) 5 F 1 22 4 23

G(F 26 8 10 15

G 2 F 3 5

21 0G)

5 F 1 22 4 23

GF 29 3 11 15

G 5 F 1(29) 1 (22)(11) 1(3) 1 (22)(15)

4(29) 1 (23)(11) 4(3) 1 (23)(15)G 5 F 231 227

269 233G

23. 3 22 4 1 5 3(1) 2 (4)(22) 5 11

24. 24 5 2 21 5 (24)(21) 2 (5)(2) 5 26

25. 21 3 1 0 2 23

5 1 22 21 3

0 2

5 1

5 (4 2 45 1 0) 2 (10 1 3 1 0)

5 254

26. 2 0 21 5 23 2

1 4 6 2 0

5 23

1 4

5 (236 1 0 2 20) 2 (3 1 16 1 0) 5 275

27. F 3 4 4 5GF x

yG 5 F 6 7G

A21 5 2 1 } 1 F 5 24

24 3G 5 F 25 4

4 23G X 5 A21B 5 F 25 4

4 23GF 6 7G 5 F 22


28. F 2 27 1 23GF x

yG 5 F 236

216G A21 5

1 } 1 F 23 7

21 2G 5 F 23 7

21 2G X 5 A21B 5 F 23 7

21 2GF 236

216G 5 F 24 4G


29. F 5 3

29 26GF x yG 5 F 25

12G A21 5 2

1 } 3 F 26 23

9 5G 5 F 2 1

23 2 5 } 3 G X 5 A21B 5 F 2 1

23 2 5 } 3 GF 25

12G 5 F 2


30. F 3 2

21 4GF x yG 5 F 15

233G A21 5

1 } 14 F 4 22

1 3G 5 F 2 } 7 2 1 } 7

1 } 14

3 } 14

G X 5 A21B 5 F 2 } 7 2 1 } 7

1 } 14

3 } 14

GF 15

233G 5 F 9



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187Algebra 2


31. f 5 amount invested in 5% interest bond

s 5 amount invested in 7% interest bond

f 1 s 5 15,000

0.05f 1 0.07s 5 880

F 1 1 0.05 0.07GF f

sG 5 F 15,000

880G A21 5

1 } 0.02 F 0.07 21

20.05 1G 5 F 3.5 250

22.5 50G X 5 A21B 5 F 3.5 250

22.5 50GF 15,000

880G 5 F 8500

6500G The investor should invest $8500 in 5% interest bonds

and $6500 in 7% interest bonds.

32. c 5 number of children’s tickets

a 5 number of adult tickets

s 5 number of senior citizen tickets

c 1 a 1 s 5 800

3c 1 8a 1 5s 5 3775

c 5 2a

2a 1 a 1 s 5 800

3a 1 s 5 800

3(2a) 1 8a 1 5s 5 3775

14a 1 5s 5 3775

3a 1 s 5 800 3 (25) 215a 2 5s 5 24000

14a 1 5s 5 3775 14a 1 5s 5 3775

2a 5 2225

a 5 225

c 5 2(225) 5 450

450 1 225 1 s 5 800 → s 5 125

There were 450 children’s tickets, 225 adult tickets, and 125 senior citizen tickets sold.

33. b 5 speed of boat

c 5 speed of current

b 1 c 5 34

b 2 c 5 28

2b 5 62

b 5 31

31 1 c 5 34 → c 5 3

The boat travels 31 miles per hour in still water. The speed of the current is 3 miles per hour.

Standardized Test Preparation (p. 229)

1. Substitute these values into the fi rst equation:

21 1 2 1 (23) 0 2

22 Þ 2

Because 22 Þ 2, choice B can be eliminated.

2. Substitute the matrix in choice D for X in the equation:

F 3 4

2 3GF 1 0

0 1G 0 F 4 210

1 26G F 3 4

2 3G Þ F 4 210

1 26G Because the resulting matrices are not equal, choice D

can be eliminated.

Standardized Test Practice (pp. 230–231)

1. C;

d 1 (d 2 39.9) 5 600.9

2d 5 640.8

d 5 320.4

The top DVD grossed $320.4 million.

2. C;

3x 1 2y 5 8 Eq. 1 x 2 4y 5 216 Eq. 2

3(0) 1 2(4) 0 8 0 2 4(4) 0 216

8 5 8 ✓ 216 5 216 ✓

3. B;

Eliminate other possibilities.

4. A;

s 5 number of scarves

g 5 number of pairs of gloves

s 1 g 5 8 → g 5 8 2 s

6s 1 9g 5 66

6s 1 9(8 2 s) 5 66

23s 5 26

s 5 2

5. D;

A21 5 2 1 } 1 F 7 3

5 2G 5 F 27 23

25 22G X 5 A21B 5 F 27 23

25 22GF 4 6 0 21G 5 F 228 239

220 228G


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6. B;

d 5 driving time

c 5 classroom instruction

b 5 observing

d 1 c 1 b 5 46

c 5 3d

b 5 d 2 4

d 1 3d 1 (d 2 4) 5 46

5d 5 50

d 5 10

7. C;

2x 2 3 1 2 } 3 x 1 2 2 5 9

0x 5 15

0 5 15

No solution

8. D;

Area of triangle with vertices

(1, 25), (2, 3), and (12, 2):

Area 5 6 1 } 2

1 25 1 2 3 1

12 2 1 1 25

2 3

12 2

5 6 1 } 2 [(3 2 60 1 4) 2 (36 1 2 2 10)]

5 40.5

Area of triangle with vertices

(21, 0), (5, 9), and (8, 0):

Area 5 6 1 } 2

21 0 1 5 9 1

8 0 1 21 0

5 9

8 0

5 6 1 } 2 [(29 1 0 1 0) 2 (72 1 0 1 0)]

5 40.5

9. B;

p 5 number of peony bulbs

h 5 number of phlox bulbs

l 5 number of lily bulbs

p 1 h 1 l 5 6

3.99p 1 2.61h 1 2.24l 5 19.43

h 5 2l

p 1 2l 1 l 5 6

p 1 3l 5 6

3.99p 1 2.61(2l) 1 2.24l 5 19.43

3.99p 1 7.46l 5 19.43

p 1 3l 5 6 3 (23.99) 23.99p 2 11.97l 5 223.94

3.99p 1 7.46l 5 19.43 3.99p 1 7.46l 5 19.43

24.51l 5 24.51

l 5 1

10. B; 4 6

26 29 5 4(29) 2 (26)(6) 5 0

Because the determinant is 0, the matrix has no inverse.

11. 2x 2 5y 5 210

x 1 4y 5 21 → x 5 21 2 4y

2(21 2 4y) 2 5y 5 210

213y 5 252

y 5 4

12. 21 1 2(22) 1 2z 5 7

2z 5 12

z 5 6

13. 2 1 4 3 25 0

1 0 21 2 1

3 25

1 0

5 (10 1 0 1 0) 2 (220 1 0 2 3) 5 33

14. 2F 5x 0

22 3G 2 F 3 21 4y 22G 5 F 17 1

210 8G F 10x 0

24 6G 2 F 3 21 4y 22G 5 F 17 1

210 8G F 10x 2 3 1

24 2 4y 8G 5 F 17 1

210 8G 10x 2 3 5 17 24 2 4y 5 210

10x 5 20 24y 5 26

x 5 2 y 5 3 } 2

x 1 4y 5 2 1 4 1 3 } 2 2 5 8

15. F 48 21 11 2

46 22 8 6

43 21 12 6

41 19 15 7

GF 2

0

1

1

G 5 F 109

106

104

104

G Detroit earned 109 points. Tampa Bay earned 106 points

and San José and Boston each earned 104 points.

16. m 5 number of matineé tickets

r 5 number of regular tickets

4m 1 6r 5 6000

r 5 890 1 m

4m 1 6(890 1 m) 5 6000

10m 5 660

m 5 66

r 5 890 1 66 5 956

The theater sold 66 matineé tickets and 956 regular tickets.


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189Algebra 2


17. a. y 5 3x 1 12.87 where x represents the number of years since 1996

b. y 5 1.86x 1 26.97 where x represents the number of years since 1996

c.

x

y

200

4 6 8 10 12 14 16 18

16

8

24

32

40

48

56

64

72

Years since 1996

Sp

end

ing

(d

olla

rs)

y 5 1.86x 1 26.97

y 5 3x 1 12.87

Sample answer: If you look at the graphs of the 2 equations, the two lines intersect at (12.5, 50.4). This means that about one-half of the way into the twelfth year, the video game spending will equal that of the box offi ce spending.

18 a. Area 5 6 1 } 2

0 0 1 20 0 1

x y 1 5 100

b. 6 1 } 2

0 0 1 20 0 1

x y 1 0 0

20 0

x y

5 100

6 1 } 2 [(0 1 0 1 20y) 2 (0 1 0 1 0)] 5 100

6 1 } 2 (20y) 5 100

20y 5 6200

y 5 610

Because (x, y) is in the fi rst quadrant, y must be positive. So, y 5 10.

c. The value of x does not matter as long as y 5 10.

Cumulative Review, Chs. 1–3 (pp. 232–233)

1. 3x2 2 5x2 2 8x 1 12x 1 3x 5 22x2 1 7x

2. 15x 2 6x 1 4x 1 10y 2 3y 5 13x 1 7y

3. 3x 1 6 2 4x2 1 3x 1 9 5 24x2 1 6x 1 15

4. 6x 2 7 5 22x 1 9

8x 5 16

x 5 2

Check: 6(2) 2 7 0 22(2) 1 9

5 5 5 ✓

5. 4(x 2 3) 5 16x 1 18

4x 2 12 5 16x 1 18

212x 5 30

x 5 2 5 } 2

Check: 4 1 2 5 } 2 2 3 2 0 16 1 2

5 } 2 2 1 18

222 5 222 ✓

6. 1 }

3 x 1 3 5 2

7 } 2 x 2

3 } 2

1 }

3 x 1

7 } 2 x 5 2

3 } 2 2 3

2 }

6 x 1

21 } 6 x 5 2

3 } 2 2

6 } 2

23

} 6 x 5 2

9 } 2

x 5 2 9 } 2 1 6 } 23

2 x 5 2

27 } 23

Check: 1 }

3 1 2

27 } 23 2 1 3 0 2

7 } 2 1 2

27 } 23 2 2

3 } 2

60

} 23

5 60

} 23 ✓

7. x 1 3 5 5

x 1 3 5 5 or x 1 3 5 25

x 5 2 or x 5 28

8. 4x 2 1 5 27

4x 2 1 5 27 or 4x 2 1 5 227

4x 5 28 or 4x 5 226

x 5 7 or x 5 2 13

} 2

9. 9 2 2x 5 41

9 2 2x 5 41 or 9 2 2x 5 241

22x 5 32 or 22x 5 250

x 5 216 or x 5 25

10. 6(x 2 4) > 2x 1 8

6x 2 24 > 2x 1 8

4x > 32

x > 8

0 2 4 6 8 10 12

11. 3 ≤ x 2 2 ≤ 8 3 1 2 ≤ x ≤ 8 1 2

5 ≤ x ≤ 10

1 3 5 7 9 1121

12. 2x < 26 or x 1 2 > 5

x < 23 or x > 3

1 3 5212325

13. x 2 4 < 5

25 < x 2 4 < 5

25 1 4 < x < 5 1 4

21 < x < 9

0 2 4 6 8 1022

14. x 1 3 ≥ 15

x 1 3 ≤ 215 or x 1 3 ≥ 15

x ≤ 218 or x ≥ 12

0 6 1226212218


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15. 6x 1 1 < 23

223 < 6x 1 1 < 23

223 2 1 < 6x < 23 2 1

2 24

} 6 < x < 22

} 6

24 < x < 11

} 3

0 2 4 62426 22

113

16. (3, 2), (21, 25), m 5 25 2 2

} 21 2 3 5

7 } 4

The line rises.

17. (27, 4), (5, 23), m 5 23 2 4

} 5 2 (27)

5 2 7 } 12

The line falls.

18. (24, 26), (24, 4), m 5 4 2 (26)

} 24 2 (24)

5 10

} 0

No slope; the line is vertical.

19. 1 2 5 } 4 , 3 2 , 1 2 }

3 , 3 2 , m 5

3 2 3 }

2 }

3 2 1 2

5 } 4 2 5

0 }

23

} 12

5 0

The line is horizontal.

20.

1

x

y

21

21.

1

x

y

21

22.

1

x

y

22

23.

2

x

y

21

24.

1

x

y

21

25.

1

x

y

22

26.1

x

y

21

27.

1

x

y

21

28.

1

x

y

21

29.

1

x

y

21

30.

1

x

y

21

The relation is a function. The relation is not a function.

31. 4x 2 3y 5 32 4x 2 3y 5 32

22x 1 y 5 214 3 2 24x 1 2y 5 228

2y 5 4

y 5 24

4x 2 3(24) 5 32 → x 5 5


32. 5x 2 2y 5 24 3 3 15x 2 6y 5 212

3x 1 6y 5 36 3x 1 6y 5 36

18x 5 24

x 5 4 } 3

5 1 4 } 3 2 2 2y 5 24 → y 5

16 } 3

The solution is 1 4 } 3 ,

16 }

3 2 .

33. x 2 y 1 2z 5 24 x 2 y 1 2z 5 24

2x 1 3y 1 z 5 9 3 (22) 24x 2 6y 2 2z 5 218

23x 2 7y 5 222

3x 1 y 2 4z 5 26 3x 1 y 2 4z 5 26

2x 1 3y 1 z 5 9 3 4 8x 1 12y 1 4z 5 36

11x 1 13y 5 30

23x 2 7y 5 222 3 11 233x 2 77y 5 2242

11x 1 13y 5 30 3 3 33x 1 39y 5 90

238y 5 2152

y 5 4

23x 2 7(4) 5 222 → x 5 22

22 2 4 1 2z 5 24 → z 5 1


34. B 2 3A 5 F 3 21 5 2G 2 3F 22 6

1 4G 5 F 3 2 3(22) 21 2 3(6)

5 2 3(1) 2 2 3(4)G 5 F 9 219

2 210G


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191Algebra 2


35. 2(A 1 B) 2 C

5 2(F 22 6 1 4G 1 F 3 21

5 2G) 2 F 24 8

27 12G 5 2F 1 5

6 6G 2 F 24 8

27 12G 5 F 2(1) 2 (24) 2(5) 2 8

2(6) 2 (27) 2(6) 2 12G 5 F 6 2 19 0G

36. (C 2 A)B 5 (F 24 8 27 12G 2 F 22 6

1 4G)F 3 21 5 2G

5 F 22 2 28 8GF 3 21

5 2G 5 F 22(3) 1 2(5) 22(21) 1 2(2)

28(3) 1 8(5) 28(21) 1 8(2)G 5 F 4 6

16 24G37. (B 1 C)D

5 (F 3 21 5 2G 1 F 24 8

27 12G)F 1 0 24

22 3 21G 5 F 21 7

22 14GF 1 0 24

22 3 21G 5 F 21(1) 1 7(22) 21(0) 1 7(3) 21(24) 1 7(21)

22(1) 1 14(22) 22(0) 1 14(3) 22(24) 1 14(21)G 5 F 215 21 23

230 42 26G38. A21 5 2

1 } 1 F 3 24

24 5G 5 F 23 4

4 25G39. A21 5

1 } 3 F 24 29

3 6G 5 F 2 4 } 3 23

1 2G

40. A21 5 2 1 } 10 F 1 22

24 22G 5 F 2 1 } 10 1 } 5

2 } 5 1 } 5 G 41. A21 5

1 } 24 F 28 28

22 25G 5 F 2 1 } 3 2 1 } 3

2 1 } 12 2 5 } 24 G

42. Area 5 6 1 } 2

0 0 1 15 10 1

8 25 1 0 0

15 10

8 25

5 6 1 } 2 [(0 1 0 1 375) 2 (80 1 0 1 0)]

5 147.5

The area of the playground is 147.5 square yards.

43. a. W

} T 5 R2

} R2 1 A2

W 5 TR2

} R2 1 A2

b. W 5 162(949)2

}} (949)2 1 (768)2

W ø 97.9

The estimate shows that the Red Sox won about 98 games, which is about the same as the actual number of games won.

44. a. g 5 21 2 4t

b.

Time (hours)

Gas

olin

e (g

allo

ns)

0 2 4 6 8 t

g

0

6

12

18

24(0, 21)

(5.25, 0)

c. Domain: 0 ≤ t ≤ 5.25

Range: 0 ≤ g ≤ 21

45. c 5 kp

3900 5 k(78,000)

k 5 0.05

c 5 0.05p

When p 5 125,000: c 5 0.05(125,000)

c 5 6250

If a house sells for $125,000, the real estate agent’s commission will be $6250.


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46.

Years since 1994

Rec

ove

red

mat

eria

l(m

illio

ns

of

ton

s)

0 2 4 6 8 t

m

050

55

60

65

70

Sample answer:

Best-fitting line approximation: m 5 2.5t 1 52

When t 5 16: m 5 2.5(16) 1 52

m 5 92

In 2010, about 92 million tons of material willbe recovered.

47. s > 213, j ≤ 263, s 1 j > 472.5

0 100 200 300 4000

100

200

300

400

Snatch weight (kg)

Cle

an a

nd

jerk

wei

gh

t (k

g)

s

j


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Aat Solutions - Ch03

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