F G

Chapter 3,
continued
matrix to a 2 3 3 matrix.
2
13. } C 5
3
5
F
F
2
} (26)
3
2
} (22)
3
2
2
3
}3(1)
2
3
}(24)
4
24
2}3
2
3
2}3
}(21)
6
8
}
2
} (9)
3
2
2}3
G
G
3. AB 5
5
5
F
F
F
1 22
1(5) 1 (22)(22)
g 5 21
7
2
23
0
4
1
21
2
5 23
0
1.4e 1 1.1r 1 1.3g 5 25
25
2e 1 2r 1 2g 5 42
22e 1 r 2 2g 5 0
20.3(14) 2 0.1g 5 24.4 l g 5 2
25 213
5. AB 2 AC 5
2
e 1 14 1 2 5 21 l e 5 5
You should buy 5 pounds of Empire, 14 pounds of Red
Delicious, and 2 pounds of Golden Delicious apples.
2.
3.
F
F
F
8
8
G
32.24 17.68 23.12
22.08 14.56 28.32
G
26 11 25
21 11
7
18 20 28
5.
Hardcover
Paperback
F
7
23
4.
F G
G
R
M
S
C
44
36
38
21
76
44
22
50
12
3.6 Guided Practice (pp. 195–198)
1. AB is defined and has dimensions 5 3 2.
2. AB is not defined because the number of columns in A
does not equal the number of rows in B.
5
1
0
G
21
2
23
0
4
1
21
2
23
0
4
1
4(23) 1 1(21)
G F G)
G F G)
3
22 21
24
5
1
0
23(2) 1 0(21)
14(2) 1 1(21)
21(24) 1 2(1)
21(5) 1 2(0)
2 23(24) 1 0(1)
23(5) 1 0(0)
5
27
24
29
26
10
7
GF G
G
G
213
1
5 221
9
1
1
6. 2}(AB) 5 2}
2
2
5
F
4(5) 1 1(0)
6
G
G
25
12 215
2
20
215
25 213
27 24
29
26
10
7
1
2}2 (27)
1
2}2 (29)
1
2}2 (10)
G
2
4(3) 1 1(22)
38 239
211
24
23(23) 1 0(21)
4(24) 1 1(1)
G
2
21(2) 1 2(21)
22 225
Algebra 2
Worked-Out Solution Key
22 21
G F G)
G
21(3) 1 2(22)
Lesson 3.6
138
2
5 23(3) 1 0(22)
Graphing Calculator Activity 3.5 (p. 194)
1.
9
5 221
22e 1 r 2 2g 5 0
3r
5 42
r 5 14
19 25
3
4(7) 1 1(23)
213
1
20.3r 2 0.1g 5 24.4
32
G (F
GF
5 23(7) 1 0(23)
21.4e 2 1.4r 2 1.4g 5 229.4
e 1 r 1 g 5 21
F
F
F
F
(F
(F
F
F
F
F
F
9
21
3 (21.4)
1.4e 1 1.1r 1 1.3g 5
G
212 221
G
23 21
4 1
21(7) 1 2(23) 21(23) 1 2(21)
Equation 3: r 5 2(g 1 e), or 22e 1 r 2 2g 5 0
r1
23 22
1(1) 1 (22)(23)
Equation 2: 1.4e 1 1.1r 1 1.3g 5 25
e1
G
5
23(5) 1 3(22)
r 5 pounds of Red Delicious apples
g 5 pounds of Golden Delicious apples
Equation 1: e 1 r 1 g 5 21
1
23(1) 1 3(23)
4. A(B 2 C) 5
14. e 5 pounds of Empire apples
GF
3
23
1
2}2 (24)
1
2}2 (26)
1
2}2 (7)
GF G
5
}
7
2
2
9
2
3
}
7
25 2}2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
12. The sum 3A 1 C is not possible. You cannot add a 2 3 2
Chapter 3,
7.
F
14
30
16
5
GF
18
25 20
F
continued
G
75
1
45
15.
14(75) 1 30(1) 1 18(45)
16(75) 1 25(1) 1 20(45)
GF G
F
5
2
0
24
2125
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.
2. Sample answer: To find the element in the first row and
first column of AB, multiply each element in the first row
of A by the corresponding element in the first column of
B, then add the products.
3. AB is defined and has dimensions 2 3 2.
4. AB is defined and has dimensions 3 3 2.
equal the number of rows in B.
6. AB is defined and has dimensions 1 3 3.
5(7) 1 2(0)
0(7) 1 (24)(0)
1(3) 1 6(22)
1(7) 1 6(0)
equal the number of rows in B.
8. AB is defined and has dimensions 2 3 5.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
9. A; Matrix A has 2 rows and matrix B has 2 columns,
so AB has dimensions 2 3 2.
11.
F GF
1
FG
5
7
22 1
4
F
G5F 8G
F
GF G
5 3(5) 1 (21)(7)
G5
1(22) 1(1)
4(22) 4(1)
5
22
1
28
4
12. Not defined; The number of columns in the first matrix
does not equal the number of rows in the second matrix.
13.
F
9 23
0
5
F
GF
0
1
4 22
0(0) 1 2(4)
0(1) 1 2(22)
5
0
23
2
24
1
6
2
5
F
F
212
15
GF
5(23) 1 0(6)
10
18 26
0
29
7
G
F
GF
F
F G
F GF
1
17.
5
5
3
0
2 12
24
9
1
4
23
22
4
G
G
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)
21 28
74 250
F
F
2
5
21
4
0
3 27
1
5
23 10
24
G
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)
G
3(0) 1 (27)(23) 3(1) 1 (27)(10) 3(5) 1 (27)(24)
52 210
215
39 221
5 212
21 267
43
G
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.
G
21. B;
5
G
5
5(2) 1 0(2)
24(23) 1 1(6) 24(2) 1 1(2)
215
8
2(4) 1 5(3) 5 23
9(1) 1 (23)(22)
G
5
G
9(0) 1 (23)(4)
8 24
5
14.
F
F
2
G
does not equal the number of rows in the second matrix.
18.
7. AB is not defined. The number of columns in A does not
G
0
11 35
5
5. AB is not defined. The number of columns in A does not
21
22
16. Not defined; The number of columns in the first matrix
number of columns in A is equal to the number of rows
in B.
F3
6
5(3) 1 2(22)
5
1. The product of matrices A and B is defined provided the
10.
7
5 0(3) 1 (24)(22)
3.6 Exercises (pp. 199–202)
Skill Practice
3
F
F G
1
1890
5
GF G
G
F GF G
F
F G
1 24
4 21
3 22
0 23
1(4) 1 (24)(0)
1(21) 1 (24)(23)
3(4) 1 (22)(0)
3(21) 1 (22)(23)
4
11
12
3
G
Algebra 2
Worked-Out Solution Key
139
Chapter 3,
5
5
5
23
0
1
22
4
4
22
15
29
0
1
26
12
4
22
15(0) 1 (29)(4)
15(1) 1 (29)(22)
26(0) 1 12(4)
26(1) 1 12(22)
236
33
48
230
1
1
23. 2} AC 5 2}
2
2
5 23
26 3
4
4 1
22
1
5
1
5
5
25. AB 2 BA 5
140
3
4
21
23
1
4
7
0
22
3
4
21
F
G (F
GF
1 3
2
23 1
4
2 1 22
22
4
6
4
1
2
5
5
23
GF
G
1
2}2 (12)
1
21 26
5
214
5 23
0
1
4
4 22
5 23
26 3
4
4 1
22
22
1
5(0) 1 (23)(4)
5(1) 1 (23)(22)
22(0) 1 4(4)
22(1) 1 4(22)
5(26) 1 (23)(4)
5(3) 1 (23)(1)
22(26) 1 4(4)
22(3) 1 4(1)
11
212
16 210
1
242
4
22
0
1
0
4 22
4
5(0) 1 (23)(4)
5(1) 1 (23)(22)
22(0) 1 4(4)
22(1) 1 4(22)
0(5) 1 1(22)
0(23) 1 1(4)
4(5) 1 (22)(22)
4(23) 1 (22)(4)
212
11
16 210
210
7
28 10
Algebra 2
Worked-Out Solution Key
2
22
4
24
220
4
21
3(22) 1 4(4) 1 (21)(5)
3(4) 1 4(1) 1 (21)(5)
23(6) 1 1(2) 1 4(23)
F
(F
F
F
9
228
5 224 18
48
5 11
29
30
27. (D 1 E)D
5
5
5
1 3
2
23 1
4
2 1 22
G
G F G) F
GF G
23
1
4
1 3
2
7
0
22
23 1
4
3
4
21
1
22
4
6
1
3
2
4
1
2
23
1
4
5
5
23
2
1
22
F
2 1 22
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)
4(2) 1 1(4) 1 2(22)
5
5(2) 1 5(4) 1 (23)(22)
4
0
5 15
8
216 17
36
22
5 22
5
G
G
G
F GF G
F
F
GF
28. 22(BC) 5 22
G)
G
22(2) 1 4(4) 1 6(22)
4 22
22
3
23(4) 1 1(1) 1 4(5)
1
5 23
22
7(4) 1 0(1) 1 (22)(5)
44 212
5 23
4
0
23(22) 1 1(4) 1 4(5)
23
254
1
7
5 7(22) 1 0(4) 1 (22)(5)
12
28 22
23
1
3(6) 1 4(2) 1 (21)(23)
24. AB 1 AC 5
5
22
22(3) 1 4(1)
1
5
4
0
5(3) 1 (23)(1)
2}2 (28) 2}2 (22)
2
1
7
22(26) 1 4(4)
1
5
5
F
F
23
1 5(26) 1 (23)(4)
2}2 (242)
2
5
7(6) 1 0(2) 1 (22)(23)
5 2}2
5
26. E(D 1 E)
0
1
26
3
4
22
4
1
0(26) 1 1(4)
0(3) 1 1(1)
4(26) 1 (22)(4)
4(3) 1 (22)(1)
22(4)
22(1)
22(232) 22(10)
5
28
22
64
220
G
G
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
F GF G
F GF G
F
G
F G
F GF G
F
G
F
GF G
F GF G
F GF G
F
G
F
G
F GF G
F G
F GF G
F GF G
F
G
F
G
F GF G
F G
22. 3AB 5 3
5
continued
Chapter 3
F
F
F
F
F
F
F
F
29. 4AC 1 3AB 5 4
continued
GF G
GF G
5 23
4
22
5 23
1 3 22
4
54
13
5
1
5
30.
3
4
1
0
1
4
22
33. A2 5
5
5(26) 1 (23)(4)
5(3) 1 (23)(1)
(22)(26) 1 4(4)
(22)(3) 1 4(1)
G
G
5(0) 1 (23)(4)
5(1) 1 (23)(22)
(22)(0) 1 4(4)
(22)(1) 1 4(22)
4(242)
4(12)
4(28)
4(22)
G
G
3(212)
3(11)
3(16)
3(210)
112 1 48
28 1 (230)
160 238
G
A3 5
5
48 1 33
81
5
5
2168 2 36
2204
5
26
G
5
2
34. A 5
3(1) 1 2(x) 1 4(3) 5 19
2x 5 4
5
x52
0(1) 1 (22)(x) 1 4(3) 5 y
0(1) 1 (22)(2) 1 4(3) 5 y
A3 5
85y
The solution is x 5 2 and y 5 8.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
31. 22(9) 1 x(2) 1 1(21) 5 213
2x 5 6
5
x53
4(9) 1 1(2) 1 3(21) 5 y
35 5 y
5
The solution is x 5 3 and y 5 35.
32. A2 5
5
5
3
A 5
5
5
5
F
F
F
F
F
F
F
1 21
0
2
GF G
1 21
0
G
1(21) 1 (21)(2)
0(1) 1 2(0)
0(21) 1 2(2)
0
4
1 21
0
2
1 23
0
4
G
GF GF G
GF G
1 21
1 21
0
0
2
2
1(21) 1 (23)(2)
0(1) 1 4(0)
0(21) 1 4(2)
27
0
8
G
GF
G
(24)(1) 1 1(21)
2(1) 1 (21)(21)
G
GF GF
GF G
G
18 25
3
210
1
24
2 21
1
24
2 21
G
1
24
3
1
24
2 21
18 25
210
2 21
18(24) 1 (25)(2)
18(1) 1 (25)(21)
210(24) 1 3(2)
210(1) 1 3(21)
23
282
F
F
46 213
G
2
0 21
1
3
2
22 21
0
6
1 22
1
7
5
25 23
GF
GF
G
0
0 21
1
3
2
22 21
0
6
1 22
1
7
5
25 23
0
17
5
21
16
13
213 29
21
2
4
6
5
0 21
1
3
2
22 21
0
GF
G
G
2
0 21
2
0 21
1
3
2
1
3
2
22 21
0
22 21
0
2
0 21
1
3
2
22 21
0
F G
GF G)
G
B5
9 ,
G
a
b
c
d
1
0
0
1
e
f
g
h
ae 1 bg af 1 bh
ce 1 dg
cf 1 dh
kae 1 kbg kaf 1 kbh
kce 1 kdg kcf 1 kdh
(kA)B 5 k
5
GF
G
2
G
24
F G
(F
F
F
(F
F
F
5
G
2 21
2(24) 1 (21)(2)
2
A5
1
24
(24)(24) 1 1(2)
5k
2
1(1) 1 (23)(0)
1
2 21
36. k(AB) 5 k
1 21
0
1
24
35. Sample answer:
2
1(1) 1 (21)(0)
1 23
F
F
F
F
F
F
F
F
F
F
a
c
b
d
ka
kb
kc
kd
G) F G
GF G
e
g
G
f
h
e
f
g
h
kae 1 kbg kaf 1 kbh
kce 1 kdg kcf 1 kdh
G
Algebra 2
Worked-Out Solution Key
141
Chapter 3,
5
5
F G ( F G)
F GF G
F
G
a
b
e
f
c
d
k g
h
a
b
ke
kf
c
d
kg
kh
F
kae 1 kbg kaf 1 kbh
Problem Solving
Equipment
F
G
Bats
(dollars)
21
Balls
4
Total cost 5
F
45 15
G
FG
F
Class 2
F
Cost
20
14
FG
15
F
17
20 14
15
GF
3.35
1.75
4.50
G
24(3.35) 1 12(1.75) 1 17(4.50)
29
17
14
F
27
38
Students
Adults
GF G
5
177.9
159.00
Seniors
42.
G
120
150
40
Ted
192
215
54
4
Algebra 2
Worked-Out Solution Key
32(3) 1 17(2) 1 14(1)
27(3) 1 27(2) 1 38(1)
GFG
212
5 144
173
equal
825 1050
G
F G
21 16
40 33
5
This matrix shows that dealer A made a profit of $62,400
and dealer B made a profit of $57,575.
Jean
5
1
F650(21)1825(40)11050(15) 650(16)1825(33)11050(19) G
5 F62,400 57,575 G
Seniors
FG
2
15 19
Adults
2
GF G
3
35(3) 1 39(2) 1 29(1)
F
Students
Ticket cost (dollars)
142
39
Attendance
F
1
PS 5 650
The supplies for class 1 cost $177.90 and the supplies for
class 2 cost $159.00.
Saturday
FG
The matrix PS is defined.
20(3.35) 1 14(1.75) 1 15(4.50)
Friday
14
38
not equal
24 12
39.
17
27
41. SP: (3 3 2)(1 3 3) PS: (1 3 3)(3 3 2)
4.50
Total cost 5
32
27
USA scored 212 points, China scored 144 points, and
Russia scored 173 points.
1.75
Canvases
29
35
3.35
Brushes
F
17
39
32
27
G
35
2
5
(dollars)
Paint
5
G
12
1150
1675
Bronze
Silver
F
G 5 F 882 G
24
5
Silver
Points
3
Bronze
Canvases
GF G
Gold
Gold
Art Supplies
Class 1
F
Russia
4
Brushes
4
192(2) 1 215(5) 1 54(4)
China
21
Paint
5
120(2) 1 150(5) 1 40(4)
USA
The total cost of equipment for the softball team is $882.
38.
F
GF G
40. Medals Won
30
5 12(21) 1 45(4) 1 15(30)
54
215
30
Uniforms
12
192
2
The income from Friday night’s play was $1150 and the
income from Saturday night’s play was $1675.
FG
Cost
Bats Balls Uniforms
12 45
15
40
5
kce 1 kdg kcf 1 kdh
All of the matrices are equal, so k(AB) 5 (kA)B 5 A(kB)
37.
120 150
Pat
Al
Matt
F
Grades (G)
G
Homework
Quizzes
Tests
82
88
86
92
88
90
82
73
81
74
75
78
88
92
90
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
A(kB) 5
continued
Chapter 3,
continued
FG
b.
Weights (W)
Homework
Quizzes
0.3
F
Tests
0.5
82(0.2) 1 88(0.3) 1 86(0.5)
92(0.2) 1 88(0.3) 1 90(0.5)
GW 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)
Year 1
GF G
Year 2
Year 3
85.8
89.8
5 78.8
c. SC 5
76.3
5
F
F
F
F
12p
q
p
12q
0.8
0.05
0.2
0.95
b. M1 5
GF
5
G
GF G
0.8
0.05
0.2
0.95
G
5
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5
M3 5
5
M4 5
5
F
F
F
F
F
F
0.8
0.05
0.95
GF G
4400
8600
0.2(4400) 1 0.95(8600)
0.8
0.05
0.2
0.95
GF G
0.2(3950) 1 0.95(9050)
0.05
0.2
0.95
GF G
Mascot
30
20 300 100
50
GF G
10
15
20
10(10) 1 100(15) 1 50(20) 1 30(20)
G
F
GF G
F
G
F G
20 100
15
0
50
30
20 300 100
50
20
25
30
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)
GF G
5
2900
4300
GF G
SP represents the total price received for the scarves
each year.
GF G
5
3612.5
9387.5
3612.5
9387.5
GF
3359.375
5
9640.625
FG FG
Cost (C)
School name
0
50
SP 5 10 100
F GF G
F GF G
2900
d. SP 2 SC 5
G
M2, M3, and M4 represent the number of commuters
after 2, 3, and 4 years, respectively.
Class year
50
10 100
0
3950
5
9050
0.2(3612.5) 1 0.95(9387.5)
Plain
100
2300
4400
5
8600
0.8(3612.5) 1 0.05(9387.5)
44. a.
300
10,300
3950
9050
0.8(3950) 1 0.05(9050)
0.8
20
5 3200
5
0.8(4400) 1 0.05(8600)
30
SC represents the cost of making the scarves
each year.
5000
8000
0.2(5000) 1 0.95(8000)
0.2
0
50
20(10) 1 300(15) 1 100(20) 1 50(20)
This matrix represents the number of commuters who
will drive and use public transportation after one year.
c. M2 5
100
7700
0.8(5000) 1 0.05(8000)
5
0.05
0.2 1 2 0.05
20
100
20
0(10) 1 20(15) 1 100(20) 1 0(20)
90.2
1 2 0.2
0
20 100
G
Mascot
10
F
F
F G
0
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
Sales (S)
Plain Class year School name
0.2
5
2300
4300
2 3200
10,300
7700
2900 2 2300
4300 2 3200
10,300 2 7700
5
600
1100
2600
This matrix represents the profit that is made
each year.
Price (P)
10
Plain
15
15
Class year
20
20
School name
25
20
Mascot
30
Algebra 2
Worked-Out Solution Key
143
Chapter 3,
45. a. AB 5
5
continued
F GF
F
G
0 21
1
27 24
0
4
24
8
2
G
Lesson 3.7
3.7 Guided Practice (pp. 204–207)
3 2 5 3(1) 2 6(22) 5 15
1.
6 1
{
24 28 22
27 24 24
1
2.
y
x
21
{
4 21
2
4 21
23 22 21 23 22
0
5
1
0
5
{
{
5 (28 1 0 2 30) 2 (0 2 20 1 3)
5 238 2 (217) 5 221
(22, 24)
3.
(24, 27)
b. For the triangle rotated 1808:
5
F GF GF
F GF
G
F
G
0 21
0 21
1
1
0
0
21
0
4
27 24
0 21
7
5
27 24
4
4
24
8
2
G
5 198 1 272 5 470
1
4. Area 5 6}
2
2
4
1
The area of the triangle is 34 square units.
F GF GF GF
G
F GF GF
G
F GF
G
F G
5
5
0 21
0 21
1
1
0
0
21
0 21
0 21
0
21
1
0
0
1
27 24
4
4
8
2
7
4
4
27 24
0
4
27 24
0
4
8
8
24
2
The value of the book at 5 years was $6, which is twice
the value at 2 years ($3).
47. F;
height of large triangle
base of large triangle
}} 5 }}
2
5
x12
30
}5}
5(x 1 2) 5 2 + 30
5x 5 50
x 5 10
The distance across the river is 10 meters.
144
Algebra 2
Worked-Out Solution Key
23
23
3 215
{ 2 13 { 5 }
39 2 (230)
y 5 }
53
2
46. C;
3 24
215 24
2
Mixed Review for TAKS
{ 2 5 { 5 15 2 (28) 5 23
{ 13 5 { 5 }}
275 2 (252)
5 21
x 5 }
24
The vertices of this triangle are (4, 7), (8, 4),
and (2, 4).
height of small triangle
base of small triangle
5.
24
8
{
5 11
9 2
1 3
5 6}2 (268) 5 34
For the triangle rotated 2708:
1
1
1
1
1
24 28 22
0 21
{
5 11
9 2
1 3
5 6}2 [(10 1 11 1 27) 2 (2 1 15 1 99)]
The vertices of this triangle are (7, 24), (4, 28),
and (4, 22).
5
{
5 (240 1 0 2 42) 2 (0 2 280 1 8)
24
8
{
10 22
3 10 22
2 212
4 2 212
0 27 22 0 27
23
23
The solution is (21, 3).
6.
4
7
{23 22{ 5 28 2 (221) 5 13
2
7
28 22
24 2 (256)
x 5 }
5}
54
13
{
13
4
2
23 28
{
{
{
232 2 (26)
y 5 }
5}
5 22
13
13
The solution is (4, 22).
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
(28, 24)