Ch. 8 Systems of Equations and Inequalities
8.1 Systems of Linear Equations: Substitution and Eliminatio
n
1 Solve Systems of Equations by Substitution
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Verify that the values of the variables listed are solutions of the system of equations.
1)
x + y = 7
x – y = 5
x = 6, y = 1
A) solution B) not a solution
2)
x + y = –2
x – y = 10
x = –4, y = –6
A) solution B) not a solution
3)
2x + y = –3
3x + 2y = –3
x = –3, y = 3
A) solution B) not a solution
4)
3x + y = –6
2x + 3y = 10
x = –4, y = –6
A) solution B) not a solution
Solve the system of equations by substitution.
5)
x + y = –6
x – y = 19
A) x = 6.5
,
y = –12.5; (6.5
,
–12.5) B) x =6.5
,
y =12.5; (6.5
,
12.5)
C) x = 6
,
y = –12.5; (6
,
–12.5) D) x =6
,
y =6.5; (6
,
6.5)
6)
x + 7y =-
2
3x + y =34
A) x = 12, y = –2; (12, –2) B) x = –2, y =3; (–2, 3)
C) x = 3, y = 7; (3, 7) D) x =7, y =12; (7, 12)
7)
x + 2y =2
8x – 7y =-
7
A) x = 0, y = 1; (0, 1) B) x = 1, y =0; (1, 0) C) x =1, y =1; (1, 1) D) x =0, y =0; (0, 0)
Page 1
8)
5x – 2y = –1
x + 4y = 35
A) x = 3, y = 8; (3, 8) B) x = 3, y =9; (3, 9) C) x =2, y =8; (2, 8) D) x =2, y =9; (2, 9)
9)
4x + 6y =0
2x =-
6
A) x = –3
,
y = 2; (–3
,
2) B) x =2
,
y = –3; (2
,
–3)
C) x = –3
,
y = –2; (–3
,
–2) D) x = –2
,
y =0; (–3
,
0)
10)
5x + 3y = 80
2x + y = 30
A) x = 10, y = 10; (10, 10) B) x =0, y =10; (0, 10)
C) x = 10, y = 0; (10, 0) D) x =0, y =0; (0, 10)
11)
5x + y = 0
–5x + y = –10
A) x = 1, y = –5; (1, –5) B) x = –1, y =5; (–1, 5)
C) x = 1, y = 10; (1, 10) D) x = –1, y = –5; (–1, –5)
12)
x + y = 0
2x + 3y = –7
A) x = 7, y = –7; (7, –7) B) x = –7, y =7; (–7, 7)
C) x = 6, y = –6; (6, –6) D) x = –6, y =6; (–6, 6)
13)
3x + y =13
2x – 7y =24
A) x = 5, y = –2; (5, –2) B) x = –5, y =2; (–5, 2)
C) x = 5, y = 2; (5, 2) D) x = –5, y = –2; (–5, –2)
14)
1
5x – y = – 4
5
x + 8y = –4
A) x = –4, y = 0; (–4, 0) B) x =4, y =0; (4, 0)
C) x = 0, y = –4; (0, –4) D) x =0, y =4; (0, 4)
15)
1
2x + 2
3y =32
1
4x – 5
9y =40
A) x = 100, y = –27; (100, –27) B) x =100, y =27; (100, 27)
C) x = –100, y = –27; (–100, –27) D) x = –100, y =27; (–100, 27)
Page 2
16)
2x + 9y = –8
x + 1
3y = 13
3
A) x = 5, y = –2; (5, –2) B) x = –5, y =2; (–5, 2)
C) x = 5, y = 2; (5, 2) D) x = –5, y = –2; ( –5, –2)
2 Solve Systems of Equations by Elimination
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations by elimination.
1)
4x + 56y =56
5x – 8y =-
8
A) x = 0, y = 1; (0, 1) B) x = 1, y =0; (1, 0) C) x =1, y =1; (1, 1) D) x =0, y =0; (0, 0)
2)
5x – 2y = –1
x + 4y = 35
A) x = 3, y = 8; (3, 8) B) x = 3, y =9; (3, 9) C) x =2, y =8; (2, 8) D) x =2, y =9; (2, 9)
3)
x + y =-
7
x – y =12
A) x = 2.5
,
y = –9.5; (2.5
,
–9.5) B) x =2.5
,
y =9.5; (2.5
,
9.5)
C) x = 7
,
y = –9.5; (7
,
–9.5) D) x =7
,
y =2.5; (7
,
2.5)
4)
3x + 4y = 42
3x + 2y = 48
A) x = 18
,
y = –3; (18
,
–3) B) x = –18
,
y =3; ( –18
,
3)
C) x = –18
,
y = 4; (–18
,
4) D) x = –3
,
y =18; (–3
,
18)
5) 6x + 3y =51
2x – 6y =38
A) x = 10, y = –3; (10, –3) B) x = –3, y =10; (–3, 10)
C) x = –10, y = 3; (–10, 3) D) x =3, y = –10; (3, –10)
6) 3x – 5y =-
12
6x + 8y =-
24
A) x = –4, y = 0; (–4, 0) B) x =4, y =0; (4, 0)
C) x = 0, y = –4; (0, –4) D) x =0, y =4; (0, 4)
7)
2x + 20y =-
144
11x + 5y =48
A) x = 8
,
y = –8; (8
,
–8) B) x = –8
,
y =8; (–8
,
8)
C) x = 11
,
y = –11; (11
,
–11) D) x = –5
,
y =8; (–5
,
8)
Page 3
8)
5x + 3y = 80
2x + y = 30
A) x = 10, y = 10; (10, 10) B) x =0, y =10; (0, 10)
C) x = 10, y = 0; (10, 0) D) x =0, y =0; (0, 0)
9)
7
12x – y =10
5
9x + 2y =11
A) x = 18, y = 1
2; 18, 1
2B) x = 16, y = 1
2; 16, 1
2
C) x = –18, y = – 1
2; –18, – 1
2D) x = –16, y = – 1
2; –16, – 1
2
10)
2
5x + 7
10y = 18
5
4x + 2y = 56
A) x = 16
,
y = –4; (16
,
–4) B) x = –16
,
y =4; (–16
,
4)
C) x = –16
,
y = 7; (–16
,
7) D) x = –4
,
y =16; (–4
,
16)
Solve the system of equations. [Hint: Let u = 1
x and v = 1
y, and solve for u and v. Then let x = 1
u, and y = 1
v.]
11)
2
x + 4
y =7
1
x – 2
y =4
A) x = 4
15, y = –8; 4
15, –8 B) x = 15
4, y = – 1
8; 15
4, – 1
8
C) x = –8, y = 4
15; –8, 4
15 D) x = – 1
8, y = 15
4; – 1
8, 15
4
Solve the problem.
12) A flat rectangular piece of aluminum has a perimeter of 50 inches. The length is 9 inches longer than the
width. Find the width.
A) 8 in. B) 17 in. C) 26 in. D) 25 in.
13) The Family Fine Arts Center charges $25 per adult and $14 per senior citizen for its performances. On a
recent weekend evening when 526 people paid admission, the total receipts were $9333. How many who
paid were senior citizens?
A) 347 senior citizens B) 179 senior citizens C) 257 senior citizens D) 269 senior citizens
14) A retired couple has $210,000 to invest to obtain annual income. They want some of it invested in safe
Certificates of Deposit yielding 6%. The rest they want to invest in AA bonds yielding 12% per year. How
much should they invest in each to realize exactly $21,600 per year?
A) $150,000 at 12% and $60,000 at 6% B) $150,000 at 6% and $60,000 at 12%
C) $140,000 at 6% and $70,000 at 12% D) $160,000 at 12% and $50,000 at 6%
Page 4
15) A tour group split into two groups when waiting in line for food at a fast food counter. The first group
bought 8 slices of pizza and 7 soft drinks for $34.51. The second group bought 7 slices of pizza and 7 soft
drinks for $31.36. How much does one slice of pizza cost?
A) $3.15 per slice of pizza B) $1.33 per slice of pizza
C) $2.65 per slice of pizza D) $1.83 per slice of pizza
16) A movie theater charges $8.00 for adults and $5.00 for children. If there were 40 people altogether and the
theater collected $272.00 at the end of the day, how many of them were adults?
A) 24 adults B) 10 adults C) 16 adults D) 29 adults
17) An 8–cylinder Crown Victoria gives 18 miles per gallon in city driving and 21 miles per gallon in highway
driving. A 300–mile trip required 15.5 gallons of gasoline. How many whole miles were driven in the city?
A) 153 mi B) 147 mi C) 132 mi D) 168 mi
18) The Paperback Trader is a book store that takes in used paperbacks for 20% of their cover price and sells
them for 50% of their cover price. Pat brings in a stack of 15 paperback books to trade and gets $15.65
credit. Some of the books had a cover price of $5.95, the rest $4.95. She wants to get some Tom Clancy
books having a cover price of $5.95. How many $5.95 books did she bring in and how many Clancy books
can she get without paying any additional cash?
A) 4 $5.95 books, 5 Clancy books B) 11 $5.95 books, 2 Clancy books
C) 4 $5.95 books, 6 Clancy books D) 4 $5.95 books, 2 Clancy books
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
19) The perimeter of a parking lot is 500 yards. Find the dimensions of the lot if the length is 50 yards more
than three times the width.
20) A store has a sale on workout gear. Mark bought three pairs of shorts and three T–shirts for $70.35 (before
tax). Later, he went back and bought two more pairs of shorts and four more T–shirts for $63.90 (before
tax). How much did the shorts and T–shirts cost?
21) A tea shop owner is mixing a blend of two teas, one of which costs $6.50 per pound, the other costing $4.00
per pound. The owner wants to have 20 pounds of a mixture that will sell for $5.50 per pound. How much
of each type of tea should be used?
22)
J
aya has $16,000 to invest. She invests part of it in an account paying 6% simple interest and the rest in an
account paying 7% simple interest. If her annual income is $1080, how much does she have invested in
each account?
23) A boat on a river goes 77 miles downstream in 2 hours and 45 minutes. The return trip takes 3 hours and
30 minutes. If the boat’s speed is the same in each direction, find the speed of the boat and the speed of the
current.
3 Identify Inconsistent Systems of Equations Containing Two Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations. If the system has no solution, say that it is inconsistent.
1)
x + y = –6
x + y = –4
A) x = 0, y = 0; (0, 0) B) x = –6
,
y = –4; (–6
,
–4)
C) x = 0, y = –10; (0, –10) D) inconsisten
t
Page 5
2)
x – 4y = –10
2x – 8y = –17
A) x = 2, y = 4; (2, 4) B) x = 2, y =3; (2, 3) C) x =4, y =2; (4, 2) D) inconsisten
t
3)
3x – 7y = –2
3x – 7y = 8
A) x = 3
,
y = 7; (3
,
7) B) x = –2
,
y =8; (–2
,
8)
C) x = –2
7, y = 8
3; –2
7, 8
3D) inconsisten
t
4)
9x – 6y = 4
18x – 12y = 16
A) x = 2
,
y = 4; (2
,
4) B) x = 20
27, y = – 10
9; 20
27, 10
9
C) x = 4
,
y = 16; (4
,
16) D) inconsisten
t
5)
9x – 3y = 5
–18x + 6y = –15
A) x = 9
,
y = 5; (9
,
5) B) x = 9
10, y = – 3
10; ( 9
10, – 3
10)
C) x = 2
,
y = 3; (2
,
3) D) inconsisten
t
4 Express the Solution of a System of Dependent Equations Containing Two Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations. If the system has no solution, say that it is inconsistent.
1)
x + 3y = –8
2x + 6y = –16
A) y = – x
3 – 8
3, where x is any real number
or {(x, y) | y = – x
3 – 8
3, where x is any real number}
B) x = 0, y = 0; (0, 0)
C) x = –8
,
y = 0; (–8
,
0)
D) inconsisten
t
Page 6
2)
x
2 + y
3 = 4
x
4 + y
6 = 2
A) y = – 3
2x + 12, where x is any real number
or {(x, y) | y = – 3
2x + 12, where x is any real number}
B) x = –1, y = 15
2; –1, 15
2
C) x = 0, y = 12; (0, 12)
D) inconsisten
t
3)
3x +y= 4
–9x –3y =-
12
A) y = –3x + 4
,
where x is any real number
or {(x, y) | y = –3x + 4, where x is any real number}
B) x = –3y + 4
,
where y is any real number
or {(x, y) |x = –3y + 4, where y is any real number}
C) y = 3x + 4
,
where x is any real number
or {(x, y) | y = 3x + 4, where x is any real number}
D) inconsisten
t
5 Solve Systems of Three Equations Containing Three Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Verify that the values of the variables listed are solutions of the system of equations.
1)
x + y + z = 3
x – y + 4z = –1
5x + y + z = –13
x = 2, y = 5, z = –4
A) solution B) not a solution
2)
x – y + 2z = 5
3x + z = 5
x + 5y + z = 30
x = 0, y = 5, z = 5
A) solution B) not a solution
Page 7
Solve the system of equations.
3)
x + 4y + 4z =-
3
5y + 5z =0
z =-
1
A) x = –3
,
y = 1
,
z = –1; (–3
,
1
,
–1) B) x = –3
,
y = –1
,
z = 1; (–3
,
–1
,
1)
C) x = –1
,
y = 1
,
z = –3; (–1
,
1
,
–3) D) inconsisten
t
4)
x – y + 2z =9
3x + z =3
x + 3y + z =-
6
A) x = 0
,
y = –3
,
z = 3; (0
,
–3
,
3) B) x =3
,
y =0
,
z = –3; (3
,
0
,
–3)
C) x = 3
,
y = –3
,
z = 0; (3
,
–3
,
0) D) inconsisten
t
5)
x – y + 3z =3
3x + z =0
x + 3y + z =-
9
A) x = 0
,
y = –3
,
z = 0; (0
,
–3
,
0) B) x =0
,
y =0
,
z = –3; (0
,
0
,
–3)
C) x = 0
,
y = –3
,
z = 3; (0
,
–3
,
3) D) inconsisten
t
6)
2x + 4y + z = –3
3x – 4y – z = 28
2x + y + 4z = 18
A) x = 5
,
y = –4
,
z = 3; (5
,
–4
,
3) B) x =5
,
y =3
,
z = –4; (5
,
3
,
–4)
C) x = 3
,
y = –4
,
z = 5; (3
,
–4
,
5) D) inconsisten
t
7)
7x + 7y + z =1
x + 8y + 8z =8
9x + y + 9z =9
A) x = 0, y = 0, z = 1; (0, 0, 1) B) x =1, y = –1, z =1; (1, –1, 1)
C) x = 0, y = 1, z = 0; (0, 1, 0) D) x = –1, y =1, z =1; (–1, 1, 1)
8)
x + y + z =-
8
x – y + 3z =-
14
2x + y + z =-
9
A) x = –1
,
y = –2
,
z = –5; (–1
,
–2
,
–5) B) x = –5
,
y = –1
,
z = –2; (–5
,
–1
,
–2)
C) x = –5
,
y = –2
,
z = –1; (–5
,
–2
,
–1) D) inconsisten
t
9)
x – y + z = –3
x + y + z = –9
x + y – z = –7
A) x = –5
,
y = –3
,
z = –1; (–5
,
–3
,
–1) B) x = –5
,
y = –1
,
z = –3; (–5
,
–1
,
–3)
C) x = –3
,
y = –1
,
z = –5; (–3
,
–1
,
–5) D) inconsisten
t
Page 8
10)
x + y + z =-
2
x – y + 4z =-
13
3x + y + z =8
A) x = 5
,
y = –2
,
z = –5; (5
,
–2
,
–5) B) x = –5
,
y =5
,
z = –2; (–5
,
5
,
–2)
C) x = –5
,
y = –2
,
z = 5; (–5
,
–2
,
5) D) inconsisten
t
11)
3x + 2y + z =19
5x – 3y – z =-
8
3x + y + 2z =17
A) x = 2
,
y = 5
,
z = 3; (2
,
5
,
3) B) x =2
,
y =3
,
z =5; (2
,
3
,
5)
C) x = 3
,
y = 5
,
z = 2; (3
,
5
,
2) D) inconsisten
t
12)
x + y + z = 7
x – y + 2z = 7
5x + y + z = 11
A) x = 1, y = 2, z = 4; (1, 2, 4) B) x =1, y =4, z =2; (1, 4, 2)
C) x = 4, y = 2, z = 1; (4, 2, 1) D) x =4, y =1, z =2; (4, 1, 2)
13)
x – y + z = 8
x + y + z = 6
x + y – z = –12
A) x = –2, y = –1, z = 9; (–2, –1, 9) B) x =2, y = –1, z =9; (2, –1, 9)
C) x = –2, y = –1, z = –9; (–2, –1, –9) D) x =2, y = –1, z = –9; (2, –1, –9)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
14) Find real numbers a, b, and c such that the graph of the function y = ax2 + bx + c contains the points (1, 1),
(2, 4), and (–3, 29).
15) Find real numbers a, b, and c such that the graph of the function y = ax2 + bx + c contains the points (1, 2),
(2, 11), and (–3, –14).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
16) The Family Arts Center charges $23 for adults, $18 for senior citizens, and $8 for children under 12 for
their live performances on Sunday afternoon. This past Sunday, the paid revenue was $12,835 for 800
tickets sold. There were 40 more children than adults. How many children attended?
A) 273 children B) 233 children C) 294 children D) 263 children
17) Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake
to 157 g protein, 137 g fat, and 177 g carbohydrate. According to the health conscious hostess, the
marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g
protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g
carbohydrate. How many of each snack can he eat to obtain his goal?
A) 7 mushrooms, 6 meatballs, 4 eggs B) 6 mushrooms, 4 meatballs, 7 eggs
C) 4 mushrooms, 7 meatballs, 6 eggs D) 8 mushrooms, 7 meatballs, 5 eggs
Page 9
18) A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to
prepare, 2 hours to paint, and 9 hours to fire. A tree takes 15 hours to prepare, 3 hours to paint, and 4
hours to fire. A sleigh takes 4 hours to prepare, 17 hours to paint, and 7 hours to fire. If the workshop has
111 hours for preparation time, 82 hours for painting, and 113 hours for firing, how many of each can be
made?
A) 8 wreaths, 5 trees, 3 sleighs B) 5 wreaths, 3 trees, 8 sleighs
C) 3 wreaths, 8 trees, 5 sleighs D) 9 wreaths, 6 trees, 4 sleighs
19) Three shrimp boats supply the shrimp wholesalers on Hilton Head with fresh catch. The Annabelle takes
50% of its catch to Hudson’s, 20% to Captain J’s, and 30% to Mainstreet. The Curly Q takes 40% of its catch
to Hudson’s, 40% to Captain J’s, and 20% to Mainstreet. The SloJoe takes 30% of its catch to Hudson’s, 40%
to Captain J’s, and 30% to Mainstreet. One week Hudson’s received 238.4 pounds of shrimp, Captain J’s
received 208.2 pounds, and Mainstreet received 158.4 pounds. How many pounds of shrimp did each boat
catch?
A) Annabelle 169 lbs, Curly Q 231 lbs, SloJoe 205 lbs
B) Annabelle 231 lbs, Curly Q 205 lbs, SloJoe 169 lbs
C) Annabelle 205 lbs, Curly Q 231 lbs, SloJoe 169 lbs
D) Annabelle 205 lbs, Curly Q 169 lbs, SloJoe 231 lbs
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
20) To raise money, a group of three friends is each trying to sell 100 candles. Each friend has a package
containing the same number of tapers, votives, and tealights. The tapers sell for $6.00 each, the votives sell
for $1.50 each, and the tealights sell for $0.75 each. Maya sold all of the candles to raise $431.25, Lorrin sold
all of the tapers and tealights to raise $378.75, while Sara sold all the votives, all the tealights, and half of
the tapers to raise $251.25. How many of each type of candle did each package contain?
21) Meisha has $25,000 that she wants to invest. She invests it in accounts paying 12%, 7%, and 6% simple
interest. The account paying 12% is a higher–risk account, so she wants the amount in that account to be
half of the amount she has in the account paying 6% simple interest. If her annual interest is $1945, how
much is invested at each rate?
22) Lexie wants to have an income of $9000 per year from investments. To that end she is going to invest
$90,000 in three different accounts. These accounts pay 7%, 10%, and 14% simple interest. If she wants to
have $10,000 more in the account paying 7% simple interest than she has in the account paying 14% simple
interest, how much should go into each account?
23) Craig, Jenni, and Jade go to a store having a sale on workout gear. Craig buys three pairs of socks, two
T–shirts, and one pair of shorts for $35.90. Jenni buys four pairs of socks, three T–shirts, and three pairs of
shorts for $71.35. Jade buys one pair of socks, four T–shirts, and 2 pairs of shorts for $59.30. What is the
price of each item?
24) A health shop owner made trail mix containing dried fruit, nuts, and carob chips. The dried fruit sells for
$5.50 per pound, the nuts sell for $7.50 per pound, and the carob chips sell for $8.50 per pound. The shop
owner mixed 50 pounds of the trail mix and sells it for $6.70 per pound. If the amount of nuts is five
pounds more than the amount of carob ships, how much of each item was used for the trail mix?
Page 10
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
25) An application of Kirchoff’s Rules to the circuit shown results in the following system of equations:
I2 + I3 =I1
20 + 40 – 10I1 – 10I2=0
40 – 10I1 – 5I3=0
Find the currents I1, I2, and I3.
20 V 40 V
8 Ω
5 Ω3 Ω
2 Ω7 Ω
A) I1 = 3.5, I2 = 2.5, and I3 = 1B)I
1=5.5, I2=4.5, and I3 = 1
C) I1 = 4.5, I2 = 3.5, and I3 = 1D)I
1=2.5, I2=1.5, and I3 = 1
6 Identify Inconsistent Systems of Equations Containing Three Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations. If the system has no solution, say that it is inconsistent.
1)
x
+ y + z =-
2
x
– y + 5z =-
20
2x + 2y + 2z =-
7
A) x = –5
,
y = 5
,
z = –2; (–5
,
5
,
–2) B) x = –2
,
y =5
,
z = –5; (–2
,
5
,
–5)
C) x = –2
,
y = –5
,
z = 5; (–2
,
–5
,
5) D) inconsisten
t
2)
x – y + 4z =3
2x + z =0
–x + y – 4z =-
6
A) x = 0
,
y = –3
,
z = 0; (0
,
–3
,
0) B) x =4
,
y = –3
,
z =0; (4
,
–3
,
0)
C) x = 0
,
y = 0
,
z = –3; (0
,
0
,
–3) D) inconsisten
t
Page 11
7 Express the Solutions of a System of Dependent Equations Containing Three Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations.
1)
x + 4y – z =3
x + 5y – 2z =5
3x + 12y – 3z =9
A) x = –3z – 5, and y = z + 2, where z is any real number
or {(x, y, z) |x = –3z – 5, and y = z + 2, where z is any real number}
B) x = z – 2, and y = –3z – 5, where z is any real number
or {(x, y, z) |x = z – 2, and y = –3z – 5, where z is any real number}
C) x = 3z + 5, and y = z – 2, where z is any real number
or {(x, y, z) |x = 3z + 5, and y = z – 2, where z is any real number}
D) inconsisten
t
2)
–x + y + 2z =0
x + 2y + z =6
–2x – y + z =-
6
A) x = z + 2, and y = 2 – z, where z is any real number
or {(x, y, z) |x = z + 2, and y = 2 – z, where z is any real number}
B) x = 2 – z, and y = z + 2, where z is any real number
or {(x, y, z) |x = 2 – z, and y = z + 2, where z is any real number}
C) x = z + 2, and y = z – 2, where z is any real number
or {(x, y, z) |x = z + 2, and y = z – 2, where z is any real number}
D) inconsisten
t
3)
2x – y + 5z =-
7
x + y – 2z =-
2
x – y + 4z =-
4
A) x = –3 – z, and y = 3z + 1, where z is any real number
or {(x, y, z) |x = –3 – z, and y = 3z + 1, where z is any real number}
B) x = 3z + 1, and y = z – 3, where z is any real number
or {(x, y, z) |x = 3z + 1, and y = z – 3, where z is any real number}
C) x = z + 3, and y = 3z + 1, where z is any real number
or {(x, y, z) |x = z + 3, and y = 3z + 1, where z is any real number}
D) inconsisten
t
Page 12
4)
2x + 4y – 2z = 0
3x + 5y = 1
A) x = 2 – 5z, and y = 3z – 1, where z is any real number
or {(x, y, z) |x = 2 – 5z, and y = 3z – 1, where z is any real number}
B) x = 2 + 7z, and y = –3z – 1, where z is any real number
or {(x, y, z) |x = 2 + 7z, and y = –3z – 1, where z is any real number}
C) x = –2 – 7z, and y = –3z –1, where z is any real number
or {(x, y, z) |x = –2 – 7z, and y = –3z –1, where z is any real number}
D) x = 1 – 5z, and y = 3z – 1
2, where z is any real number
or (x, y, z) |x = 1 – 5z, and y = 3z – 1
2, where z is any real number
8.2 Systems of Linear Equations: Matrices
1 Write the Augmented Matrix of a System of Linear Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the augmented matrix for the system.
1) 6x + 4y =-
8
–2x + 5y =9
A) 64–8
–25 9 B) –846
9–25 C) 6–2–8
459 D) 649
5–2–8
2) 5x + 5y =15
9y =36
A) 5515
0936 B) 5515
936 0 C) 9036
55 5 D) 15 5 5
36 0 9
3)
6x – y =0
–6x + y – 12 = 0
A)
6
–1 0
–6 1 12
B)
6
–1 0
–6 1
– 12
C)
6
–6 0
–1 1 12
D)
6
–6 12
–1 1 0
4)
4x + 0.5y =14.5
0.8x + 1.5y =-
1.3
A)
4 0.5 14.5
0.8 1.5 –1.3
B)
4 14.5 0.5
0.8 –1.3 1.5
C)
4 0.8 14.5
1.5 0.5 –1.3
D)
14.5 0.5 4
–1.3 1.5 0.8
Page 13
5)
5
2x + 2
11y = – 1
8
4
11x – 1
2y = 1
2
A)
5
2 2
11
– 1
8
4
11 – 1
2 1
2
B)
5
2 2
11
1
8
4
11 1
2 1
2
C)
5
2 2
11
– 1
8
4
11 1
2 – 1
2
D)
5
2
2
11
– 1
80
4
11
– 1
2
1
20
6)
7x +3y +2z =98
4x +2y +7z =99
6x –2y +6z =82
A)
7 3298
4 2799
6–2682
B)
732
427
6–26
C)
74 698
32–299
27 682
D)
98 2 3 7
99 7 2 4
82 6 –26
7)
4x +2z =48
4y +8z =44
5x –2y +3z =65
A)
4 0248
0 4844
5–2365
B)
40 548
04–244
28 365
C)
4 2048
4 8044
5–2365
D)
402
048
5–23
8)
4x +6y +5z =55
7x +8y +7z =78
3x +4y +8z =79
6x +7y +4z =3
A)
4 6 5 55
7 8 7 78
3 4 8 79
6 7 4 3
B)
4 7 3 6
6 8 4 7
5 7 8 7
55 78 79 3
C)
4 6 5 3
7 8 7 79
3 4 8 78
6 7 4 55
D)
4 7 3 7
6 8 4 3
5 7 8 7
55 78 79 6
Page 14
9)
5x + 7y – 10z + w = –8
12y + z = –4
x – y – 6z = –3
2x – 2y + 8z = –2
A)
57
–10 1 –8
012 1 0
–4
1–1–60
–3
2–280
–2
B)
57
–10 –8
012 1 6
1–1–6–3
2–28
–2
C)
5 019
712
–1–2
–10 1 –68
1 000
–8–4–3–2
D)
5 7 10 1 –8
012 1 0
–4
1160
–3
2280
–2
2 Write the System of Equations from the Augmented Matrix
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the system of equations associated with the augmented matrix. Do not solve.
1) 57
–9
8–38
A)
5x + 7y = –9
8x – 3y = 8 B)
5x +7y =0
8x – 3y = 0 C)
7x +5y = –9
8x – 3y = 8 D)
5x +7y = –9
–3x + 8y = 8
2) 17 16 10
715 7
A)
17x + 16y = 10
7x + 15y = 7 B)
17x +16y = –10
7x + 15y = – 7 C)
16x +17y =10
7x + 15y = 7 D)
17x +16y =10
15x + 7y = 7
3)
2 7 –6
22 14
A)
2x + 7y = –6
2x + 2y = 14
B)
2x + 2y = –6
7x + 2y = 14
C)
–6x + 7y = –2
14x + 2y = –2
D)
2x + 14y = –7
7x + –6y = –2
4)
100 9
010 3
001–6
A)
x = 9
y = 3
z = –6
B)
x = –9
y = –3
z = 6
C)
x =0
y = 12
z = 3
D)
x =15
y = 9
z = 0
Page 15
5)
–7
–4 2 5
6 –5 0 8
40 –4 9
A)
–7x – 4y + 2z = 5
6x – 5y = 8
4x – 4z = 9
B)
–7x – 4y + 2z = 5
6x – 5y + 8z = 0
4x – 4y + 9z = 0
C)
–7x – 4y + 2z = 5
6x + –5z = 8
4x – 4y = 9
D)
–7x – 4y + 2z = 5
6x – 5y = 8
4x – 4y = 9
6)
4 6 5–2
20 3 4
4 5 0 2
A)
4x + 6y + 5z = –2
2x + 3z = 4
4x + 5y = 2
B)
4x –6y +5z = –2
2x + 3z = –4
4x + 5y = –2
C)
4x + 6y + 5z = –2
2x + 3z = 4
4x + 5z = 2
7)
1 0 0 0 3
0 1 0 0 –4
0 0 1 0 –4
0 0 0 1 0
A)
x1 = 3
x2 = –4
x3 = –4
x4 = 0
B)
x1 = –3
x2 = 4
x3 = 4
x4 = 0
C)
x1 = 0
x2 = –1
x3 = –1
x4 = –3
D)
x1 = 7
x2 = 0
x3 = 0
x4 = 0
Determine whether the system corresponding to the given augmented matrix is consistent or inconsistent. If it is
consistent, give the solution.
8)
1 0 0 8
010 10
000 8
A) consistent; x = 8
,
y = 10; (8
,
10) B) consistent; x = –8
,
y = –10; (–8
,
–10)
C) consistent; x = 8
,
y = 10
,
z = 8; (8
,
10
,
8) D) inconsisten
t
9)
1 0 0 0
010 0
000 10
A) consistent; x = 0, y = 0; (0, 0) B) consistent; x =0, y = 10; (0, 10)
C) consistent; z = 10; (–9) D) inconsisten
t
Page 16
10)
1 54 37
0 34 32
0 01 5
A) consistent; x = –3
,
y = 4
,
z = 5; (–3
,
4
,
5) B) consistent; x = –3
,
y = 5
,
z = 4; (–3
,
5
,
4)
C) consistent; x = 5
,
y = 4
,
z = –3; (5
,
4
,
–3) D) inconsisten
t
11)
10–57
012 –6
000 0
A) consistent; x = 7 +5z, y = –6 – 2z, z any real number
or {(x, y, z)|x = 7 + 5z, y = –6 – 2z, z any real number}
B) consistent; x = 7 –5z, y = –6 + 2z, z any real number
or {(x, y, z)|x = 7 – 5z, y = –6 + 2z, z any real number}
C) consistent; x = 7
,
y = –6
,
z = –5; (7
,
–6
,
–5)
D) inconsisten
t
12)
1001 1
0102 –4
001–50
0000 0
A) consistent; x1 = 1 – x4
,
x2 = –4 –2x4
,
x3=5x4
,
x4any real number
or {(x, y, z)|x1 = 1 – x4, x2 = –4 – 2x4, x3 = 5x4, x4 any real number}
B) consistent; x1 = 1 + x4
,
x2 = –4 +2x4
,
x3= –5x4
,
x4any real number
or {(x, y, z)|x1 = 1 + x4, x2 = –4 + 2x4, x3 = –5x4, x4 any real number}
C) consistent; x1 = 2
,
x2 = –5
,
x3 = –1
,
x4= –5; (2
,
–5
,
–1
,
–5)
D) inconsisten
t
3 Perform Row Operations on a Matrix
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Perform the row operation(s) on the given augmented matrix.
1) R1= 1
2r1
2 10 –4
53
–7
A) 1 5 –2
5 3 –7B)
1 5 –2
5
2 3
2 – 7
2C) 1 5–2
6 8 –9D) 1 5–4
5 3 –7
2) R2 = –2r1 + r2
1 7 10
22–7
A) 1 7 10
0
–12 –27 B) 1 710
0 16 13 C) –2–14 –20
0
–12 –27 D) –2–14 –20
2 2 –7
Page 17
3) R3 = 4r1 + r3
–7
–5
–1
–10
6 –2 9 5
28 –6 6 18
A)
–7
–5–1
–10
6 –2 9 5
0 –26 2 –22
B)
–7
–5–1–10
6 –2 9 5
0 16 2 –22
C)
–7–5–1–10
6 –2 9 5
0 –26 10 –22
D)
–7–5–1–10
6 –2 9 5
0 16 10 –22
4) R2 = –2r2 + r1
2 4 6 –8
1 2 3 6
4 6 7 1
A)
2 4 6 –8
0 0 0–20
4 6 7 1
B)
2 4 6 –8
0 0 0 0
4 6 7 1
C)
24 6
–8
–4
–8
–12 6
4 6 7 1
D)
24 6
–8
–4
–8
–12 –20
4 6 71
5) R2= –5r1 + r2
1–415
–50 5–3
–13
–2–1
A)
1–41 5
–10 20 0 –28
–13
–2–1
B)
1–415
0–20 10 22
–13
–2–1
C)
26 –4–24 20
–50 5
–3
–13
–2–1
D)
–10 20 0 –28
–505
–3
–13
–2–1
6) (a) R2 = –5r1 + r2
(b) R3 = –4r1 + r3
(c) R3 = 2r2 + r3
1–3–5–2
5–5–4 5
4 5 4 6
A)
1–3–5–2
0102115
0376644
B)
1–3–5–2
0–8–93
0168
C)
1–3–5–2
0102115
0274529
D)
1–3–5–2
0202910
0578234
7) (a) R2 = –4r1 + r2
(b) R3 = 2r1 + r3
(c) R3 = 4r2 + r3
1–3–5 2
4–5 2 5
–2 –5 4 6
A)
1–3–52
0722
–3
01782–2
B)
1–3–52
0722
–3
0–79 –78 –2
C)
1–3–52
0722
–3
0 29 102 –2
D)
1–3–52
01722–3
0 6 16 7
Page 18
8) (a) R2 = 3r1 + r2
(b) R3 = –2r1 + r3
(c) R3 = 4r2 + r3
1 2 –3 4
–3
–5 8 –10
2 0 –1 4
A)
1 2 –3 4
0 1 –1 2
0 0 1 4
B)
1 2 –3
–4
0 1 –1 2
0 0 1 –4
C)
1 2–3 4
0 1 1 –2
0 0 1 4
D)
1 2 –3– 4
0 1 –1 2
0 0 1 4
9) (a) R3 = –3r1 + r3
(b) R4 = 5r1 + r4
11
–11 2
0–25
–20
40
–1–34
–340 1
–4
A)
11
–11 2
0–25
–20
1–32
–6–2
29
–56 6
B)
11
–112
0–25
–20
1–32
–6–2
–34 0 1
–4
C)
11
–112
0–25
–20
73
–4010
29
–566
D)
11
–11 2
0–25
–20
1–32
–61
29
–566
4 Solve a System of Linear Equations Using Matrices
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations using matrices (row operations). If the system has no solution, say that it is
inconsistent.
1) 4x + 6y =-
2
5x + 5y =-
10
A) x = –5
,
y = 3; (–5
,
3) B) x =3
,
y = –5; (3
,
–5)
C) x = –5
,
y = –3; (–5
,
–3) D) inconsisten
t
2) 2x – 4y =-
4
6x – 12y =-
8
A) inconsisten
t
B) x =2
,
y =-8; (2
,
–8)
C) x =-4
,
y =-8; (–4
,
–8) D) x =2
,
y =2; (2
,
2)
Page 19
3)
x – y = 2
x + y = 1
A) x = 3
2, y = – 1
2; 3
2, – 1
2B) x = 1
2, y = 1
2; 1
2, 1
2
C) x = 3
,
y = –1; 3
,
–1 D) inconsisten
t
4)
3x – 2y = 6
x + y = 1
4
A) x = 13
10, y = – 21
20; 13
10, – 21
20 B) x = 11
10, y = – 17
20; 11
10, – 17
20
C) x = 13
2, y = – 25
4; 13
2, – 25
4D) inconsisten
t
5) 5x + 6y =14
3x =-
6
A) x = –2
,
y = 4; (–2
,
4) B) x =4
,
y = –2; (4
,
–2)
C) x = –2
,
y = –4; (–2
,
–4) D) inconsisten
t
6) 5x + y =22
7x + 2y =29
A) x = 5
,
y = –3; (5
,
–3) B) x = –3
,
y =5; (–3
,
5)
C) x = –3
,
y = –5; (–3
,
–5) D) inconsisten
t
7)
8x + 8y – z =103
x – 3y + 2z =-
13
7x + y + z =50
A) x = 6
,
y = 7
,
z = 1; (6
,
7
,
1) B) x =6
,
y =1
,
z =7; (6
,
1
,
7)
C) x = –6
,
y = 7
,
z = 12; (–6
,
7
,
12) D) inconsisten
t
8)
8x – y – 8z =17
9x – 5z =47
8y + z =61
A) x = 8
,
y = 7
,
z = 5; (8
,
7
,
5) B) x =8
,
y =5
,
z =7; (8
,
5
,
7)
C) x = –8
,
y = 7
,
z = 16; (–8
,
7
,
16) D) inconsisten
t
9)
–2x + 2y – z =-3
x – 9y – 3z =-
18
–8x + y + z =-
21
A) x = 3
,
y = 2
,
z = 1; (3
,
2
,
1) B) x =3
,
y =1
,
z =2; (3
,
1
,
2)
C) x = –3
,
y = 2
,
z = 6; (–3
,
2
,
6) D) inconsisten
t
Page 20
10)
5x – 6y + 2z =-
2
–25x + 30y – 10z =10
10x – 12y + 4z =-
4
A) x = 6
5y – 2
5z – 2
5, y is any real number, z is any real number
or (x, y, z) | x = 6
5y – 2
5z – 2
5, y is any real number, z is any real number
B) x = 10
,
y = 12
,
z = 4; (10
,
12
,
4)
C) x = 2
,
y = 5
,
z = 9; (2
,
5
,
9)
D) inconsisten
t
11)
x – y + 2z = 15
3x + z = 5
x + 4y + z = –15
A) x = 0
,
y = –5
,
z = 5; (0
,
–5
,
5) B) x =0
,
y =5
,
z = –5; (0
,
5
,
–5)
C) x = 5
,
y = –5
,
z = 0; (5
,
–5
,
0) D) x =5
,
y =0
,
z = –5; (5
,
0
,
–5)
12)
x + y + z = 0
x – y + 5z = –22
3x + y + z = 8
A) x = 4
,
y = 1
,
z = –5; (4
,
1
,
–5) B) x =4
,
y = –5
,
z = 1; (4
,
–5
,
1)
C) x = –5
,
y = 1
,
z = 4; (–5
,
1
,
4) D) x = –5
,
y =4
,
z = 1; (–5
,
4
,
1)
13)
3x – 2y + z = –7
x + y – 2z = 12
3x + y – z = 10
A) x = 1, y = 3, z = –4; (1, 3, –4) B) x =1, y = –3, z = –16; (1, –3, –16)
C) x = 5, y = 13
2, z = –3; 5, 13
2, –3 D) x =2, y =3, z = –7; (2, 3, –7)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14)
–5x + 2z = 13
x + 3y – 2z = –15
4x + 5y – z = –18
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
15)
x – y + 2z + w = 4
y + z = 2
z – w = 1
A) x = 3 – 4w, y = 1 –w, z = 1 + w, where w is any real number
B) inconsisten
t
C) x = 3
,
y = 1
,
z = 1
,
w = 0; (3
,
1
,
1
,
0)
D) x = –1
,
y = 0
,
z = 2
,
w = 1; (–1
,
0
,
2
,
1)
Page 21
16)
x + 3y – 2z – w = 6
4x + y + z + 2w = 17
–3x – y – 3z – 2w = –24
x – y – 3z –2w = –20
A) x = 1
,
y = 5
,
z = 4
,
w = 2; (1
,
5
,
4
,
2)
B) x = 6
,
y = 17
,
z = –24
,
w = –20; (6
,
17
,
–24
,
–20)
C) x = 1 + w, y = 5 – 2w, z = 4 + 2w, where w is any real number
D) x = 0
,
y = 7
,
z = 0
,
w = 4; (0
,
7
,
0
,
4)
17)
5x + y =2
4x – y + z – w =7
z+ w =6
A) x = 1
3 + 2
9w, y = 1
3 – 10
9w, z = 6 – w, where w is any real number
B) x = 1
3 – 10
9w, y = 1
3 + 2
9w, z = 6 – w, where w is any real number
C) x = 1
3 – 2
9w, y = 1
3 + 10
9w, z = 6 – w, where w is any real number
D) inconsisten
t
18)
3x + 5y – 2w=-13
2x + 7z – w =-
1
4y + 3z + 3w= 1
–x + 2y + 4z =-
5
A) x = 1, y = –2, z = 0, w = 3; (1, –2, 0, 3) B) x = 4
3, y = – 13
20, z = 0, w = 5
2; 4
3, – 13
20, 0, 5
2
C) x = 3
4, y = –2, z = 0, w = 3
4; 3
4, –2, 0, 3
4D) x = –1, y = – 20
13, z = 0, w = 2
5; –1, – 20
13, 0, 2
5
19)
x + y + z – w=6
2x – y + 3z + 4w =-
4
4x + 2y – z – w =-13
–x – 2y + 4z + 3w=12
A) x = –4, y = 3, z = 5, w = –2; (–4, 3, 5, –2) B) x =4, y = –3, z = –5, w = 2; (4, –3, –5, 2)
C) x = – 1
4, y = 1
3, z = 1
5, w = – 1
2; – 1
4, 1
3, 1
5, – 1
2D) x = 1
4, y = – 1
3, z = – 1
5, w = 1
2; 1
4, – 1
3, – 1
5, 1
2
20)
3x + 5y + 2w = –12
2x + 6z – w = –5
–2y + 3z – 3w =- 3
–x + 2y + 4z + w = –2
A) x = –1, y = –3, z = 0, w = 3; (–1, –3, 0, 3) B) x =1, y = –3, z = 0, w = 3; (1, –3, 0, 3)
C) x = –1, y = 3, z = 0, w = –3; (–1, 3, 0, –3) D) x =1, y =3, z = 0, w = –3; (1, 3, 0, –3)
Page 22
21)
x + y – z + w = –5
3x – y + 3z – 2w = 7
–2x + 2y + z – w =16
–x – 2y – 3z + 3w = –22
A) x = –2, y = 3, z = 4, w = –2; (–2, 3, 4, –2)
B) x = –2, y = –3, z = 5, w = 1
2; –2, –3, 5, 1
2
C) x = 2, y = –3, z = –4, w = –2; (2, –3, –4, –2)
D) x = 1
2, y = – 1
3, z = – 1
4, w = – 1
2; 1
2, – 1
3, – 1
4, – 1
2
Solve the problem using matrices.
22) Find the function f(x) = ax3 + bx2 + cx + d for which f(0) = –2, f(1) = 5, f(–1) = 3,
f(2) = 4.
A) f(x) = – 10
3x3 + 6x2 + 13
3x – 2 B) f(x) = 10x3 – 18x2 – 13x + 6
C) f(x) = 8
3x3 + 4x2 + 5
3x – 2 D) f(x) = –8x3 +12x2+ 5x – 6
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
23) Find real numbers a, b, and c such that the graph of the function y = ax2 + bx + c contains the points
(–2, –4), (1, –1), and (3, –19).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
24) An application of Kirchoff’s Rules to the circuit shown results in the following system of equations:
I2 + I3 =I1
12 + 24 – 6I1 – 6I2=0
24 – 6I1 – 3I3=0
Find the currents I1, I2, and I3.
12 V 24 V
5 Ω
3 Ω2 Ω
1 Ω4 Ω
A) I1 = 3.5, I2 = 2.5, and I3 = 1B)I
1=5.5, I2=4.5, and I3 = 1
C) I1 = 4.5, I2 = 3.5, and I3 = 1D)I
1=2.5, I2=1.5, and I3 = 1
Page 23
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
25) Melody has $45,000 to invest and wishes to receive an annual income of $4290 from this money. She has
chosen investments that pay 5%, 8%, and 12% simple interest. Melody wants to have the amount invested
at 12% to be double the amount invested at 8%. How much should she invest at each rate?
26) A company manufactures three types of wooden chairs: the Kitui, the Goa, and the Santa Fe. To make a
Kitui chair requires 1 hour of cutting time, 1.5 hours of assembly time, and 1 hour of finishing time. A Goa
chair requires 1.5 hours of cutting time, 2.5 hours of assembly time and 2 hours of finishing time. A Santa
Fe chair requires 1.5 hours of cutting time, 3 hours of assembly time, and 3 hours of finishing time. If 41
hours of cutting time, 70 hours of assembly time, and 58 hours of finishing time were used one week, how
many of each type of chair were produced?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
27) The table below shows the number of birds for three selected years after an endangered species protectio
n
program was started.
x (Number of years after 1980) 1 5 10
y (Number of birds) 40 160 445
Use the quadratic function y = ax2 + bx + c to model the data. Solve the system of linear equations
involving a, b, and c using matrices. Find the equation that models the data.
A) y = 3x2 + 12x + 25 B) y = 4x2 + 24x + 20 C) y = 5x2 – 12x + 28 D) y = 6x2 – 36x + 21
28) A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to
prepare, 2 hours to paint, and 9 hours to fire. A tree takes 14 hours to prepare, 3 hours to paint, and 4
hours to fire. A sleigh takes 4 hours to prepare, 15 hours to paint, and 7 hours to fire. If the workshop has
105 hours for prep time, 63 hours for painting, and 115 hours for firing, how many of each can be made?
A) 9 wreaths; 5 trees; 2 sleighs B) 5 wreaths; 2 trees; 9 sleighs
C) 2 wreaths; 9 trees; 5 sleighs D) 10 wreaths; 6 trees; 3 sleighs
29) There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sale
s
by age group for marketing purposes. The percentage of 36– to 59–year–old buyers and the percentage of
buyers 60 and older combined exceeds the percentage of buyers 35 and younger by 38%. If the percentage
of buyers in the oldest group is doubled, it is 12% less than the percentage of users in the middle group.
Find the percentage of buyers in each of the three age groups.
A) 31% 35 and younger; 50% 36–59 year olds; 19% 60 and older
B) 33% 35 and younger; 47% 36–59 year olds; 20% 60 and older
C) 25% 35 and younger; 52% 36–59 year olds; 23% 60 and older
D) 19% 35 and younger; 50% 36–59 year olds; 31% 60 and older
30) Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake
to 137 g protein, 117 g fat, and 183 g carbohydrate. According to the health conscious hostess, the
marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g
protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g
carbohydrate. How many of each snack can he eat to obtain his goal?
A) 9 mushrooms, 6 meatballs, 2 eggs B) 6 mushrooms, 2 meatballs, 9 eggs
C) 2 mushrooms, 9 meatballs, 6 eggs D) 10 mushrooms, 7 meatballs, 3 eggs
Page 24
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
31) The perimeter of a picture frame is 11 feet. If three times the height is equal to eight times the width, wha
t
are the dimensions of the frame?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
32)
J
enny receives $1270 per year from three different investments totaling $20,000. One of the investments
pays 6%, the second one pays 8%, and the third one pays 5%. If the money invested at 8% is $1500 less
than the amount invested at 5%, how much money has Jenny invested in the investment that pays 6%?
A) $1500 B) $8500 C) $4500 D) $10,000
8.3 Systems of Linear Equations: Determinants
1 Evaluate 2 by 2 Determinants
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the value of the determinant.
1)
44
35
A) 8 B) 32 C) –8D)1
2)
–69
28
A) –66 B) –30 C) 66 D) –70
3)
13
22
A) –4 B) 8 C) 4 D) –1
4)
12 –7
–4 3
A) 8 B) 64 C) –8D)4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve for x.
5)
8 x
2 5
= 32
6)
5 9
–2 x = 8
Page 25
2 Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.
1)
3x + 2y =-
1
5x + y =-
11
A) x = –3
,
y = 4; (–3
,
4) B) x =4
,
y = –3; (4
,
–3)
C) x = –4
,
y = –3; (–4
,
–3) D) x =3
,
y = –4; (3
,
–4)
2)
2x – 2y =8
4x + 3y =37
A) x = 7
,
y = 3; (7
,
3) B) x =3
,
y =7; (3
,
7)
C) x = –3
,
y = 7; (–3
,
7) D) x = –7
,
y = –3; (–7
,
–3)
3)
3x + 3y =33
3x – 2y =8
A) x = 6
,
y = 5; (6
,
5) B) x =5
,
y =6; (5
,
6)
C) x = –5
,
y = 6; (–5
,
6) D) x = –6
,
y = –5; (–6
,
–5)
4)
4x – 7y =5
2x + 5y =-
3
A) x = 2
17, y = – 11
17; 2
17, – 11
17 B) x = 23
3, y = – 11
3; 23
3, – 11
3
C) x = – 2
17, y = 11
17; – 2
17, 11
17 D) x = 2
3, y = 1
3; 2
3, 1
3
5)
7x – 2y = 7
14x – 4y = 21
A) x = 2
,
y = 3; (2
,
3) B) x = 4
3, y = – 14
3; 4
3, – 14
3
C) x = 7
,
y = 21; (7
,
21) D) not applicable
6)
4x + 2y = 8
5
5x – 5y = 5
A) x = 3
5, y = – 2
5; 3
5, – 2
5B) x = 3
5, y = 8
5; 3
5, 8
5
C) x = – 3
5, y = – 2
5; – 3
5, – 2
5D) not applicable
Page 26
7)
x + 2y = 6
1
4x – y = –20
A) x = – 68
3, y = 43
3; – 68
3, 43
3B) x = – 68
,
y =3; –68
,
3
C) x = 92
3, y = 83
3; 92
3, 83
3D) not applicable
3 Evaluate 3 by 3 Determinants
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the value of the determinant.
1)
–53 1
–22–3
43–3
A) –83 B) –11 C) 83 D) 59
2)
800
775
385
A) –40 B) 600 C) 40 D) –35
3)
124
243
124
A) 0 B) 76 C) 1 D) –12
4)
–5–2–2
–124
11
–1
A) 30 B) 46 C) –30 D) –22
5)
412
323
464
A) –20 B) 180 C) 20 D) –44
6)
1–1–5
3–3–4
–4–4–5
A) 88 B) 120 C) –88 D) 30
Page 27
7)
–2 5 4
3–2 1
1 6 –3
A) 130 B) 80 C) –90 D) –12
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve for x.
8)
x –4
–1
–2 2 0
–1 –2 8
= 10
9)
5
–3 1
–2
–2 x
82
–1
= 28
10)
x1 3
1x–4
01 3
= 5x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
11) The equation of the line passing through the distinct points (x1
,
y1) and (x2
,
y2) is given by
x y 1
x1 y1 1
x2 y2 1
= 0. Find the equation of the line passing through the points (3, 5) and (–1, 4).
A) x – 4y + 17 = 0B)x + 7y +17 =0C)
–x +4y –17 =0D)x
+ 4y +17 =0
12) The equation of a line passing through two distinct points (x1
,
y1) and (x2
,
y2) is given by
x y 1
x2 y2 1
x3 y3 1
= 0. Use the determinant to write an equation for the line passing through (–4, 4) and
(8, –9). Express the line’s equation in standard form.
A) 13x + 12y + 4 = 0 B) 4x +8y +36 =0C)
–13x –12y +4=0D)
–9x –4y +32 =0
13) Determinants are used to show that three points lie on the same line (are collinear). I
f
x1 y1 1
x2 y2 1
x3 y3 1
= 0,
then the points (x1, y1), (x2, y2), and (x3, y3) are collinear. If the determinant does not equal 0, then the
points are not collinear. Are the points (–8, 4), (0, –3), and (–24, 18) collinear?
A) Yes B) No
Page 28
14) Determinants are used to show that three points lie on the same line (are collinear). I
f
x1 y1 1
x2 y2 1
x3 y3 1
= 0,
then the points (x1, y1), (x2, y2), and (x3, y3) are collinear. If the determinant does not equal 0, then the
points are not collinear. Are the points (–3, 5), (0, –9), and (–6, 21) collinear?
A) No B) Yes
15) The area of a triangle with vertices (x1
,
y1), (x2
,
y2), and (x3
,
y3) is
Area = ± 1
2
x1 y1 1
x2 y2 1
x3 y3 1
,
where the symbol ± indicates that the appropriate sign should be chosen to yield a positive area. Use this
formula to find the area of a triangle whose vertices are (9, 10), (5, –8), and (–2, –3).
A) 73 B) 146 C) 7 D) 14
4 Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.
1)
8x – 4y – z =-
5
x + 8y + 6z =47
4x + y + z =11
A) x = 1
,
y = 2
,
z = 5; (1
,
2
,
5) B) x =2
,
y =5
,
z =2; (2
,
5
,
2)
C) x = 1
,
y = –2
,
z = –5; (1
,
–2
,
–5) D) x =2
,
y =0
,
z =5; (2
,
0
,
5)
2)
–5x – 2y – 3z =-
47
5x + 3y + 5z =62
3x – 4y + 5z =-
9
A) x = 4
,
y = 9
,
z = 3; (4
,
9
,
3) B) x =9
,
y =3
,
z =9; (9
,
3
,
9)
C) x = 4
,
y = –9
,
z = –3; (4
,
–9
,
–3) D) x =5
,
y =7
,
z =3; (5
,
7
,
3)
3)
7x – 2z =5
6x + 9y + 7z =67
7x – 4y =-
17
A) x = 1
,
y = 6
,
z = 1; (1
,
6
,
1) B) x =6
,
y =1
,
z =6; (6
,
1
,
6)
C) x = 1
,
y = –6
,
z = –1; (1
,
–6
,
–1) D) x =2
,
y =4
,
z =1; (2
,
4
,
1)
4)
x – y + 2z =4
4x + z =0
–x + y – 2z =-12
A) x = 0
,
y = –4
,
z = 0; (0
,
–4
,
0) B) x =2
,
y = –4
,
z =0; (2
,
–4
,
0)
C) x = 0
,
y = 0
,
z = –4; (0
,
0
,
–4) D) not applicable
Page 29
5 Know Properties of Determinants
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use properties of determinants to find the value of the second determinant, if the value of the first is known.
1)
xyz
uvw
1–2–2
= –14
1–2–2
uvw
xyz
= ?
A) 14 B) –14
C) 0 D) cannot be determined
2)
xy z
uvw
11 2
= –26
xyz
uvw
–2–2–4
= ?
A) 52 B) 26 C) –52 D) –26
3)
xy z
uv w
12–2
= 24
uvw
–3–66
xyz
= ?
A) –72 B) 72 C) –24 D) 24
4)
xyz
uvw
1–2–2
= 19
1–2–2
–2u –2v –2w
x – 1 y + 2 z + 2
= ?
A) 38 B) –19 C) –38 D) 19
5)
xyz
uvw
1–21
= 29
xy z
– x
u v w – u
1–20
= ?
A) 29 B) –29
C) 0 D) cannot be determined
6)
xy z
uv w
13–3
= 178
x – 2y – 6z + 6
–2u – 1 –2v – 3 –2w + 3
13
–3
= ?
A) –356 B) –178 C) 356 D) 178
7)
s t u
v wx
4 2 8
= 3
32 – s 16 – t 64 – u
vw x
42 8
= ?
A) –3B)
–24 C) 24 D) 3
Page 30
8)
x y z
a b c
2 4 5
= 3
2 4 5
3a 3b 3c
x – 2 y – 4 z – 5
= ?
A) –9B)9 C)0 D)6
8.4 Matrix Algebra
1 Find the Sum and Difference of Two Matrices
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Perform the indicated operation(s), whenever possible.
1)
Let A = [5 4] and B = –2
8. Find A + B.
A)
[3 12]
B)
3
12
C)
5–2
48
D) not defined
2)
Let A =
3
–2
–4
and B =
–5
2
5
. Find A + B.
A)
–2
0
1
B)
[–2 0 1]
C)
3–5
–22
–45
D)
2
1
2
3)
Let A =
–56
6–4
85
and B =
–31
–18
–36
. Find A + B.
A)
–87
54
511
B)
–25
7–12
11 14
C)
–87
–5–4
5–11
D)
–8–4
54
511
4)
Let A =
–1 4
04
7 –4
and B =
72
17 4
32
. Find A – B.
A)
–8 2
–17 0
4 –6
B)
1 5
78
10 0
C)
1 2
70
4 –2
D)
3–3
70
–46
5)
Let A = –11
25 and B = 62
5–2. Find A + B.
A)
53
73
B)
34
–13
C)
51
4–3
D)
18
Page 31
6)
Let A = –10
63 and B = –16
31 . Find A – B.
A)
0–6
32
B)
–26
94
C)
06
–3–2
D)
[–1]
7)
Let A = 7–9
–7–7, B = –25
69 and C = –92
3–3. Find A + B – C.
A)
14 –6
–45
B)
–4–2
2–1
C)
18 –16
–16 –13
D)
0–12
–10 –19
8) Let A =
7–48
–65
–1
06
–3
and B =
–2–6–1
–7–43
–3–9–5
. Find A – B.
A)
929
19
–4
315 2
B)
929
192
315–4
C)
5–10 7
–13 1 –8
–3–32
D)
5–10 7
–13 1 2
–3–3–8
9) Let A =
–4 6 7
3 –5 12
7–11 14
and B =
6 10 –4
–5 6 – 8
3 11 7
. Find A – B.
A)
–10 –4 11
8 –11 20
4 –22 7
B)
2 16 3
–2 1 4
10 0 21
C)
–10 –4 3
–2 –11 4
4 –22 7
D)
10 4 –11
–8 11 –20
–4 22 –7
10)
Let A =
5 8 1
10 6 –1
–52 3
and B =
10 –8 1
–50 4
4 3 –2
. Find A + B.
A)
15 0 2
56 3
–1 5 1
B)
5–16 2
15 6 3
–151
C)
5–16 2
56
–5
151
D)
15 0 2
15 6 –5
15 1
11)
Let A =
6 10 1
12 8 –1
–32 3
and B =
9 –10 1
–30 5
2 3 –2
. Find A – B.
A)
–320 0
15 8 –6
–5 –1 5
B)
–300
15 8 –6
1–1 1
C)
–3–20 0
15 8 –6
–515
D)
15 0 2
98 4
–15 1
Page 32
2 Find Scalar Multiples of a Matrix
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the given matrices to compute the given expression.
1) Let A = –32
02 . Find 4A.
A)
–12 8
08
B)
–12 8
02
C)
–12 2
02
D)
16
46
2) Let B = –155–3. Find –3B.
A) 3–15 –15 9 B) 355–3C) –31515–9D) –333–5
3) Let C =
6
–2
12
. Find 1/2 C.
A)
3
–1
6
B)
12
–4
24
C)
3
–2
12
D)
6
–1
12
4) Let A =33
25 and B =04
–16 . Find 4A + B.
A)
12 16
726
B)
12 28
444
C)
12 16
111
D)
12 7
711
5) Let C =
1
–3
2
and D =
–1
3
–2
. Find C – 4D.
A)
5
–15
10
B)
–3
9
4
C)
–5
15
–10
D)
5
–6
4
6) Let A = –32 and B =10 . Find 3A +4B.
A) –56 B) –94 C) –54 D) 02
7) If A = 2–1
79
and B = 5–3
47
, find –2A + 4B.
A)
–24 –18
–30 –34
B)
16 –10
210
C)
74
11 13
D)
–3–6
35
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
8) If A = 7–5
–81
and B = –9–3
2–8, find –5A + 4B.
Page 33
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
9) Let A =
–5–24
–2–6–5
86
–8
and B =
32 5
–15 8
–31–2
. Find –3A – 4B.
A)
3–2–32
10 –2–17
–12 –22 32
B)
18 8 –7
52323
–27 –17 22
C)
–20 9
–3–13
57
–10
D)
–2–35
0–17
93
–10
3 Find the Product of Two Matrices
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Compute the product.
1)
–23
22
–20
–14
A)
112
–68
B)
40
–28
C)
4–6
–25
D)
12 1
8–6
2)
13–2
20 4
30
–21
04
A)
–3–5
616
B)
–5–3
16 6
C)
3–60
0016
D) not defined
3)
0 –3 1
5 –1 0
1 2
0 1
1
–1
A)
1 –4
59
B)
1 5
–4 9
C)
0 10
0
–1
00
D)
10 –5 1
5
–1 0
–5
–2 0
4)
72–4
–34 9
1
–9
9
A)
–47
42
B)
72
–4
–34 9
1–99
C)
–47 42
D) not defined
Page 34
5)
–326
5
0
–3
A) –33 B) 84 C) –15 0 –18 D)
–15
0
–18
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6)
–2
1
–1
1–10
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
7)
87–5
42 2
–2–61
–91
–3
1–5–4
A)
–84 –16 7
–24 –32 –10
B)
–84 –24
–16 –32
7–10
C)
8 7–5
4 2 2
–22 2
–91
–3
11
–4
D)
–16 –42 –5
–36 2 –6
4–10 –8
8)
–5–7–8
–1–9–8
5 –3 9
–5 1 1
–3–8 4
–2–9–6
A)
62 123 15
48 143 11
–34 –52 –61
B)
62 48 –34
123 48 –52
15 11 –61
C)
–5–7–8
–1–9–8
5–39
–51 1
–3–84
–2–9–6
D)
25 –7–8
372
–32
–10 27 –54
9)
2–8
1 6
0 1
–3 2
1 –1
6 1
A)
4 10 –24
12 –5 9
1 –1 2
B)
–24 10 4
9 –5 12
2 –1 1
C)
9 –1 4
–24 –5 1
2 10 12
D) not defined
Page 35
10)
3–21
04
–1
50
–21
A)
15 0
04
B)
15 –10 5
–68
–3
C)
15 –6
–10 8
5–3
D) not defined
11)
–13
12 0–26
1–32
A)
3–70
2–810
B) not defined C)
32
–7–8
010
D)
0–618
1–64
12)
[2 –3 –4]
78
–4
–9–35
46
–1
A) 25 1 –19 B)
25
1
–19
C)
2–3–4
78
–4
–9
–35
46
–1
D)
14 –24 16
–18 9 –20
8–18 4
Perform the indicated operations and simplify.
13) Let A = 3 –4
–2 5 , B = 5 –2 8
1 0 –3 , and C =
7 –9 0
3 –5 1
–1 6 2
. Find AB + BC.
A) 32 7 50
5–23 –37 B) 68 3 31
8
–2 –5C) 32 19 40
–15 31 –37 D) –10 –19 12
–15 31 –25
14)
Let A =
0 –1
2 0
2 5
, B =
–5 0
1 1
2 2
, and C = –2 1 2
0 –2–1. Find C(A – B).
A) –97
–2–1B) 12 8
2–5C) 2–5
12 8 D) –12 2
9–3
15)
Let A =
25
1 0
2 –1
, B =
–5 1
0 2
–1–5
, and C = 525
1 –2 0. Find BC + 5ℐ3.
A)
–19 –12 –25
2 1 0
–10 8 0
B)
–24 –12 –25
2–4 0
–10 8 –5
C)
31 8 25
2 9 0
10 12 10
D)
26 8 25
2 4 0
10 12 5
Page 36
Solve the problem.
16) State University has a College of Arts & Sciences, a College of Business, and a College of Engineering. The
percentage of students in each category are given by the following matrix.
Freshman Sophomore Junior Senior
Arts & Sciences
Business
Engineering
60% 50% 40% 70%
20% 40% 30% 10%
20% 10% 30% 20%
The student population is distributed by class and age as given in the following matrix.
Female Male
Freshman
Sophomore
Junior
Senior
490 740
550 750
810 630
630 480
How many female students are in the College of Business? How many male students are in the College of
Arts & Sciences?
A) 624 students; 1407 students B) 522 students; 685 students
C) 685 students; 508 students D) 1334 students; 624 students
17) The final grade for an algebra course is determined by grades on the midterm and final exam. The grades
for four students and two possible grading systems are modeled by the following matrices.
Midterm Final
Student 1
Student 2
Student 3
Student 4
73 79
44 62
85 90
98 96
System
1
System
2
Midterm
Final 0.3 0.5
0.7 0.5
Find the final course score for Student 3 for both grading System 1 and System 2.
A) System 1: 88.5; System 2: 87.5 B) System 1: 70.5; System 2: 104.5
C) System 1: 77.2; System 2: 76 D) System 1: 44.2; System 2: 53
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
18) A company manufactures three types of wooden chairs at two different locations. In one week, the main
location produces 18 Kitui chairs, 12 Goa chairs, and 6 Santa Fe chairs while the secondary location
produces 10 Kitui chairs, 3 Goa chairs, and 5 Santa Fe chairs.
(a) Find a 2 by 3 matrix representing the above data.
(b) If each Kitui chair requires 25 board–feet of wood, a Goa chair requires 29 board–feet of wood, and a
Santa Fe requires 30 board–feet of wood, find a 3 by 1 matrix representing the amount of material.
(c) Multiply the 2 by 3 matrix found in part a and the 3 by 1 matrix found in part b to get a 2 by 1 matrix
showing the week’s usage of material at both locations.
Page 37
19) The matrix 922
435 represents one week’s usage of wood (in board–feet) at two locations of a company that
manufactures chairs. Due to differences in transportation costs, the first location pays $0.89 per board–foot
and the second location pays $0.94 per board–foot.
(a) Find a 1 by 2 matrix representing the cost of the wood.
(b) Multiply the given matrix and the matrix found in part a to determine the total cost of wood for that
week.
4 Find the Inverse of a Matrix
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Each matrix is nonsingular. Find the inverse of the matrix. Be sure to check your answer.
1)
5 3
32
A) 2–3
–35 B) 23
35 C)
1
2
–3
–31
5
D)
1
23
31
5
2)
10 1
–10
A) 0–1
110 B) 01
–110 C) 01
110 D) 0–1
–110
3)
–24
4–4
A)
1
2
1
2
1
2
1
4
B)
1
2
1
4
1
2
1
4
C)
– 1
2
1
4
1
2
– 1
4
D)
1
2
– 1
2
– 1
2
1
4
4)
–5–1
60
A)
0
1
6
–1– 5
6
B)
0
1
6
–15
6
C)
0
– 1
6
1– 5
6
D)
0
5
6
–1– 1
6
5)
43
1–5
A)
5
23
3
23
1
23
– 4
23
B)
5
23
– 3
23
1
23
4
23
C)
– 5
23
– 3
23
– 1
23
4
23
D)
4
23
3
23
1
23
– 5
23
Page 38
6)
–1–1
–21
A)
– 1
3
– 1
3
– 2
3
1
3
B)
– 1
3
1
3
2
3
1
3
C)
– 2
3
1
3
– 1
3
– 1
3
D)
1
3
– 1
3
– 2
3
– 1
3
7)
0–2
–2–5
A)
5
4
– 1
2
– 1
20
B)
5
4
1
2
1
20
C)
– 1
20
5
4
– 1
2
D)
0– 1
2
– 1
2
5
4
8)
55
0–3
A)
1
5
1
3
0– 1
3
B)
1
5
– 1
3
0– 1
3
C)
0– 1
3
1
5
1
3
D)
– 1
3
1
3
01
5
9)
6–5
–10
A)
0– 1
– 1
5
– 6
5
B)
01
1
5
– 6
5
C)
– 1
5
– 6
5
0– 1
D)
– 6
5
– 1
– 1
50
10)
–5 a
–7a
A)
a
2
– a
2
7
2
– 5
2
B)
a
2 1
2
– 7
2 a
2
C)
a
2
– 7
2
a
2
5
2
D)
a
2 1
2
7
2 a
2
Page 39
11)
2–10
–11
–2
10
–1
A)
1–12
–3–2–4
–1–1 1
B)
1–12
–3–24
–11 1
C)
1–22
3–2–4
–1–1 2
D)
1–12
–2–1–4
–1–1 1
12)
100
110
111
A)
1 00
–1 10
0 –11
B)
1 0 0
1 11
0 –11
C)
1 0
–1
–1 1 0
0 –1 1
D)
1 0 0
–1 –10
0 1 1
13)
3 –31
–22
–1
–45
–2
A)
1–11
0–2 1
–2–30
B)
1–3 1
0–21
–2–30
C)
1 –1 1
–2
–3 0
0–21
D)
1 –1 0
–2
–3 0
0–2–1
14)
100
610
0–71
A)
100
–610
–42 7 1
B)
10 0
61 0
00–7
C)
1–742
011
001
D)
100
–7–10
42 6 1
15)
100
–110
111
A)
100
110
–2–11
B)
111
011
001
C)
1–11
01
–1
001
D)
–100
–1–10
–1–1–1
16)
111
211
223
A)
–1 1 0
4
–1
–1
–2 0 1
B)
–1–1–1
–2
–1
–1
–2
–2
–3
C)
11 1
1
21 1
1
2 1
2 1
3
D)
1–11
1
21 1
1
2 1
2 – 1
3
Page 40
17)
1 3 2
133
278
A)
–310
–3
2
–4 1
–1 1 0
B)
–1–3–2
–1
–3
–3
–2
–7
–8
C)
1 1
3 1
2
1
1
3 1
3
1
2 1
7 1
8
D)
1– 1
3 1
2
1– 1
3 1
3
1
7 1
2 1
8
18)
108
123
253
A)
9
–40 16
–3 13 –5
–1 5 –2
B)
112
025
833
C)
–10
–8
–1
–2
–3
–2
–5
–3
D)
–1–12
02
–5
823
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Show that the matrix has no inverse.
19)
14 –10
–75
20)
4208
–3–11
–1 74
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a graphing utility to find the inverse, if it exists, of the matrix. Round answers to two decimal places.
21)
–16 3 28
5 –14 15
34 25 2
A)
–0.02 0.03 0.02
0.02 –0.04 0.02
0.02 0.02 0.01
B)
–0.01 0.02 0.01
0.02 –0.03 0.01
0.02 0.02 0.01
C)
–0.01 0.03 0.02
0.02 –0.03 0.01
0.02 0.02 0.01
D)
–0.02 0.03 0.02
0.02 –0.04 0.01
0.02 0.02 0.00
Page 41
5 Solve a System of Linear Equations Using an Inverse Matrix
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system using the inverse matrix method.
1)
–2x – 6y =-
2
2x – y =-
5
A) x = –2, y = 1; (–2, 1) B) x =1, y = –2; (1, –2)
C) x = 2, y = –1; (2, –1) D) x = –1, y =2; (–1, 2)
2)
x + 3y =-
8
21x + 6y =3
A) x = 1, y = –3; (1, –3) B) x = –3, y =1; (–3, 1)
C) x = –1, y = 3; (–1, 3) D) x =3, y = –1; (3, –1)
3)
–2x + 6y =6
3x + 2y =13
A) x = 3, y = 2; (3, 2) B) x =2, y =3; (2, 3)
C) x = –3, y = –2; (–3, –2) D) x = –2, y = –3; (–2, –3)
4)
–5x + 3y =8
3x – 6y =-
30
A) x = 2, y = 6; (2, 6) B) x =6, y =2; (6, 2)
C) x = –2, y = –6; (–2, –6) D) x = –6, y = –2; (–6, –2)
5)
bx + 6y = 10
bx + 3y = 7 b ≠ 0
A) x = 4
b, y = 1; 4
b, 1 B) x = 1, y = 4
b; 1, 4
b
C) x = –4
b, y = –1; –4
b, –1 D) x = –4
b, y = 1; –4
b, 1
6)
x + 2y + 3z =-
1
x
+ y + z =8
2x + 2y + z =-
2
The inverse of
123
111
221
is
–1 4–1
1 –5 2
0 2 –1
.
A) x = 35
,
y = –45
,
z = 18; (35
,
–45
,
18) B) x =9
,
y = –48
,
z = 17; (9
,
–48
,
17)
C) x = –4
,
y = –16
,
z = –2; (–4
,
–16
,
–2) D) x =29
,
y =35
,
z = 14; (29
,
35
,
14)
Page 42
7)
x + 2y + 3z =4
x + y + z =-
9
x
– 2z =9
The inverse of
12 3
11 1
10–2
is
–2 4–1
3 –5 2
–1 2–1
.
A) x = –53
,
y = 75
,
z = –31; (–53
,
75
,
–31) B) x = –44
,
y =79
,
z = –31; (–44
,
79
,
–31)
C) x = 4
,
y = 0, z = 0; (4
,
0, 0) D) x = –19
,
y = –15
,
z = –5; (–19
,
–15
,
–5)
8)
x + 2y + 3z =-
6
x + y + z =11
–x + y + 2z =-
9
The inverse of
1 2 3
1 1 1
–112
is
1 –1–1
–3 5 2
2 –3–1
.
A) x = –8
,
y = 55
,
z = –36; (–8
,
55
,
–36) B) x = –57
,
y =88
,
z = 37; (–57
,
88
,
37)
C) x = –6
,
y = 44
,
z = 18; (–6
,
44
,
18) D) x = –4
,
y =19
,
z = 12; (–4
,
19
,
12)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
9)
4x +2z =-
20
x–y–5z =5
–3x –2y –z=12
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10)
2x +4y –5z =-
8
x+5y +2z =-
1
3x +3y +3z =15
A) x = 5, y = –2, z = 2; (5, –2, 2) B) x =2, y =5, z =2; (2, 5, 2)
C) x = 5, y = 2, z = –2; (5, 2, –2) D) x = –5, y = –2, z = –2; (–5, –2, –2)
Use a graphing utility to solve the system of equations. Round answers to two decimal places.
11)
25x + 61y –12z =15
18x – 12y + 7z =-
3
3x + 4y – z =12
A) x = 4.56, y = –6.06, z = –22.55; (4.56, –6.06, –22.55)
B) x =-22.55, y = 4.56, z = –6.06; (–22.55, 4.56, –6.06)
C) x = 4.56, y = –22.55, z = –6.06; (4.56, –22.55, –6.06)
D) x = –22.55, y = –6.06, z = 4.56; (–22.55, –6.06, 4.56)
Page 43
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual
order.
12) Use the coding matrix A = 37
25 to encode the message LIFE.
A) 99 53
69 37 B) 78 62
54 43 C) 54 28
129 67 D) –3–5
3 3
13) Use the coding matrix A = –1–3
2 5 to encode the message CARE.
A) –6–33
11 61 B) –57 –16
96 27 C) 18 105
–7 –4D) –1
–8
–4–29
14) Use the coding matrix A =
10–2
12 3
11 1
to encode the message COME_HERE.
A)
–23 –11 –5
72 29 56
31 13 28
B)
3 5 5
15 0 18
13 8 5
C)
19 27 –21
–14 –30 39
8 11 –13
D)
–7–21 3
28 69 44
13 33 26
15) Use the coding matrix A = 1 –4
–2 9 and its inverse A–1 = 94
21 to decode the cryptogram –7–8
16 21 .
A) ABLE B) ACTS C) ARMS D) ALAS
16) Use the coding matrix A = 21
53 and its inverse A–1 = 3 –1
–5 2 to decode the cryptogram 9 6
25 17 .
A) BEAD B) CARE C) DARE D) CURB
17) Use the coding matrix A =
1 1 1
–112
1 2 3
and its inverse A–1 =
–1–1 1
5 2 –3
–3–1 2
to decode the cryptogram
37 16 35
38 20 4
82 40 60
.
A) GOOD_LUCK B) STAY_CALM C) LOOK_DOWN D) HELP_THEM
8.5 Partial Fraction Decompositio
n
1 Decompose P/Q, Where Q Has Only Nonrepeated Linear Factors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Tell whether the given rational expression is proper or improper.
1) x2 – 12x + 35
(x2 – 8x + 15)(x + 5)
A) proper B) improper
2) x3 – 14x + 48
x2 – 11x + 30
A) improper B) proper
Page 44
Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial
and a proper rational expression.
3) x
7 – x
A) improper; –1 + 7
7 – x B) proper
C) improper; 1 + 7
7 – x D) improper; 1 – 7
7 – x
4) x – 4
x2 + 9
A) proper B) improper; x – 4
(x + 3)(x – 3)
C) improper; 4
x + 3 + 4
x – 3 D) improper; 4
x + 3 – 4
x – 3
Write the partial fraction decomposition of the rational expression.
5) x
(x – 2)(x – 3)
A) –2
x – 2 + 3
x – 3 B) 2
x – 2 + –3
x – 3 C) –3
x – 2 + 2
x – 3 D) –2
x – 2 + –3
x – 3
6) x
x2 + 7x + 12
A) 4
x + 4 + –3
x + 3 B) –4
x + 4 + 3
x + 3 C) 3
x + 4 + –4
x + 3 D) 4
x + 4 + 3
x + 3
7) x – 4
(x – 2)(x – 3)
A) 2
x – 2 + –1
x – 3 B) –1
x – 2 + 2
x – 3 C) 2
x – 2 + 1
x – 3 D) 1
x – 2 + –2
x – 3
8) 2x – 13
(x + 4)(x – 3)
A) 3
x + 4 – 1
x – 3 B) 3
x + 4 + 1
x – 3 C) 1
x – 3 – 3
x + 4 D) 2
x + 4 – 13
x – 3
9) 7x2 – x – 14
x(x + 1)(x – 1)
A) 14
x + –3
x + 1 + –4
x – 1 B) 14
x + 3
x + 1 + –4
x – 1 C) 14
x + –3
x + 1 + 4
x – 1 D) 14
x + –4
x + 1 + 3
x – 1
10) 12x2 + 162x + 384
(x + 8)(x + 2)(x + 11)
A) 8
x + 8 + 2
x + 2 + 2
x + 11 B) – 8
x + 8 – 2
x + 2 – 2
x + 11
C) – 8
x + 8 + 2
x + 2 + 2
x + 11 D) 8
x + 8 + 2
x + 2 – 2
x + 11
Page 45
11) 2x – 5
x2 – 5x – 6
A) 1
x – 6 + 1
x + 1 B) 1
x – 3 + 1
x – 2 C) 9
x + 2 + 1
x – 3 D) 17
x + 6 – 3
x – 1
12) 8x + 18
x2 + 4x + 3
A) 5
x + 1 + 3
x + 3 B) 5
x + 1 + –3
x + 3 C) 6
x + 1 + 2
x + 3 D) 5
x – 1 + 3
x – 3
13) 13x2 – 11x + 60
x(x + 6)(x – 5)
A) –2
x + 9
x + 6 + 6
x – 5 B) –2
x + 10
x + 6 + 5
x – 5 C) –2
x + 8
x + 6 + 7
x – 5 D) x +9
x2 + 6
+ 6
x – 5
2 Decompose P/Q, Where Q Has Repeated Linear Factors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the partial fraction decomposition of the rational expression.
1) 8x2 + 24x + 19
(x + 2)(x + 1)2
A) 3
x + 2 + 5
x + 1 + 3
(x + 1)2B) –3
x + 2 + 5
x + 1 + –3
(x + 1)2
C) 3
x + 2 + –5
x + 1 + 3
(x + 1)2D) 3
x + 2 + 5
x + 1 + –3
(x + 1)2
2) x + 2
x3 – 2x2 + x
A) 2
x + –2
x – 1 + 3
(x – 1)2B) 2
x + 3
x – 1 + –2
(x – 1)2
C) 2
x + –2
x – 1 + 5
(x – 1)2D) –2
x + 2
x – 1 + 3
(x – 1)2
3) 36 – 7x
x3 – 6x2 + 9x
A) 4
x + –4
x – 3 + 5
(x – 3)2B) 4
x + 5
x – 3 + –4
(x – 3)2
C) 4
x + –4
x – 3 + 10
(x – 3)2D) –4
x + 4
x – 3 + 5
(x – 3)2
Page 46
4) x + 1
(x – 2)2(x + 4)
A)
1
12
x – 2
+
1
2
(x – 2)2 +
– 1
12
x + 4
B)
1
2
(x – 2)2 +
– 1
12
x + 4
C)
–1
x – 2
+
1
4x
(x – 2)2 +
– 1
4
x + 4
D) 12
x – 2
+ 2
(x – 2)2 + –12
x + 4
5) 7x3 – 2
x2(x + 1)3
A) 6
x – 2
x2 – 6
x + 1 + 3
(x + 1)2 – 9
(x + 1)3B) 2
x2 – 6
x + 1 + 3
(x + 1)2 – 9
(x + 1)3
C) 6
x – 6
x + 1 + 3
(x + 1)2 – 9
(x + 1)3D) 6
x + 2
x2 + 6
x + 1 – 3
(x + 1)2 + 9
(x + 1)3
6) 3x + 9
(x – 6)2
A) 3
x – 6 + 27
(x – 6)2B) 3
x – 6 + 36
(x – 6)2C) 3
x – 6 + x +27
(x – 6)2D) 1
x – 6 + x +27
(x – 6)2
7) 4x2 – 4x + 4
(x – 1)3
A) 4
x – 1 + 4
(x – 1)2 + 4
(x – 1)3B) 4
x – 1 + 4
(x – 1)2
C) 4
x – 1 + x + 4
(x – 1)2 + x2 + 4
(x – 1)3D) 4
x – 1 – 4
(x – 1)2 + 4
(x – 1)3
3 Decompose P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the partial fraction decomposition of the rational expression.
1) 14x + 1
(x – 1)(x2 + x + 1)
A) 5
x – 1 + –5x + 4
x2 + x + 1
B) 5
x – 1 + –5
x + 1 + 4
x – 1
C) –5
x – 1 + 5x + 4
x2 + x + 1
D) 5
x – 1 + 4x –5
x2 + x + 1
Page 47
2) x2 – 111
x4 – x2 – 72
A) 1
x + 3 – 1
x – 3 + 7
x2 + 8
B) 1
x + 3 + 1
x – 3 – 7
x2 + 8
C) 1
x + 3 – 1
x – 3 – 7
x2 + 8
D) 1
x + 3 + 1
x – 3 + 7
x2 + 8
3) 3x – 2
x3 – 1
A)
1
3
x – 1
+
– 1
3x + 7
3
(x2 + x + 1)
B) 3
(x – 1)2 + 1
(x – 1)3
C) 3
x – 1
+ –3(x – 7)
x2 + x + 1
D)
1
2
x – 1
+
5
2
x + 1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
4) x2 – 56
x4 + 5x2 – 36
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
5) 12x + 3
x3 – 1
A) 5
x – 1 + –5x + 2
x2 + x + 1
B) 5
x – 1 + –5
x + 1 + 2
x – 1
C) –5
x – 1 + 5x + 2
x2 + x + 1
D) 5
x – 1 + 2x –5
x2 + x + 1
6) 8x2 + 37x + 16
(x + 6)(x2 + 5)
A) 2
x + 6 + 6x + 1
x2 + 5
B) 2
x + 6 + 6
x2 + 5
C) 2
x + 6 + 6x – 1
x2 + 5
D) 2
x + 6 + 6
x + 5 + 1
(x + 5)2
7) 9x2 + 20x – 13
x3 + 5x2 + 3x + 15
A) 4
x + 5 + 5x – 5
x2 + 3
B) 4
x + 5 + 5
x2 + 3
C) 4
x + 3 + 5x – 5
x2 + 5
D) 4
x + 5 + 5
x + 3 + –5
(x + 3)2
Page 48
8) 3x + 64
(x + 3)2(x2 + 2)
A) 3
x + 3 + 5
(x + 3)2 + –3x + 4
x2 + 2
B) 3
x + 3 + 5
(x + 3)2 + –3
x2 + 2
C) 3
x + 3 + 9
(x + 3)2 + –3x – 4
x2 + 2
D) 9x –4
(x + 3)2 + –3x +4
x2 + 2
4 Decompose P/Q, Where Q Has a Repeated Irreducible Quadratic Factor
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the partial fraction decomposition of the rational expression.
1) 4x3 + 2x2
(x2 + 5)2
A) 4x + 2
x2 + 5
+ –20x – 10
(x2 + 5)2B) 4x –2
x2 + 5
+ –20x +10
(x2 + 5)2C) 4x +2
x2 + 5
+ 20x +10
(x2 + 5)2D) 4x +2
x2 + 5
+ 20x –10
(x2 + 5)2
2) x2 + 2x – 2
(x2 + 3)2
A) 1
x2 + 3
+ 2x – 5
(x2 + 3)2B) x + 1
x2 + 3
+ 2x –5
(x2 + 3)2C) 1
x2 + 3
+ –x –5
(x2 + 3)2D) –1
x2 + 3
+ 2x –5
(x2 + 3)2
3) 3x3 + 18x – 4
(x2 + 5)2
A) 3x
x2 + 5
+ 3x – 4
(x2 + 5)2B) 3x +1
x2 + 5
+ 3x –4
(x2 + 5)2C) 3x
x2 + 5
+ –3x –4
(x2 + 5)2D) 3x
x2 + 5
+ 3x +4
(x2 + 5)2
4) 5x3 + 3x2 + 19x + 10
(x2 + 3)3
A) 5x + 3
(x2 + 3)2 + 4x + 1
(x2 + 3)3B) x +1
x2 + 3
+ 5x +3
(x2 + 3)2 + 4x + 1
(x2 + 3)3
C) 5x – 3
(x2 + 3)2 + 4x – 1
(x2 + 3)3D) x
x2 + 3
+ 5x +3
(x2 + 3)2 + 4x + 1
(x2 + 3)3
Page 49
8.6 Systems of Nonlinear Equations
1 Solve a System of Nonlinear Equations Using Substitution
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graph of two equations along with the points of intersection are given. Substitute the points of intersection
into the systems of equations. Are the points of intersection solutions to the system of equations (Y/N)?
1)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
35
30
25
20
15
10
5
-5
(1, 2)
(-3, 10)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
35
30
25
20
15
10
5
-5
(1, 2)
(-3, 10)
x2 = y –1
y = –2x + 4
A) Yes B) No
2)
x
–10–8-6-4-2 2 4 6 8 10
y
100
50
-50
(0.8, –3.4)
(-4.8, 41.4)
x
–10–8-6-4-2 2 4 6 8 10
y
100
50
-50
(0.8, –3.4)
(-4.8, 41.4)
2x2 = y + 5
y = –8x + 3
A) No B) Yes
Page 50
3)
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
y
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
(3, 4)
(-5, 0)
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
y
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
(3, 4)
(-5, 0)
x2 + y2 = 25
2y + x = 5
A) No B) Yes
4)
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
y
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
(4, –6)
(-4, 6)
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
y
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
(4, –6)
(-4, 6)
x2 + y2 = 52
2y+3x = 0
A) Yes B) No
Page 51
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Graph the equations of the system. Then solve the system to find the points of intersection.
5)
y = x2 – 10x + 25
y = –x + 11
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
6)
x2 + y2 = 64
x2 – y2 = 64
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 52
7)
x2 + y2 =64
y =x2 – 8
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
8)
y
= x
y
= 6 – x
x
510
y
10
5
x
510
y
10
5
Page 53
9)
(x – 2)2 + (y + 4)2 = 4
y2 + 8y – x + 14 = 0
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations using substitution.
10)
x2 + y2 = 113
x + y = –15
A) x = –7
,
y = –8; x = –8
,
y = –7
or (–7, –8), (–8, –7)
B) x =7
,
y = –8; x = 8
,
y = –7
or (7, –8), (8, –7)
C) x = –7
,
y = 8; x = –8
,
y = 7
or (–7, 8), (–8, 7)
D) x =7
,
y =8; x =8
,
y = 7
or (7, 8), (8, 7)
11)
xy = 6
x + y = –5
A) x = –2
,
y = –3; x = –3
,
y = –2
or (–2, –3), (–3, –2)
B) x =2
,
y = –3; x = 3
,
y = –2
or (2, –3), (3, –2)
C) x = –2
,
y = 3; x = –3
,
y = 2
or (–2, 3), (–3, 2)
D) x =2
,
y =3; x =3
,
y = 2
or (2, 3), (3, 2)
12)
x2 + y2 = 25
x – y = 1
A) x = –3
,
y = –4; x = 4
,
y = 3
or (–3, –4), (4, 3)
B) x =3
,
y = –4; x = 4
,
y = –3
or (3, –4), (4, –3)
C) x = –3
,
y = 4; x = –4
,
y = 3
or (–3, 4), (–4, 3)
D) x =3
,
y =4; x =4
,
y = 3
or (3, 4), (4, 3)
Page 54
13)
y = x2 – 12x + 36
x + y = 18
A) x = 2
,
y = 16; x = 9
,
y = 9
or (2, 16), (9, 9)
B) x = –2
,
y =20; x = –9
,
y = 27
or (–2, 20), (–9, 27)
C) x = 6
,
y = 12 or (6
,
12) D) x =2
,
y =20; x = 9
,
y = 9
or (2, 20), (9, 9)
14)
y = –x2 + 3
x2 + y2 = 5
A) x = 2
,
y = –1; x = 1
,
y = 2; x = –1
,
y =2; x = –2
,
y = –1
or (2, –1), (1, 2), (–1, 2), (–2, –1)
B) x = 1
,
y = 2; x = –1
,
y = 2
or (1, 2), (–1, 2)
C) x = 2
,
y = –1; x = –2
,
y = –1
or (2, –1), (–2, –1)
D) x = 1
,
y = 4; x = 4
,
y = 19
or (1, 4), (4, 19)
15)
x2 + y2 = 4
x + y = 2
A) x = 0, y = 2; x = 2, y = 0
or (0, 2), (2, 0)
B) x =0, y =0; x =2, y = –2
or (0, 0), (2, –2)
C) x = 0, y = –2; x = –2, y = 0
or (0, –2), (–2, 0)
D) x =2, y = –2; x = –2, y = –2
or (2,–2), (–2, –2)
16)
xy = 20
x + y = 9
A) x = 5, y = 4; x = 4, y = 5
or (5, 4), (4, 5)
B) x =6, y =3; x =3, y = 6
or (6, 3), (3, 6)
C) x = 10, y = 2; x = 2, y =10
or (10, 2), (2, 10)
D) x =20, y =1; x = 1, y = 20
or (20, 1), (1, 20)
17)
x2 + y2 = 169
x + y = 17
A) x = 12, y = 5; x = 5, y = 12
or (12, 5), (5, 12)
B) x = –12, y =5; x = –5, y = 12
or (–12, 5), (–5, 12)
C) x = 12, y = –5; x = 5, y = –12
or (12, –5), (5, –12)
D) x = –12, y = –5; x = –5, y = –12
or (–12, –5), (–5, –12)
Page 55
18)
xy – x2 = –20
x – 2y = 3
A) x = 5, y = 1; x = –8, y = – 11
2
or (5, 1), –8, – 11
2
B) x = –5, y = –1; x = 8, y = 11
2
or (–5, –1), 8,
11
2
C) x = 5, y = 1; x = – 11
2, y = –8
or (5, 1), – 11
2, –8
D) x = –5, y = –1; x = 11
2, y = 8
or (–5, –1), – 11
2, 8
19)
x2 – y2 = 39
x – y = 3
A) x = 8, y = 5 or (8, 5) B) x =8, y = –5 or (8, –5)
C) x = –8, y = 5 or (–8, 5) D) x = –8, y = –5 or (–8, –5)
20)
y =6x2 – 5x
y =2x + 3
A) x = 3
2, y = 6; x = – 1
3, y = 7
3
or 3
2, 6 ), – 1
3, – 7
3
B) x = 1
3, y = 11
3; x = – 3
2, y = 0
or 1
3, 11
3, – 3
2, 0
C) x = 1
6, y = 10
3; x = 1, y = 5
or 1
6, 10
3, (1, 5)
D) x = – 1
2, y = 2; x = 1, y = 5
or – 1
2, 2 , (1, 5)
21)
ln x =3ln y
3x =27y
A) x = 33, y = 3 or (3 3, 3)B)x
= 3, y = 33 or ( 3, 3 3)
C) x = 9, y = 3
or (9, 3)D)x
= 3, y = 9 or ( 3, 9)
22)
–5x – y = 8
y = x2 – 2
A) x = –2
,
y = 2; x = –3
,
y = 7
or (–2, 2), (–3, 7)
B) x =2
,
y =2; x =3
,
y = 7
or (2, 2), (3, 7)
C) x = –2
,
y = 6; x = –3
,
y = 11
or (–2, 6), (–3, 11)
D) x =2
,
y =2; x = –3
,
y = 7
or (2, 2), (–3, 7)
Page 56
23)
y = x + 4
y2 = 16x
A) x = 4
,
y = 8 or (4
,
8) B) x =4
,
y = 8; x = –4
,
y = 0
or (4, 8), (–4, 0)
C) x = 4
,
y = 8; x = 4
,
y = –8
or (4, 8), (4, –8)
D) x =4
,
y = 8; x =4
,
y = –8; x = –4
,
y = 0
or (4, 8), (4, –8), (–4, 0)
24)
xy = 1
–40x – y = 13
A) x = – 1
8 , y = –8; x = – 1
5, y = –5
or – 1
8, –8 , – 1
5, –5
B) x = –8 , y = – 1
8; x = –5, y = – 1
5
or –8, – 1
8, –5, – 1
5
C) x = 8 , y = –8; x = 5
,
y = –5
or (8, –8), (5, –5)
D) x = – 1
5 , y = –5 or – 1
5, –5
25)
7x2 + 5y2 = 80
y = x + 4
A) x = 0, y = 4; x = – 10
3, y = 2
3
or 0, 4 , – 10
3, 2
3
B) x = 0, y = –4; x = – 10
3, y = 2
3
or 0,
–4 , – 10
3, 2
3
C) x = 0, y = 4; x = 10
3, y = 22
3
or 0, 4 , 10
3, 22
3
D) x = 0, y = –4; x = 10
3, y = 22
3
or 0,
–4 , 10
3, 22
3
26)
xy = 30
x2 + y2 = 61
A) x = 6
,
y = –6; x = –6
,
y = –5; x = 5
,
y =6; x = –5
,
y = –6
or (6, –6), (–6, –5), (5, 6), (–5, –6)
B) x = 6
,
y = 5; x = –6
,
y = –5; x = 6
,
y = –5; x = –6
,
y =5
or (6, 5), (–6, –5), (6, –5), (–6, 5)
C) x = 6
,
y = 5; x = 5
,
y = 6; x = 6
,
y = –5; x =5
,
y = –6
or (6, 5), (5, 6), (6, –5), (5, –6)
D) x = –6
,
y = –5; x = –5
,
y = –6; x = –6
,
y =5; x = –5
,
y =6
or (–6, –5), (–5, –6), (–6, 5), (–5, 6)
Page 57
27)
y = (x + 5)2 + 1
2x – y + 10 = 0
A) x = –4, y = 2 or (–4, 2) B) x = –4, y =2; x =4, y = 18
or (–4, 2), (4, 18)
C) x = 0, y = 10; x = 0, y = 26
or (0, 10), (0, 26)
D) x = –5, y =0 or (–5, 0)
28)
2y – x = 10
x2 + y2 – 100 = 0
A) x = –10
,
y = 0; x = 6
,
y = 8
or (–10, 0), (6, 8)
B) x =10
,
y =0; x = –10
,
y = 0; x = 6
,
y =8
or (10, 0), (–10, 0), (6, 8)
C) x = 0, y = 5; x = 8
,
y = 9
or 0, 5 , 8, 9
D) x =0, y =5; x =0, y = – 5; x = 8
,
y =9
or 0, 5 , 0,
– 5 , 8, 9
29)
y = x2 – 4x – 4
y = –x2 – 10x + 4
A) x = 1
,
y = –7; x = –4
,
y = 28
or (1, –7), (–4, 28)
B) x = –1
,
y =1; x =4
,
y = –4
or (–1, 1), (4, –4)
C) x = 1
,
y = –7; x = 0, y = –4
or (1, –7), (0, –4)
D) x = –1
,
y =1; x =0, y = –4
or (–1, 1), (0, –4)
30)
x + y = 8
x2 + y2 = –16y + 96
A) x = 12
,
y = –4; x = 4
,
y = 4
or (12, –4), (4, 4)
B) x = –4
,
y =12; x = 4
,
y = 4
or (–4, 12), (4, 4)
C) x = 4
,
y = 4; x = 12
,
y = –4
or (4, 4), (12, –4)
D) x =4
,
y =4; x = –4
,
y = 12
or (4, 4), (–4, 12)
2 Solve a System of Nonlinear Equations Using Elimination
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve using elimination.
1)
x2 + y2 = 61
x2 – y2 = 11
A) x = 6
,
y = 5; x = –6
,
y = 5; x = 6
,
y = –5; x = –6
,
y = –5
or (6, 5), (–6, 5), (6, –5), (–6, –5)
B) x = 6
,
y = 5; x = 5
,
y = 6; x = –6
,
y = –5; x = –5
,
y = –6
or (6, 5), (5, –6), (–6, –5), (–5, –6)
C) x = 6
,
y = –5; x = 6
,
y = 5 or (6
,
–5), (6
,
5)
D) x = –6
,
y = –5; x = –5
,
y = –6 or (–6
,
–5), (–5
,
–6)
Page 58
2)
2x2 – 5y2 = –37
3x2 + 5y2 = 57
A) x = 2
,
y = 3; x = –2
,
y = 3 ; x = 2
,
y = –3; x = –2
,
y = –3
or (2, 3), (–2, 3), (2, –3), (–2, –3)
B) x = 2
,
y = 3; x = 3
,
y = 2; x = –2
,
y = –3; x = –3
,
y = –2
or (2, 3), (3, 2), (–2, –3), (–3, –2)
C) x = 2
,
y = –3; x = 2
,
y = 3 or (2
,
–3), (2
,
3)
D) x = –2
,
y = –3; x = –3
,
y = –2 or (–2
,
–3), (–3
,
–2)
3)
x2 + y2 = 49
x2 – y2 = 49
A) x = 7
,
y = 0; x = –7
,
y = 0 or (7
,
0), (–7
,
0) B) x =7
,
y =7; x = –7
,
y = 7 or (7
,
7), (–7
,
7)
C) x = 7
,
y = 0; x = 7
,
y = 7 or (7
,
0), (7
,
7) D) x = –7
,
y =0; x = –7
,
y = 7 or (–7
,
0), (–7
,
7)
4)
x2 + y2 = 64
x2
64 + y2
9 = 1
A) x = –8, y= 0; x = 8, y = 0 or (–8, 0), (8, 0) B) x =0, y= –8; x =0, y = 8 or (0, –8), (0, 8)
C) x = 0, y= –3; x = 0, y = 3 or (0, –3), (0, 3) D) No real solution exists.
5)
3x2 + 2y2 = 89
x2 – 2y2 = –21
A) x = 17, y = 19; x = – 17, y = 19; x = ,17, y = – 19 ; x = – 17, y = – 19
or ( 17, 19), (– 17, 19), ( 17, – 19), (– 17, – 19)
B) x = 19, y = 17; x = – 19, y = 17; x = ,19, y = – 17 ; x = – 19, y = – 17
or ( 17, 19), (– 17, 19), ( 17, – 19), (– 17, – 19)
C) x = 17, y = 19; x = ,17, y = – 19 ;
or ( 17, 19), ( 17, – 19)
D) x = – 17, y = 19; x = – 17, y = – 19
or (– 17, 19), (– 17, – 19)
6)
2x2 + y2 = 17
3x2 – 2y2 = –6
A) x = 2, y = 3; x = 2, y = –3; x = –2, y =3; x = –2, y = –3
or (2, 3), (2, –3), (–2, 3), (–2, –3)
B) x= 1, y = 3; x = 1, y = –3; x = –1, y =3; x = –1, y = –3
or (1, 3), (1, –3), (–2, 3), (–2, –3)
C) x = 2, y = –3; x = –2, y = 3 or (2, –3), (–2, 3)
D) x = 1, y = 3; x = –1, y= –3 or (1, 3), (–1, –3)
Page 59
7)
2x2 + xy – y2 = 3
x2 + 2xy + y2 = 3
A) x = 23
3, y = 3
3; x = – 23
3, y = – 3
3
or 23
3, 3
3, – 23
3, – 3
3
B) x = 23
3, y = – 3
3; x = – 23
3, y = 3
3
or 23
3,– 3
3, – 23
3, 3
3
C) x = –2
3, y = 1
3; x = 23
3, y = – 1
3
or –2
3, 1
3, 23
3, – 1
3
D) x = –2
3, y= – 1
3; x = 23
3, y = 1
3
or –2
3, – 1
3, 23
3, 1
3
Use a graphing utility to solve the system of equations. Express the solution rounded to two decimal places.
8)
3x3 + y2 = 6
x4y = 2
A) x = 1.18, y = 1.03; x = 1.07, y = 1.52; x = –0.91, y =2.88
or (1.18, 1.03), (1.07, 1.52), (–0.91,2.88)
B) x = 1.99, y = 0.13; x = 1.04, y = 1.70; x = –0.91, y =2.95
or (1.99, 0.13), (1.04, 1.70), (–0.91, 2.95)
C) x = 1.81, y = 0.28; x = 0.20, y = 2.45; x = –0.20, y = –2.45
or (1.81, 0.28), (0.20, 2.45), (–0.20, –2.45)
D) No real solution exists.
9)
x3 + y2 =2
x2y =4
A) x = –1.37, y = 2.14 or (–1.37, 2.14) B) x =2.14, y =1.37 or (2.14, 1.37)
C) x = 1.37, y = 2.14 or (1.37, 2.14) D) x =2.14, y = –1.37 or (2.14, –1.37)
10)
y = x–4/3
y = ex
A) x = 0.64, y = 1.89 or (0.64, 1.89) B) x =0.48, y =1.63 or (0.48, 1.63)
C) x = 0.22, y = 0.98 or (0.22, 0.98) D) x =1.14, y =0.81 or (1.14, 0.81)
11)
x2 + y2 = 25
y = –ln x
A) x = 4.76, y = –1.52; x = 0.01, y = 5.00
or (4.76, –1.52), (0.01, 5.00)
B) x = –4.76, y =1.52; x = –0.01, y =5.00
or (–4.76, 1.52), (–0.01, 5.00)
C) x = 5.12, y = –3.67; x = 0.76, y = 2.05
or (5.12, –3.67), (0.76, 2.05)
D) x =5.12, y =3.67; x = –0.76, y =2.05
or (5.12, 3.67), (–0.76, 2.05)
Solve the problem.
12) The sum of the squares of two numbers is 85. The sum of the two numbers is –1. Find the two numbers.
A) –7 and 6 B) –7 and 6 or –6 and 7
C) –6 and 7 D) 6 and 7 or –7 and –6
Page 60
13) The sum of the squares of two numbers is 34. The difference of the two numbers is –2. Find the two
numbers.
A) 3 and 5 or –5 and –3 B) 3 and 5
C) –5 and –3D)
–5 and 3 or –3 and 5
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) The difference of two numbers is 5 and the difference of their squares is 55. Find the numbers.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
15) A right triangle has an area of 36 square inches. The square of the hypotenuse is 145. Find the lengths of
the legs of the triangle. Round your answer to the nearest inch.
A) 8 in. and 9 in. B) 64 in. and 81 in. C) 4 in. and 18 in. D) 16 in. and 4.5 in.
16) The perimeter of a rectangle is 34 inches and its area is 66 square inches. What are its dimensions?
A) 6 in. by 11 in. B) 5 in. by 12 in. C) 7 in. by 10 in. D) 5 in. by 10 in.
17) A system for tracking ships indicated that a ship lies on a hyperbolic path described by 5x2 – y2 = 80. The
process is repeated and the ship is found to lie on a hyperbolic path described by y2 – 2x2 = 28. If it is
known that the ship is located in the first quadrant of the coordinate system, determine its exact location.
A) (6
,
10) B) (–6
,
–10) C) (10
,
6) D) (–10
,
–6)
18) Find the dimensions of a rectangle whose perimeter is 32 feet and whose area is 55 square feet.
A) 5 ft by 11 ft B) 4 ft by 12 ft C) 6 ft by 10 ft D) 4 ft by 10 ft
19) The area of a garden is 4860 square feet, and the length of its diagonal is 117 feet. Find the dimensions of
the garden.
A) 45 ft by 108 ft B) 5 ft by 972 ft C) 405 ft by 12 ft D) 60 ft by 81 ft
20) The area of a rectangular piece of cardboard shown is 585 square inches. The cardboard is used to make an
open box by cutting a 4–inch square from each corner and turning up the sides. If the box is to have a
volume of 868 cubic inches, find the dimensions of the cardboard that must be used.
A) 15 in. by 39 in. B) 19 in. by 43 in. C) 11 in. by 35 in. D) 7 in. by 27 in.
21) A rectangular piece of tin has an area of 874 square inches. A square of 3 inches is cut from each corner,
and an open box is made by turning up the ends and sides. If the volume of the box is 1632 cubic inches,
what were the original dimensions of the piece of tin?
A) 23 in. by 38 in. B) 26 in. by 41 in. C) 20 in. by 35 in. D) 17 in. by 29 in.
22) The diagonal of the floor of a rectangular office cubicle is 2 ft longer than the length of the cubicle and 5 ft
longer than twice the width. Find the dimensions of the cubicle. Round to the nearest tenth, if necessary.
A) width = 9.7 ft, length = 22.4 ft B) width =4 ft, length = 11 ft
C) width = 2 ft, length = 9 ft D) width =3.9 ft, length = 9.7 ft
Page 61
23) In a 1–mile race, the winner crosses the finish line 13 feet ahead of the second–place runner and 33 feet
ahead of the third–place runner. Assuming that each runner maintains a constant speed throughout the
race, by how many feet does the second–place runner beat the third–place runner? (5280 feet in 1 mile.)
A) 20.05 ft B) –13.08 ft C) –20.13 ft D) 7.02 ft
24) A person at the top of a 600 foot tall building drops a yellow ball. The height of the yellow ball is given by
the equation h = –16t2 + 600 where h is measured in feet and t is the number of seconds since the yellow
ball was dropped. A second person, in the same building but on a lower floor that is 152 feet from the
ground, drops a white ball 4 seconds after the yellow ball was dropped. The height of the white ball is
given by the equation h = –16(t – 4)2 + 152 where h is measured in feet and t is the number of seconds
since the yellow ball was dropped. Find the time that the balls are the same distance above the ground and
find this distance.
A) 5.5 sec; 116 ft B) 4.5 sec; 276 ft C) 5 sec; 200 ft D) 6 sec; 24 ft
Page 62
8.7 Systems of Inequalities
1 Graph an Inequality by Hand
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the inequality.
1) x > 8
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 63
2) y ≤ 9
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 64
3) x – y > –4
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 65
4) x + y
<
–4
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 66
5) x – y
<
–5
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 67
6) 2x + y ≤ 3
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 68
7) x + 5y ≥ –5
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 69
8) –2x – 3y ≤ 6
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 70
9) 3x + 4y ≤ 12
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 71
10) –2x – 4y ≤ –8
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 72
11) y ≥ x – 6
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 73
12) y + 4
<
x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 74
13) y ≤ 5x + 2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 75
14) x2 + y2 ≤ 49
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 76
15) x2 + y2 > 49
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 77
16) y > x2 + 7
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 78
17) y ≤ x2 – 5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 79
18) xy ≤ 3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 80
2 Graph an Inequality Using a Graphing Utility
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the inequality using a graphing utility.
1) x + y ≤ 5
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) B)
C) D)
Page 81
2) x + 2y ≥ 4
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) B)
C) D)
Page 82
3 Graph a System of Inequalities
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the solution set of the system of inequalities or indicate that the system has no solution.
1) y > 2
x ≥ 1
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page 83
2) –x + 2y ≤ –6
3x + 2y > –18
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 84
3) 2x + 3y ≤ 6
x – y ≤ 3
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 85
4) 4x + 3y ≥ 12
x ≥ y
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
Page 86
5) –2x + y > 2
–2x + y < 1
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
A)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
B)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
C)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
D) No solution
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
Page 87
6) 3x + y
<
3
3x + y > –1
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
D) No solution
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 88
7) 2x – y ≤ 6
x + 4y ≥ 8
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 89
8) y
<
–x + 7
y > 5x – 3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
–20 –10 10 20
y
20
10
-10
-20
x
–20 –10 10 20
y
20
10
-10
-20
D)
x
–20 –10 10 20
y
20
10
-10
-20
x
–20 –10 10 20
y
20
10
-10
-20
Page 90
9) x + 2y ≥ 2
x – y ≤ 0
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page 91
10) –x + 5y
<
–5
x ≥ 3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 92
11) –x + y
<
1
2x + 2y > 6
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 93
12) x + 9y ≤ 9
x + 9y ≥ 0
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) No solution
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 94
13) 9x + y ≥ 9
9x + y ≥ 0
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) No solution
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 95
14) x2 + y2 ≤ 49
–10x + 6y ≤ –60
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 96
15) x2 + y2 ≤ 81
x2 + y2 ≥ 9
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) No solution
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 97
16) y > x2
4x + 4y ≤ 16
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 98
17) y ≥ x2
x + y > 2
x
–6–5–4–3–2–1 123456
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
–6–5–4–3–2–1 123456
y
5
4
3
2
1
-1
-2
-3
-4
-5
A)
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
–7–6–5–4–3–2–1 1234567
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
–7–6–5–4–3–2–1 1234567
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
–14–12–10–8–6–4–2 2468101214
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–14–12–10–8–6–4–2 2468101214
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
-12 12
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-12 12
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 99
18) x2 + y ≤ 2
x2 – y ≤ 4
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Page 100
19) x2 + y2 ≤ 100
–3x + 5y ≤ –15
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 101
20) x2 + y2 ≤ 81
y – x2 > 0
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
–10–8-6-4–2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4–2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 102
21) x2 + y2 ≤ 36
x + y > 1
x
–10–8-6-4-2 2 4 6 8 10
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
x
–10–8-6-4-2 2 4 6 8 10
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
A)
x
–10–8–6–4–2 246810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
x
–10–8–6–4–2 246810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
B)
x
–10–8-6-4-2 2 4 6 8 10
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
x
–10–8-6-4-2 2 4 6 8 10
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
C)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
Page 103
22) xy
<
6
y ≤ x2 – 3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Graph the system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner
points.
23)
x ≥ 0
y ≥ 0
x + y ≤ 6
x + y ≥ 1
Page 104
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) bounded;
corner points (6, 0), (0, 6), (0, 1), (1, 0)
x
y
12
-12
(0, 6)
(6, 0)
(1, 0)
(0, 1)
x
y
12
-12
(0, 6)
(6, 0)
(1, 0)
(0, 1)
B) unbounded;
corner points (0, 0), (0, 1), (1, 0)
x
y
12
-12
(0, 6)
(6, 0)
(1, 0)
(0, 1)
x
y
12
-12
(0, 6)
(6, 0)
(1, 0)
(0, 1)
C) unbounded;
corner points (6, 0), (0, 6)
x
y
12
-12
(0, 6)
(6, 0)
x
y
12
-12
(0, 6)
(6, 0)
D) no solution
x
y
12
-12
(0, 6)
(6, 0)
(1, 0)
(0, 1)
x
y
12
-12
(0, 6)
(6, 0)
(1, 0)
(0, 1)
Page 105
24)
x ≥ 0
y ≥ 0
x + y ≥ 3
x + 2y ≥ 6
A) unbounded;
corner points (0, 3), (6, 0)
x
-5 5 10
y
10
5
-5
-10
(0, 3)
(6, 0)
x
-5 5 10
y
10
5
-5
-10
(0, 3)
(6, 0)
B) bounded;
corner points (0, 0), (0, 3), (3, 0)
x
-10 -5 5 10
y
10
5
-5
-10
(0, 3)
(3, 0)
(0, 0)
x
-10 -5 5 10
y
10
5
-5
-10
(0, 3)
(3, 0)
(0, 0)
C) bounded;
corner points (3, 0), (0, 3), (6, 0)
x
-10 -5 5 10
y
10
5
-5
-10
(3, 0)
(0, 3)
(6, 0)
x
-10 -5 5 10
y
10
5
-5
-10
(3, 0)
(0, 3)
(6, 0)
D) unbounded;
corner points (0, 3), (3, 0)
x
-10 -5 5 10
y
10
5
-5
-10
(0, 3)
(3, 0)
x
-10 -5 5 10
y
10
5
-5
-10
(0, 3)
(3, 0)
Write a system of linear inequalities that has the given graph.
25)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
y ≥ 0
x ≥ 0
x ≤ 4
y + x ≤ 6
B)
y ≥0
x ≥ 0
x ≤ 6
y + x ≤ 4
C) x ≤4
y + x ≤ 6 D)
y ≥0
x ≥ 0
x ≤ 4
y + x ≥ 6
Page 106
26)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
y ≥ 0
x ≥ 0
x ≤ 9
y ≤ 8
x + y ≥ 3
B)
y ≥0
x ≥ 0
y ≤ 8
x + y ≥ 3
C)
y ≥0
x ≥ 0
x ≤ 8
x + y ≥ 3
D)
x ≤9
y ≤ 8
x + y ≥ 3
27)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x ≥ 0
y ≥ 3
y ≤ x
y + x ≤ 9
B)
y ≥0
x ≥ 0
y ≤ 3
y ≤ x
y + x ≤ 9
C)
y ≥3
y ≥ x
y + x ≤ 9
D)
y ≥0
x ≥ 0
y ≤ 3
y + x ≤ 9
Page 107
28)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
y ≥ 0
y ≤ 9
y ≤ x – 5
y ≥ 5 – x
B)
x ≥0
y ≥ 0
y ≤ 9
y ≤ x – 5
y ≤ 5 – x
C)
x ≥0
y ≤ 9
y ≤ x – 5
y ≥ 5 – x
D)
y ≥0
x ≤ 9
y ≤ x – 5
y ≥ 5 – x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
29) A coffee store has available 75 pounds of A grade coffee and 120 pounds of B grade coffee. These will be
blended into 1 pound packages as follow: an economy blend that contains 4 ounces of A grade coffee and
12 ounces of B grade coffee and a superior blend that contains 8 ounces of A grade coffee and 8 ounces of
B grade coffee. Using x to denote the number of packages of the economy blend and y to denote the
number of packages of the superior blend, write a system of linear inequalities that describes the possible
number of packages of each blend. Graph the system and label the corner points.
x
y
x
y
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
30) A person with no more than $7000 to invest plans to place the money in two investments,
telecommunications and pharmaceuticals. The telecommunications investment is to be no more than 2
times the pharmaceuticals investment. Write a system of inequalities to describe the situation. Let
x = amount to be invested in telecommunications and y = amount to be invested in pharmaceuticals.
A)
x + y ≤ 7000
x ≤ 2y
x ≥ 0
y ≥ 0
B)
x +y =7000
x ≤ 2y
x ≥ 0
y ≥ 0
C)
x +y =7000
x ≥ 2y
x ≥ 0
y ≥ 0
D)
x +y ≤7000
2x ≤ y
x ≥ 0
y ≥ 0
Page 108
31) A man is planting a section of garden with tomatoes and cucumbers. The available area of the section is
120 square feet. He wants the area planted with tomatoes to be more than 35% of the area planted with
cucumbers. Write a system of inequalities to describe the situation. Let x = amount to be planted in
tomatoes and y = amount to be planted in cucumbers.
A)
x + y ≤ 120
x > 0.35y
x ≥ 0
y ≥ 0
B)
x +y =120
x ≥ 0.35y
x ≥ 0
y ≥ 0
C)
x +y ≤120
x > 35y
x ≥ 0
y ≥ 0
D)
x +y ≤120
x < 0.35y
x ≥ 0
y ≥ 0
8.8 Linear Programming
1 Set up a Linear Programming Problem
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Set up the linear programming problem.
1) A steel company produces two types of machine dies, part A and part B. The company makes a $3.00
profit on each part A that it produces and a $5.00 profit on each part B that it produces. Let x = the number
of part A produced in a week and y = the number of part B produced in a week. Write the objective
function that describes the total weekly profit.
A) z = 3x + 5y B) z =5x +3y
C) z = 8(x + y) D) z =3(x –5) +5(y – 3)
2) A dietitian needs to purchase food for patients. She can purchase an ounce of chicken for $0.25 and an
ounce of potatoes for $0.02. Let x = the number of ounces of chicken and y = the number of ounces of
potatoes purchased per patient. Write the objective function that describes the total cost per patient per
meal.
A) z = 0.25x + 0.02y B) z = 0.02x +0.25y C) z =25x +2y D) z =2y +25y
3) A steel company produces two types of machine dies, part A and part B and is bound by the following
constraints:
· Part A requires 1 hour of casting time and 10 hours of firing time.
· Part B requires 4 hours of casting time and 3 hours of firing time.
· The maximum number of hours per week available for casting and firing are 100 and 70, respectively.
· The cost to the company is $0.75 per part A and $3.00 per part B. Total weekly costs cannot exceed
$45.00.
Let x = the number of part A produced in a week and y = the number of part B produced in a week. Write
a system of three inequalities that describes these constraints.
A)
x +4y ≤100
10x +3y ≤70
0.75x +3y ≤45
B)
x +10y ≤100
4x +3y ≤70
0.75x +3y ≤45
C)
x +10y ≥100
4x +3y ≥70
0.75x +3y ≤45
D)
x +4y ≤100
10x +3y ≤70
3x +0.75y ≤45
Page 109
4) A dietitian needs to purchase food for patients. She can purchase an ounce of chicken for $0.25 and an
ounce of potatoes for $0.02. The dietician is bound by the following constraints.
· Each ounce of chicken contains 13 grams of protein and 24 grams of carbohydrates.
· Each ounce of potatoes contains 5 grams of protein and 35 grams of carbohydrates.
· The minimum daily requirements for the patients under the dietitian’s care are 45 grams of protein and
58 grams of carbohydrates.
Let x = the number of ounces of chicken and y = the number of ounces of potatoes purchased per patient.
Write a system of inequalities that describes these constraints.
A)
13x +5y ≥45
24x +35y ≥58
B)
13x +24y ≥45
5x +35y ≥58
C)
13x +5y ≥58
24x +35y ≥45
D)
13x +24x ≥45
5y +35y ≥58
5) Mrs. White wants to crochet hats and afghans for a church fundraising bazaar. She needs 8 hours to make
a hat and 3 hours to make an afghan, and she has no more than 58 hours available. She has material for no
more than 11 items, and she wants to make at least two afghans. Let x = the number of hats she makes and
y = the number of afghans she makes. Write a system of inequalities that describes these constraints.
A) 8x + 3y ≤ 58
x + y ≤ 11
y ≥ 2
B) 8x +3y ≤58
x + y ≤ 11
x ≥ 2
C) 3x +8y ≤58
x + y ≤ 11
x ≤ 2
D) 8x +3y ≥58
x + y ≤ 11
y ≥ 2
6) An office manager is buying used filing cabinets. Small file cabinets cost $4 each and large file cabinets cos
t
$8 each, and the manager cannot spend more than $72 on file cabinets. A small cabinet takes up 7 square
feet of floor space and a large cabinet takes up 8 square feet, and the office has no more than 102 square
feet of floor space available for file cabinets. The manager must buy at least 7 file cabinets in order to get
free delivery. Let x = the number of small file cabinets bought and y = the number of large file cabinets
bought. Write a system of inequalities that describes these constraints.
A) 4x + 8y ≤ 72
7x + 8y ≤ 102
x + y ≥ 7
B) 4x +8y ≤72
8x + 7y ≤ 102
x ≥ 7
C) 4x +8y ≤72
7x + 8y ≤ 102
x + y ≤ 7
D) 4x +8y ≤72
7x + 8y ≤ 102
y ≥ 7
Page 110
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
7) The Jillson’s have up to $75,000 to invest. They decide that they want to have at least $40,000 invested in
stable bonds yielding 6% and that no more than $20,000 should be invested in more volatile bonds
yielding 12%.
(a) Using x to denote the amount of money invested in the stable bonds and y the amount invested in the
more volatile bonds, write a system of linear inequalities that describes the possible amounts of each
investment.
(b) Graph the system and label the corner points.
x
y
x
y
8) Charlene baby–sits for $4 per hour. She also works as a tutor for $7 per hour. Because of school, her
parents only allow her to work 13 hours per week. How many hours can Charlene tutor and baby–sit and
still make at least $50 per week?
(a) Let x = hours spent baby–sitting and let y = hours spent tutoring. Write a system of inequalities for this
situation.
(b) Graph the solution set.
x
4 8 12 16 20
y
16
12
8
4
x
4 8 12 16 20
y
16
12
8
4
9) Eric’s Carpentry manufactures two types of bookshelves that are 4 feet tall and 3 feet wide, a basic model
and a deluxe model. Each basic bookshelf requires 1.5 hours for assembly and 1 hour for finishing; each
deluxe model requires 2.5 hours for assembly and 1 hour for finishing. Two assemblers and one finisher ar
e
employed by the company, and each works 40 hours per week.
(a) Using x to denote the number of basic bookcases and y to denote the number of deluxe bookcases,
write a system of linear inequalities that describes the possible number of each model of bookcase that can
be manufactured in a week.
(b) Graph the system and label the corner points.
Page 111
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10) The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin
C daily. A cup of dietary drink X provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A
cup of dietary drink Y provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Set up a system
of linear inequalities that describes the minimum daily requirements for calories and vitamins. Let
x = number of cups of dietary drink X, and y = number of cups of dietary drink Y. Write all the constraints
as a system of linear inequalities.
A)
60x + 60y ≥ 300
12x + 6y ≥ 36
10x + 30y ≥ 90
x ≥ 0
y ≥ 0
B)
60x + 60y > 300
12x + 6y > 36
10x + 30y > 90
x > 0
y > 0
C)
60x + 60y ≥ 300
12x + 6y > 36
10x + 30y ≥ 90
D)
60x + 60y ≤ 300
12x + 6y ≤ 36
10x + 30y ≤ 90
2 Solve a Linear Programming Problem
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the maximum or minimum value of the given objective function of a linear programming problem. The figure
illustrates the graph of the feasible points.
1) z = 3x + 6y. Find maximum and minimum.
x
y
(0, 6) (7, 6)
(7, 1)
(
2
,
0
)
(0, 2)
x
y
(0, 6) (7, 6)
(7, 1)
(
2
,
0
)
(0, 2)
A) maximum value: 57; minimum value: 6 B) maximum value: 57; minimum value: 12
C) maximum value: 27; minimum value: 6 D) maximum value: 27; minimum value: 12
Find the maximum or minimum value of the given objective function of a linear programming problem. The figure
illustrates the graph of feasible points.
2) z = x + 8y. Find maximum.
A) maximum: 42 B) maximum: 27 C) maximum: 20 D) maximum: 34
3) z = –x – 7y. Find maximum.
A) maximum: –18 B) maximum: –24 C) maximum: –37 D) maximum: –30
Page 112
4) z = 5x + 6y. Find minimum.
A) minimum: 32 B) minimum: 33 C) minimum: 40 D) no minimum
5) z = x + 5y + 5. Find minimum.
A) minimum: 19 B) minimum: 32 C) minimum: 24 D) no minimum
6) z = –6x – y. Find maximum.
A) maximum: –17 B) maximum: –21 C) maximum: –26 D) no maximum
Solve the linear programming problem.
7) Maximize and minimize z = 13x + 2y subject to:
x ≥ 0, y ≥ 0, 2x + 3y ≥ 6, x ≤ 10, y ≤ 5.
A) maximum: 140; minimum: 4 B) maximum: 130; minimum: 6
C) maximum: 10; minimum: 6 D) maximum: 140; minimum: 130
8) Minimize z = 20x + 6y + 10 subject to:
x ≥ 0, y ≥ 0, x + y ≥ 1.
A) minimum: 16 B) minimum: 30 C) minimum: 36 D) minimum: 10
9) Maximize and minimize z = 9x – 14y subject to:
x ≥ 0, y ≥ 0, 4x + 5y ≤ 30, 4x + 3y ≤ 20, x ≤ 5, y ≤ 8.
A) maximum: 45; minimum: –84 B) maximum: 45; minimum: 0
C) maximum: –58.75; minimum: –84 D) maximum: –84; minimum: 0
An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed
region. Use these values to determine the maximum value of the objective function and the values of x and y for
which the maximum occurs.
10) Objective Function z = 25x + 10y
Constraints 0 ≤ x ≤ 10
0 ≤ y ≤ 5
3x + 2y ≥ 6
A) maximum: 300; at (10, 5) B) maximum: 250; at (10, 0)
C) maximum: 50; at (0, 5) D) maximum: 30; at (0, 3)
11) Objective Function z = 6x + 7y
Constraints x ≥ 0
y ≥ 0
2x + 3y ≤ 12
2x + y ≤ 8
A) maximum 32; at (3, 2) B) maximum 32; at (2, 3)
C) maximum 52; at (4, 4) D) maximum 24; at (4, 0)
12) Objective Function z = 3x + 5y
Constraints x ≥ 0
y ≥ 0
2x + y ≤ 15
x – 3y ≥ –3
A) maximum 33; at (6, 3) B) maximum 75; at (0, 15)
C) maximum 22.5; at (7.5, 0) D) maximum 38; at (6, 4)
Page 113
13) Objective Function z = 14x – 14y
Constraints 0 ≤ x ≤ 5
0 ≤ y ≤ 8
4x + 5y ≤ 30
4x + 3y ≤ 20
A) maximum: 70; at (5, 0) B) maximum: 0; at (0, 0)
C) maximum: –52.5; at (1.25, 5) D) maximum: –84; at (0, 6)
14) Objective Function z = 5x + 5y
Constraints x ≥ 0
0 ≤ y ≤ 5
2x + 3y ≥ 12
2x + 3y ≤ 20
A) maximum: 50; at (10, 0) B) maximum: 35; at (2, 5)
C) maximum: 30; at (6, 0) D) maximum: –20; at (4, 0)
15) Objective Function z = 7x + 2y
Constraints x ≥ 0
0 ≤ y ≤ 4
x – y ≤ 4
x + 2y ≤ 10
A) maximum: 46; at (6
,
2) B) maximum: 28; at (4
,
0)
C) maximum: 22; at (2, 4) D) maximum: 8; at (0, 4)
16) Objective Function z = 7x + 6y
Constraints x ≥ 0
y ≥ 0
3x + y ≤ 21
x + y ≤ 10
x + 2y ≥ 12
A) maximum 65.5; at (5.5, 4.5) B) maximum 60; at (6, 3)
C) maximum 68; at (8, 2) D) maximum 66; at (6, 4)
Solve the linear programming problem.
17) Two kinds of crated cargo, A and B, are to be shipped by truck. The weight and volume of each type are
given in the following table:
AB
Volume 50 cubic feet 10 cubic feet
Weight 200 pounds 360 pounds
The shipping company charges $75 per crate for cargo A and $100 per crate for cargo B. The truck has a
maximum load limit of 7,200 pounds and 1,000 cubic feet. How many of each type of cargo should be
shipped to maximize profit for the shipping company?
A) 18 crates of cargo A and 10 crates of cargo B B) 20 crates of cargo A and 0 crates of cargo B
C) 0 crates of cargo A and 20 crates of cargo B D) 10 crates of cargo A and 18 crates of cargo B
Page 114
18) A vineyard produces two special wines, a white and a red. A bottle of the white wine requires 14 pounds
of grapes and 1 hour of processing time. A bottle of red wine requires 25 pounds of grapes and 2 hours of
processing time. The vineyard has on hand 2,198 pounds of grapes and can allot 160 hours of processing
time to the production of these wines. A bottle of the white wine sells for $11.00, while a bottle of the red
wine sells for $20.00. How many bottles of each type should the vineyard produce in order to maximize
gross sales?
A) 132 bottles of white and 14 bottles of red B) 14 bottles of white and 132 bottles of red
C) 76 bottles of white and 42 bottles of red D) 42 bottles of white and 59 bottles of red
19) Mrs. White wants to crochet beach hats and baby afghans for a church fund–raising bazaar. She needs 7
hours to make a hat and 2 hours to make an afghan and she has 53 hours available. She wants to make no
more than 14 items and no more than 12 afghans. The bazaar will sell the hats for $13 each and the afghans
for $7 each. How many of each should she make to maximize the income for the bazaar? What is the
maximum income?
A) 5 hats, 9 afghans, $128 B) 9 hats, 5 afghans, $152
C) 12 hats, 2 afghans, $170 D) 7 hats, 7 afghans, $140
20) A candy company has 110 pounds of cashews and 135 pounds of peanuts which they combine into two
different mixes. The deluxe mix has half cashews and half peanuts and sells for $6 per pound. The
economy mix has one third cashews and two thirds peanuts and sells for $4.70 per pound. How many
pounds of each mix should be prepared for maximum revenue?
A) 170 deluxe, 75 economy B) 255 deluxe, 50 economy
C) 85 deluxe, 25 economy D) 110 deluxe, 0 economy
21) A doctor has told a sick patient to take vitamin pills. The patient needs at least 6 units of vitamin A and a
t
least 6 units of vitamin B. The red vitamin pills cost 10¢ each and contain 1 unit of A and 2 units of B. The
blue vitamin pills cost 25¢ each and contain 2 units of A and 1 unit of B. To avoid indigestion, the patient
should take no more than 4 red pills and no more than 7 blue pills. How many pills should the patient take
each day to minimize costs?
A) 4 red and 1 blue B) 2 red and 2 blue C) 0 red and 6 blue D) 4 red and 7 blue
22) A doctor has told a patient to take vitamin pills. The patient needs at least 12 units of vitamin A and a
t
least 8 units of vitamin D. The red vitamin pills cost 20¢ each and contain 3 units of A and 1 unit of D. The
blue vitamin pills cost 35¢ each and contain 2 units of A and 2 units of D. To avoid indigestion, the patient
should take no more than 6 red pills and no more than 9 blue pills. How many pills should the patient take
each day to minimize costs?
A) 2 red and 3 blue B) 6 red and 1 blue C) 0 red and 9 blue D) 6 red and 9 blue
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
23) Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you
make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each
one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on
them. How many of what type of printer should you order if you want to minimize your cost?
24) The Jillson’s have up to $75,000 to invest. They decide that they want to have at least $25,000 invested in
stable bonds yielding 6% and that no more than $45,000 should be invested in more volatile bonds
yielding 12%. How much should they invest in each type of bond to maximize income if the amount in the
more volatile bond should not exceed the amount in the more stable bond? What is the maximum income?
Page 115
25) The Fiedler family has up to $130,000 to invest. They decide that they want to have at least $40,000
invested in stable bonds yielding 5.5% and that no more than $60,000 should be invested in more volatile
bonds yielding 11%. How much should they invest in each type of bond to maximize income if the amount
in the stable bond should not exceed the amount in the more volatile bond? What is the maximum
income?
26) Eric’s Carpentry manufactures two types of bookshelves that are 4 feet tall and 3 feet wide, a basic model
and a deluxe model. Each basic bookshelf requires 1.5 hours for assembly and 1 hour for finishing; each
deluxe model requires 2.5 hours for assembly and 1 hour for finishing. Two assemblers and one finisher
are employed by the company, and each works 40 hours per week. Eric can sell more basic models than
deluxe models, so he wants the number of basic models produced to be 50% more than the number of
deluxe models produced. If he makes $50 profit on the basic models and $65 profit on the deluxe models,
how many should he make to maximize the profit? What is the maximum profit?
27)
J
oely’s Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70
pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that
contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon
tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit
of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how
many pounds of each blend should she make to maximize profits? What is the maximum profit?
28) An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet tall and 6 fee
t
wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the mosaic have words written on
them and the artist wants the words to all be horizontal in the final mosaic. The word tiles come in two
sizes: The smaller tiles are 4 inches tall and 4 inches wide, while the large tiles are 6 inches tall and 12
inches wide. If the small tiles cost $3.50 each and the larger tiles cost $4.50 each, how many of each should
be used to minimize the cost? What is the minimum cost?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
29) The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can
produce up to 24 rings each day using up to 60 total man–hours of labor. It takes 3 man–hours to make
one VIP ring and 2 man–hours to make one SST ring. How many of each type of ring should be made
daily to maximize the company’s profit, if the profit on a VIP ring is $40 and on an SST ring is $35?
A) 12 VIP and 12 SST B) 14 VIP and 10 SST C) 16 VIP and 8 SST D) 18 VIP and 6 SST
30) An airline with two types of airplanes, P1and P2
,
has contracted with a tour group to provide
transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a
certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first class, 50 tourist class, and
110 economy class passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30
tourist class, and 44 economy class passengers. How many of each type of airplane should be used in
order to minimize the operating cost?
A) 9 P1 planes and 13 P2 planes B) 5 P1planes and 17 P2 planes
C) 11 P1 planes and 7 P2 planes D) 7 P1planes and 11 P2 planes
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31) A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average
monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100. The camp can
accommodate up to 45 staff members and needs at least 30 to run properly. They must have at least 10
aides, and may have up to 3 aides for every 2 counselors. How many counselors and how many aides
should the camp hire to minimize cost?
A) 12 counselors and 18 aides B) 27 counselors and 18 aides
C) 35 counselors and 10 aides D) 18 counselors and 12 aides
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Ch. 8 Systems of Equations and Inequalities
Answer Key
8.1 Systems of Linear Equations: Substitution and Eliminatio
n
1 Solve Systems of Equations by Substitution
2 Solve Systems of Equations by Elimination
3 Identify Inconsistent Systems of Equations Containing Two Variables
4 Express the Solution of a System of Dependent Equations Containing Two Variables
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5 Solve Systems of Three Equations Containing Three Variables
6 Identify Inconsistent Systems of Equations Containing Three Variables
7 Express the Solutions of a System of Dependent Equations Containing Three Variables
8.2 Systems of Linear Equations: Matrices
1 Write the Augmented Matrix of a System of Linear Equations
2 Write the System of Equations from the Augmented Matrix
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3 Perform Row Operations on a Matrix
4 Solve a System of Linear Equations Using Matrices
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8.3 Systems of Linear Equations: Determinants
1 Evaluate 2 by 2 Determinants
2 Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables
3 Evaluate 3 by 3 Determinants
4 Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables
5 Know Properties of Determinants
8.4 Matrix Algebra
1 Find the Sum and Difference of Two Matrices
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2 Find Scalar Multiples of a Matrix
3 Find the Product of Two Matrices
4 Find the Inverse of a Matrix
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5 Solve a System of Linear Equations Using an Inverse Matrix
8.5 Partial Fraction Decompositio
n
1 Decompose P/Q, Where Q Has Only Nonrepeated Linear Factors
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2 Decompose P/Q, Where Q Has Repeated Linear Factors
3 Decompose P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor
4 Decompose P/Q, Where Q Has a Repeated Irreducible Quadratic Factor
8.6 Systems of Nonlinear Equations
1 Solve a System of Nonlinear Equations Using Substitution
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2 Solve a System of Nonlinear Equations Using Elimination
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8.7 Systems of Inequalities
1 Graph an Inequality by Hand
2 Graph an Inequality Using a Graphing Utility
3 Graph a System of Inequalities
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8.8 Linear Programming
1 Set up a Linear Programming Problem
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2 Solve a Linear Programming Problem
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