Deck 7: Systems of Equations and Matrices

Question 1

The following system does not have a unique solution. Solve the system.
$\left\{ \begin{aligned} x + 5 y + 2 z & = - 5 \\ - 3 x + 5 y - 6 z & = 35 \end{aligned} \right.$

A)x = - 8 , y & = 1 , z = - 1

B) infinitely many solutions of the form $x = - 10 - 2 z , y = 1 , z = z$
C) infinitely many solutions of the form $x = - 10 - 2 z , y = 1 - z , z = z$
D) inconsistent system, no solution

Answer 1

B

Question 2

The following system does not have a unique solution. Solve the system.
$$\left\{ \begin{array} { l } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.$$

A) infinitely many solutions of the form $x = - \frac { 1 } { 9 } - \left( \frac { 2 } { 3 } \right) z , y = - \frac { 11 } { 9 } + \left( \frac { 2 } { 3 } \right) z , z = z$
B) inconsistent system, no solution
C) $x = 0 , y = 0 , z = \frac { 3 } { 2 }$
D) infinitely many solutions of the form $x = \frac { 19 } { 3 } + \frac { 8 } { 9 } z , y = \frac { 5 } { 6 } - \frac { 13 } { 18 } z , z = z$

Answer 2

A

Question 3

The following system does not have a unique solution. Solve the system.
$$\left\{ \begin{array} { c } - 2 x + 3 y + z = - 24 \\3 x - y + 5 z = 8 \\x + 2 y + 6 z = - 16 \\\text { A) } x = 0 , y = - 8 , z = 0\end{array} \right.$$
B) infinitely many solutions of the form $x = - \frac { 16 } { 7 } z , y = - 8 - \frac { 13 } { 7 } z , z = z$
C) inconsistent system, no solution
D) $x = - \frac { 16 } { 7 } , y = \frac { 43 } { 7 } , z = 1$

Answer 3

B

Question 4

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The total income from the 5% and 6% investments is equal to the income
From the 8% investment. Find the amount invested at each rate.

A) $10,000 at 5%, $10,000 at 6%, $5000 at 8%
B) $5000 at 5%, $10,000 at 6%, $10,000 at 8%
C) $10,000 at 5%, $5000 at 6%, $10,000 at 8%
D) $8000 at 5%, $10,000 at 6%, $7000 at 8%

Question 5

A company manufactures three types of cable. Cable A requires 3 black wires, 3 white wires, and 2 red wires. Cable B requires 1 black, 2 white, and 1 red. Cable C requires 2 black, 1 white, and 2 red. If 100 black wires, 110
White wires, and 90 red wires are required to complete an order, how many of each type of cable are in the
Order?

A) 20 cable A, 30 cable B, 10 cable C
B) 10 cable A, 30 cable B, 20 cable C
C) 10 cable A, 103 cable B, 20 cable C
D) 10 cable A, 30 cable B, 93 cable C

Question 6

$\left\{ \begin{array} { l } 5 x + 2 y + z = - 11 \\2 x - 3 y - z = 17 \\7 x + y + 2 z = - 4\end{array} \right.$

A) x=-3, y=0, z=4
B) x=3, y=0, z=-4
C) x=0, y=6, z=-1
D) x=0, y=-6, z=1

Question 7

$\left\{ \begin{aligned}x + 5 y + 2 z & = 33 \\5 y + 5 z & = 45 \\z & = 5\end{aligned} \right.$

A) x=5, y=4, z=3
B) No solution
C) x=3, y=4, z=5
D) x=3, y=5, z=4

Question 8

A $100,000 trust is to be invested in bonds paying 9%, CDs paying 7%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $9100 annual income from the
Investments, how much should the bank invest in bonds?

A) $20,000
B) $30,000
C) $28,000
D) $50,000

Question 9

$\left\{ \begin{array} { l } x + y + z = 6 \\x - z = - 2 \\y + 3 z = 11\end{array} \right.$

A) x=-1, y=2, z=-3
B) x=1, y=2, z=3
C) x=3, y=-2, z=5
D) x=0, y=4, z=2

Question 10

$\left\{ \begin{aligned}4 x - y + 3 z & = 12 \\2 x + 9 z & = - 5 \\x + 4 y + 6 z & = - 32\end{aligned} \right.$

A) x=2, y=7, z=-1
B) x=2, y=-7, z=1
C) x=2, y=7, z=1
D) x=2, y=-7, z=-1

Question 11

$\left\{ \begin{array} { r } 7 x + 7 y + z = 1 \\x + 8 y + 8 z = 8 \\9 x + y + 9 z = 9\end{array} \right.$

A) x=1, y=-1, z=1
B) x=-1, y=1, z=1
C) x=0, y=0, z=1
D) x=0, y=1, z=0

Question 12

The following system does not have a unique solution. Solve the system.
$$\left\{ \begin{array} { r } x + 3 y + 2 z = 11 \\4 y + 9 z = - 12\end{array} \right.$$

A) infinitely many solutions of the form $x = 20 + \frac { 19 } { 4 } z , y = - 3 - \frac { 9 } { 4 } z , z = z$
B) infinitely many solutions of the form $x = 20 - \frac { 19 } { 4 } z , y = 3 - \frac { 9 } { 4 } z , z = z ^ { 2 }$
C) infinitely many solutions of the form $x = 20 + \frac { 19 } { 4 } z , y = 3 - \frac { 9 } { 4 } z , z = z$
D) infinitely many solutions of the form $x = 20 + \frac { 19 } { 4 } z , y = 3 + \frac { 9 } { 4 } z , z = z ^ { 2 }$

Question 13

The sum of a student's three scores is 230. If the first is 19 points more than the second, and the sum of the first two is 23 more than twice the third, then what was the first score?

A) 50
B) 90
C) 69
D) 71

Question 14

The following system does not have a unique solution. Solve the system.
$$\left\{ \begin{array} { r } x - y + 3 z = - 8 \\x + 5 y + z = 40 \\5 x + y + 13 z = 10\end{array} \right.$$

A) $x = 4 + z , y = - 18 - 2 z , z = z$
B) $x = 0 , y = - 1 , z = - 3$
C) inconsistent system, no solution
D) $x = 0 , y = 1 , z = - 3$

Question 15

$\left\{ \begin{aligned}x - y + 8 z & = - 107 \\6 x + z & = 17 \\3 y - 5 z & = 89\end{aligned} \right.$

A) x=-5, y=8, z=13
B) x=5, y=-8, z=-13
C) x=-5, y=-8, z=13
D) x=5, y=8, z=-13

Question 16

The following system does not have a unique solution. Solve the system.
$$\left\{ \begin{array} { c } x - y + z = 8 \\x + y + z = 6 \\3 x + y + 3 z = 10\end{array} \right.$$

A) $x = - 2 , y = 1 , z = 9$
B) $x = 7 , y = - 1 , z = 0$
C) inconsistent system, no solution
D) $x = 5 , y = 3 , z = 6$

Question 17

Some people must eat a low-sodium diet with no more than 2000 mg of sodium per day. By eating 1 cracker, 1 pretzel, and 1 cookie, a person would ingest 149 mg of sodium. If a person ate 8 pretzels and 8 cookies, he or she
Would ingest 936 mg of sodium. By eating 6 crackers and 7 pretzels, a person would take in 535 mg of sodium.
Which of the following statements is true?

A) A cracker contains more sodium than a cookie.
B) A cookie contains 71 mg of sodium.
C) A cracker contains 30 mg of sodium.
D) A pretzel contains 49 mg of sodium.

Question 18

$\left\{ \begin{array} { l } x - y + z = 8 \\x + y + z = 6 \\x + y - z = - 12\end{array} \right.$

A) x=-2, y=-1, z=9
B) x=2, y=-1, z=-9
C) x=-2, y=-1, z=-9
D) x=2, y=-1, z=9

Question 19

$\left\{ \begin{array} { c } x + y + z = 7 \\x - y + 2 z = 7 \\5 x + y + z = 11\end{array} \right.$

A) x=4, y=1, z=2
B) x=1, y=2, z=4
C) x=1, y=4, z=2
D) x=4, y=2, z=1

Question 20

$\left\{ \begin{array} { c } x - y + 3 z = - 8 \\2 x + z = 0 \\x + 5 y + z = 40\end{array} \right.$

A) x=0, y=8, z=0
B) x=0, y=-8, z=-8
C) x=8, y=8, z=0
D) x=-8, y=0, z=0

Question 21

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
$$\left[ \begin{array} { l l l | r } 1 & 0 & 0 & 3 \\0 & 1 & 0 & - 14 \\0 & 0 & 1 & 1\end{array} \right]$$

A) $x = 3 , y = - 14 , z = - 1$
B) $x = 3 , y = - 14 , z = 1$
C) $x = - 3 , y = - 14 , z = - 1$
D) $x = - 3 , y = - 14 , z = 1$

Question 22

Write the augmented matrix associated with the system.
$\left\{ \begin{array} { c } - 2 x + 2 y + 7 z = 57 \\ 8 x - 2 y - 2 z = - 34 \\ - 2 x + 7 y + 9 z = 81 \end{array} \right.$

A)
$\left[ \begin{array} { r r r | r } - 2 & 2 & 7 & 57 \\ 8 & - 2 & - 2 & - 34 \\ - 2 & 7 & 9 & 81 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } - 2 & 2 & 7 \\ 8 & - 2 & - 2 \\ - 2 & 7 & 9 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 57 & 7 & - 2 & - 2 \\ - 34 & - 2 & - 2 & 8 \\ 81 & 9 & 7 & - 2 \end{array} \right]$

D)

$\left[ \begin{array} { r r r | r } - 2 & 8 & - 2 & 57 \\ 2 & - 2 & 7 & - 34 \\ 7 & - 2 & 9 & 81 \end{array} \right]$

Question 23

A psychologist studying the effects of good nutrition on the behavior of rabbits feeds one group a combination of three foods, I, II, and III. Each of these foods contains three additives, A, B, and C, that are used in the study.
The table below gives the percent of each additive that is present in each food. If the diet being used requires
19)1 g per day of A, 8.94 g of B, and 14.02 g of C, find the number of grams of food II that should be used each
Day) Let z represent the number of grams of food III. $\begin{array}{l|l|l|l} & \text { Food I| Food II } & \text { Food III } \\\hline \text { Additive A } & 10 \% & 10 \% & 11 \% \\\text { Additive B } & 6 \% & 2 \% & 15 \% \\\text { Additive C } & 8 \% & 6 \% & 13 \%\end{array}$

A) $2.7 \mathrm { z } - 60$
B) $1.8 z + 54$
C) $128 - 3.2 \mathrm { z }$
D) $2.1 z + 63$

Question 24

Write the augmented matrix associated with the system.
$$\left\{ \begin{array} { r r } - 6 x + 5 y & = 6 \\2 y + & 3 z = - 6 \\- 9 z & = 7\end{array} \right.$$

A)
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \\ 0 & 2 & 3 & - 6 \\ 0 & 0 & - 9 & 7 \end{array} \right]$

B)
$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \\ 2 & 3 & - 6 & 0 \\ - 9 & 7 & 0 & 0 \end{array} \right]$

C)
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \\ 2 & 3 & 0 & - 6 \\ 7 & 0 & 0 & 7 \end{array} \right]$

D)

$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \\ 2 & 3 & 0 & - 6 \\ - 9 & 0 & 0 & 7 \end{array} \right]$

Question 25

A bank gives three loans totaling $330,000 to a development company for the purchase of three business properties. The largest loan is $38,000 more than the sum of the other two. Find the amount of the largest loan.

A) $192,000
B) $195,000
C) $184,000
D) $180,000

Question 26

A bakery sells three types of cakes. The table below gives the number of cups of flour, cups of sugar, and eggs needed to produce each type of cake.
To fill its orders for cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 51 eggs. How many cakes of each
Type were made?

A) 19 cake I, 8 cake II, 3 cake III
B) 41 cake I, 8 cake II, 12 cake III
C) 17 cake I, 8 cake II, 3 cake III
D) 3 cake I, 8 cake II, 17 cake III

Question 27

Write the augmented matrix associated with the system.
$$\left\{ \begin{array} { l } 7 x + 5 y + 4 z = 65 \\4 x + 8 y + 4 z = 56 \\9 x + 8 y + 2 z = 93\end{array} \right.$$

A)
$\left[ \begin{array} { l l l | l } 7 & 4 & 9 & 65 \\ 5 & 8 & 8 & 56 \\ 4 & 4 & 2 & 93 \end{array} \right]$

B)
$\left[ \begin{array} { l l l } 7 & 5 & 4 \\ 4 & 8 & 4 \\ 9 & 8 & 2 \end{array} \right]$


C)
$\left[ \begin{array} { l l l | l } 7 & 5 & 4 & 65 \\ 4 & 8 & 4 & 56 \\ 9 & 8 & 2 & 93 \end{array} \right]$

D)

$\left[ \begin{array} { l l l | l } 7 & 5 & 4 & 65 \\ 4 & 8 & 4 & 56 \\ 9 & 8 & 2 & 93 \end{array} \right]$

Question 28

A trust account manager has $400,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 10%, and the goal is to earn $32,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%,
And z dollars are invested at 8%, what limits must there be on z so that all investment values are non-negative? A) $200,000 \leq z \leq 240,000$
B) $150,000 \leq \mathrm { z } \leq 180,000$
C) $210,000 \leq z \leq 250,000$
D) $150,000 \leq \mathrm { z } \leq 200,000$

Question 29

A trust account manager has $300,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $18,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and
Z dollars are invested at 8%, find x in terms of z. A) $x = 2 z$
B) $x = z$
C) $x = 100,000 + z$
D) $x = 300,000 - 3 z$

Question 30

A basketball fieldhouse seats 15,000. Courtside seats sell for $8, endzone for $7, and balcony for $5. A full house earns $88,000 in ticket revenue. If half the courtside and balcony seats and all the endzone seats are sold, the
Total ticket revenue is $51,000. How many of each type of seat are there?

A) 4000 courtside, 3000 endzone, 8000 balcony
B) 4000 courtside, 1500 endzone, 9500 balcony
C) 3200 courtside, 1800 endzone, 10,000 balcony
D) 3000 courtside, 2000 endzone, 10,000 balcony

Question 31

Write the augmented matrix associated with the system.
$$\left\{ \begin{array} { r } - 2 x - 2 z = - 6 \\- 2 y + 6 z = 10 \\9 x + 7 y + 8 z = 32\end{array} \right.$$

A)
$\left[ \begin{array} { r r r } - 2 & 0 & - 2 \\ 0 & - 2 & 6 \\ 9 & 7 & 8 \end{array} \right]$

B)
$\left[ \begin{array} { r r r } - 2 & 0 & - 2 \\ 0 & - 2 & 6 \\ 9 & 7 & 8 \end{array} \right]$
$\left[ \begin{array} { r r r | r } - 2 & 0 & - 2 & - 6 \\ 0 & - 2 & 6 & 10 \\ 9 & 7 & 8 & 32 \end{array} \right]$
C)
D)
$$\left[ \begin{array} { r r r | r } - 2 & - 2 & 0 & - 6 \\- 2 & 6 & 0 & 10 \\9 & 7 & 8 & 32\end{array} \right] \quad \left[ \begin{array} { r r r | r } - 2 & 0 & 9 & - 6 \\0 & - 2 & 7 & 10 \\- 2 & 6 & 8 & 32\end{array} \right]$$

Question 32

The Hoffman Trucking Company has an order for three products, A, B, and C, for delivery. The table below gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of
The products. If one of the company's trucks can carry 38,000 cubic feet and 57,100 pounds, and is insured to
Carry $29,750, how many units of product A can be carried on the truck? Assume that z represents the number
Of units of product C. $$\begin{array} { l | l | l | l } & \text { Product A } & \text { Product B } & \text { Product C } \\\hline \text { Unit Volume } \left( \mathrm { ft } ^ { 3 } \right) & 24 & 20 & 50 \\\text { Weight (lbs) } & 36 & 30 & 75 \\\text { Value (dollars) } & 150 & 180 & 120\end{array}$$

A) $140-3.5 z$
B) $4.5 \mathrm { z } - 18$
C) $135 - 4 \mathrm { z }$
D) There is no solution.

Question 33

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
$$\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 4 \\1 & - 1 & 4 & - 26 \\5 & 1 & 1 & - 24\end{array} \right]$$

A) $x = - 5 , y = 5 , z = - 4$
B) $x = - 4 , y = 5 , z = - 5$
C) $x = - 4 , y = - 5 , z = 5$
D) $x = - 5 , y = - 4 , z = 5$

Question 34

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?

A) 0 or 2
B) 0, 1, 2, or 3
C) 6
D) 0

Question 35

Write the augmented matrix associated with the system.
$$\left\{ \begin{array} { r } 6 x + 8 z = 64 \\4 y + 4 z = 24 \\6 x + 2 y + 3 z = 62\end{array} \right.$$

A)
$\left[ \begin{array} { l l l | l } 6 & 8 & 0 & 64 \\ 4 & 4 & 0 & 24 \\ 6 & 2 & 3 & 62 \end{array} \right]$

B)

$\left[ \begin{array} { l l l } 6 & 0 & 8 \\ 0 & 4 & 4 \\ 6 & 2 & 3 \end{array} \right]$
C)
$\left[ \begin{array} { l l l | l } 6 & 0 & 8 & 64 \\ 0 & 4 & 4 & 24 \\ 6 & 2 & 3 & 62 \end{array} \right]$

D)

$\left[ \begin{array} { l l l | l } 6 & 0 & 6 & 64 \\ 0 & 4 & 2 & 24 \\ 8 & 4 & 3 & 62 \end{array} \right]$

Question 36

A company offers three mutual fund plans for its employees. Plan I consists of 2 blocks of common stocks, 3 municipal bonds, and 4 blocks of preferred stock. Plan II consists of 2 blocks of common stocks, 2 municipal
Bonds, and 1 block of preferred stock. Plan III consists of 4 blocks of common stocks, 5 municipal bonds, and 5
Blocks of preferred stock. An employee combined these plans so that he has 22 blocks of common stock, 27
Municipal bonds, and 26 blocks of preferred stock. How many units of plan III might he have?

A) 0, 1, 2, 3, 4, or 5
B) 3, 4, 5, or 6
C) 0, 1, 2, 3, 4, 5, or 6
D) 3, 4, or 5

Question 37

A man has a total of $110,000 invested in three rental properties. One property earns 7.5% per year on its investment, the second earns 9%, and the third earns 10%. The three properties earn a total of $9600 per year.
What is the largest amount that could be invested at 7.5%?

A) $56,000
B) $110,000
C) $50,000
D) $54,000

Question 38

A trust account manager has $500,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $34,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and
Z dollars are invested at 8%, find y in terms of z. A) $y = - 600,000 + 2 z$
B) $\mathrm { y } = 900,000 - 3 \mathrm { z }$
C) $y = - 400,000 + 2 z$
D) $\mathrm { y } = 1,100,000 - 3 \mathrm { z }$

Question 39

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
$$\left[ \begin{array} { r r r | r } 1 & - 1 & 4 & - 20 \\5 & 0 & 1 & - 4 \\1 & 3 & 1 & 8\end{array} \right]$$

A) $x = 0 , y = 4 , z = - 4$
B) $x = - 4 , y = 4 , z = 0$
C) $x = 4 , y = 4 , z = - 4$
D) $x = - 4 , y = 0 , z = 4$

Question 40

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
$\left[ \begin{array} { r r r | r } 1 & 0 & 0 & - 5 \\ 0 & 1 & 0 & 9 \\ 0 & 0 & 1 & 9 \end{array} \right]$

A) $x = - 5 , y = 0 , z = 9$
B) $x = 9 , y = 9 , z = - 5$
C) $x = - 5 , y = 9 , z = 9$
D) $x = 0 , y = 9 , z = - 5$

Question 41

$\left\{ \begin{array} { l } 2 x + 4 y - 3 z + w = - 10 \\- x + y + 3 z - w = - 2 \\4 x - y + z + 3 w = 13 \\x + y + 2 z - w = 3\end{array} \right.$

A) x=1, y=-1, z=3, w=-2
B) x=-3, y=-3, z=1, w=-2
C) x=3, y=-3, z=1, w=-1
D) x=3, y=-6, z=-1, w=-1

Question 42

$\left[ \begin{array} { c c c | c } 1 & 0 & - 3 & 2 \\0 & 1 & 1 & 4 \\0 & 0 & 0 & 0\end{array} \right]$

A) infinitely many solutions of the form $x = 2 - 3 z , y = 4 - z , z = z$
B) infinitely many solutions of the form $x = 2 + 3 z , y = 4 + z , z = z$
C) infinitely many solutions of the form $x = 2 + 3 z , y = 4 - z , z = z$
D) infinitely many solutions of the form $x = 2 - 3 z , y = 4 + z , z = z$

Question 43

Find the solution or solutions, if any exist, to the system.
$$\left\{ \begin{array} { c } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.$$

A) infinitely many solutions of the form $x = - \frac { 1 } { 9 } - \left( \frac { 2 } { 3 } \right) z , y = - \frac { 11 } { 9 } + \left( \frac { 2 } { 3 } \right) z , z = z$
B) $x = 0 , y = 0 , z = 1$
C) infinitely many solutions of the form $x = 8 - \frac { 2 } { 9 } z , y = 1 - \frac { 17 } { 18 } z , z = z$
D) inconsistent system, no solution

Question 44

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
$$\left[ \begin{array} { r r r | r } 1 & - 1 & 1 & - 3 \\1 & 1 & 1 & 3 \\1 & 1 & - 1 & 1\end{array} \right]$$

A) $x = 1 , y = - 1 , z = 3$
B) $x = - 1 , y = 3 , z = 1$
C) $x = - 3 , y = 3 , z = 1$
D) $x = - 1 , y = 1 , z = 3$

Question 45

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
$$\left[ \begin{array} { r r r | r } 5 & 2 & 1 & - 8 \\2 & - 3 & - 1 & - 5 \\5 & 1 & 4 & 5\end{array} \right]$$

A) $x = - 2 , y = 4 , z = - 1$
B) $x = 4 , y = - 2 , z = - 1$
C) $x = 4 , y = - 1 , z = - 2$
D) $x = - 2 , y = - 1 , z = 4$

Question 46

Find the solution or solutions, if any exist, to the system.
$$\left\{ \begin{array} { r } x + y + z = 9 \\2 x - 3 y + 4 z = 7\end{array} \right.$$

A) infinitely many solutions of the form $x = \frac { 34 } { 5 } + \frac { 7 } { 5 } z , y = - \frac { 11 } { 5 } + \frac { 2 } { 5 } z , z = z$
B) infinitely many solutions of the form $x = \frac { 34 } { 5 } - \frac { 7 } { 5 } z , y = - \frac { 11 } { 5 } + \frac { 2 } { 5 } z , z = z$
C) infinitely many solutions of the form $x = \frac { 34 } { 5 } - \frac { 7 } { 5 } z , y = \frac { 11 } { 5 } + \frac { 2 } { 5 } z , z = z$
D) infinitely many solutions of the form $x = \frac { 34 } { 5 } + \frac { 1 } { 5 } z , y = \frac { 11 } { 5 } + \frac { 2 } { 5 } z , z = z$

Question 47

$\left[ \begin{array} { r r r | r } 1 & 0 & 1 & - 2 \\0 & 1 & 2 & - 5 \\0 & 0 & 0 & 0\end{array} \right]$

A) infinitely many solutions of the form $x = - 2 + z , y = - 5 + 2 z , z = z$
B) infinitely many solutions of the form $x = - 2 - z , y = - 5 - 2 z , z = 0$
C) $x = - 2 , y = - 5 , z = 0$
D) infinitely many solutions of the form $x = - 2 - z , y = - 5 - 2 z , z = z$

Question 48

$\left\{ \begin{aligned}6 y + w & = - 25 \\x + y + 4 z - w & = 14 \\6 x + z + 4 w & = 6 \\x + y + 4 z & = 13\end{aligned} \right.$

A) x=1, y=-4, z=4, w=-1
B) x=6, y=-24, z=16, w=-6
C) x=1, y=-24, z=8, w=-2
D) x=1, y=-4, z=16, w=-1

Question 49

$\left[ \begin{array} { l l l | l } 1 & 0 & 5 & 5 \\0 & 1 & 2 & 2 \\0 & 0 & 0 & 1\end{array} \right]$

A) $x = 0 , y = 0 , z = 1$
B) infinitely many solutions of the form $x = 5 - 5 z , y = 2 - 2 z , z = z$
C) inconsistent system, no solution
D) infinitely many solutions of the form $x = 5 z + 5 , y = 2 z + 2 , z = 1$

Question 50

Find the solution or solutions, if any exist, to the system.
$$\left\{ \begin{array} { c } x + y + z = 11 \\x - y + 5 z = 23 \\5 x + y + z = 31\end{array} \right.$$

A) $x = 4 , y = 5 , z = 2$
B) $x = 5 , y = 2 , z = 4$
C) inconsistent system, no solution
D) $x = 4 , y = 2 , z = 5$

Question 51

$\left\{ \begin{array} { c } x + y + z = 7 \\x - y + 2 z = 7 \\5 x + y + z = 11\end{array} \right.$

A) x=1, y=4, z=2
B) x=4, y=1, z=2
C) x=4, y=2, z=1
D) x=1, y=2, z=4

Question 52

Find the solution or solutions, if any exist, to the system.
$\left\{ \begin{array} { c } x - y + 3 z = - 8 \\ x + 5 y + z = 40 \\ 5 x + y + 13 z = 10 \end{array} \right.$

A) x = 0 , y = 1 , z = - 3
B) infinitely many solutions of the form $x = 4 + z , y = - 18 - 2 z , z = z$
C) $x = 0 , y = - 1 , z = - 3$
D) inconsistent system, no solution

Question 53

$\left[ \begin{array} { r r r | r } 1 & 0 & 4 & - 2 \\0 & 1 & - 3 & 1 \\0 & 0 & 0 & 0\end{array} \right]$

A) infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + 3 z , z = z$
B) infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + z , z = z$
C) infinitely many solutions of the form $x = - 2 - z , y = 1 + 3 z , z = z$
D) infinitely many solutions of the form $x = 1 + 3 z , y = - 2 - 4 z , z = z$

Question 54

$\left\{ \begin{array} { l } 2 x - 5 z + w = - 1 \\- x + 2 y + z - 2 w = 1 \\4 x - y + 5 w = - 2 \\y - 2 z - w = 0\end{array} \right.$

A) x=-2, y=4, z=3, w=-5
B) x=4, y=-2, z=-7, w=-4
C) x=3, y=0, z=-1, w=0
D) x=4, y=-2, z=1, w=-4

Question 55

Find the solution or solutions, if any exist, to the system.
$$\left\{ \begin{array} { l } 7 x + 7 y + z = 1 \\x + 8 y + 8 z = 8 \\9 x + y + 9 z = 9\end{array} \right.$$

A) $x = 0 , y = 1 , z = 0$
B) $x = - 1 , y = 1 , z = 1$
C) $x = 1 , y = - 1 , z = 1$
D) $x = 0 , y = 0 , z = 1$

Question 56

$\left\{ \begin{array} { l } 5 x + 2 y + z = - 11 \\2 x - 3 y - z = 17 \\7 x + y + 2 z = - 4\end{array} \right.$

A) x=3, y=0, z=-4
B) x=0, y=-6, z=1
C) x=0, y=6, z=-1
D) x=-3, y=0, z=4

Question 57

$\left\{ \begin{array} { l } x - y + z = 8 \\x + y + z = 6 \\x + y - z = - 12\end{array} \right.$

A) x=-2, y=-1, z=-9
B) x=2, y=-1, z=9
C) x=-2, y=-1, z=9
D) x=2, y=-1, z=-9

Question 58

Find the solution or solutions, if any exist, to the system.
$$\left\{ \begin{array} { l } x + y + z = 7 \\x - y + 2 z = 7\end{array} \right.$$

A) infinitely many solutions of the form $x = 7 - \frac { 3 } { 2 } z , y = \frac { 1 } { 2 } z , z = z$
B) infinitely many solutions of the form $x = - 7 - \frac { 3 } { 2 } z , y = \frac { 1 } { 2 } z , z = z$
C) infinitely many solutions of the form $x = - 7 - \frac { 3 } { 2 } z , y = 2 z , z = z$
D) infinitely many solutions of the form $x = 7 - \frac { 3 } { 2 } z , y = 2 z , z = z$

Question 59

$\left\{ \begin{aligned}4 x + 2 z + w & = - 18 \\- x + y + 2 z - w & = 4 \\4 x - y + 2 w & = - 21 \\y + 4 z - w & = - 3\end{aligned} \right.$

A) x=5, y=3, z=-1, w=-1
B) x=-10, y=0, z=-2, w=1
C) x=0, y=-5, z=1, w=5
D) x=-5, y=1, z=-1, w=0

Question 60

$\left\{ \begin{array} { l } 2 x + y - 4 z + w = - 17 \\- x + 2 y + z - 2 w = 7 \\4 x - y + 2 z + 4 w = 11 \\x + y - 2 z - w = - 3\end{array} \right.$

A) x=4, y=-1, z=5, w=-4
B) x=-1, y=3, z=6, w=1
C) x=4, y=-2, z=0, w=4
D) x=-4, y=-5, z=10, w=4

Question 61

Find the indicated sum or difference, if it is defined.
$\left[ \begin{array} { l l l } - 2 & 9 & 1 \end{array} \right] - \left[ \begin{array} { l l } 4 & 3 \end{array} \right]$

A) $\left[ \begin{array} { l l l } - 6 & 9 & - 2 \end{array} \right]$
B) not defined
C) $\left[ \begin{array} { l l l } - 6 & 6 & - 2 \end{array} \right]$
D) $\left[ \begin{array} { l l l } - 6 & 6 & 1 \end{array} \right]$

Question 62

Find the indicated matrix.
Let $C = \left[ \begin{array} { r } 4 \\ - 2 \\ 12 \end{array} \right]$ . Find $\frac { 1 } { 2 } \mathrm { C }$ .

A) $\left[ \begin{array} { r } 2 \\ - 2 \\ 12 \end{array} \right]$
B) $\left[ \begin{array} { r } 4 \\ - 1 \\ 12 \end{array} \right]$
C) $\left[ \begin{array} { r } 8 \\ - 4 \\ 24 \end{array} \right]$
D) $\left[ \begin{array} { r } 2 \\ - 1 \\ 6 \end{array} \right]$

Question 63

In an analysis of traffic, a certain city estimates the traffic flow as illustrated in the figure below, where the arrows indicate the flow of traffic. If x1 represents the number of cars traveling from intersection A to intersection B, x2 represents the number of cars traveling from intersection B to intersection C , and so on, we car formulate equations based on the principle that the number of vehicles entering the intersection equals the numt leaving it. Formulate an equation for the traffic at each of the four intersections.

A)
$\begin{array} { l } x _ { 1 } - x _ { 4 } = - 650 \\x _ { 1 } - x _ { 2 } = - 450 \\x _ { 2 } - x _ { 3 } = 1400 \\x _ { 3 } - x _ { 4 } = - 1100\end{array}$
B)
$\begin{array} { l } x _ { 1 } - x _ { 4 } = - 450 \\x _ { 1 } - x _ { 2 } = - 650 \\x _ { 2 } - x _ { 3 } = - 1400 \\x _ { 3 } - x _ { 4 } = 1100\end{array}$
C)
$\begin{array} { l } x _ { 1 } - x _ { 4 } = 450 \\x _ { 1 } - x _ { 2 } = 650 \\x _ { 2 } - x _ { 3 } = 1400 \\x _ { 3 } - x _ { 4 } = - 1100\end{array}$
D)
$\begin{array} { l } x _ { 1 } - x _ { 4 } = 450 \\x _ { 1 } - x _ { 2 } = 650 \\x _ { 2 } - x _ { 3 } = 1400 \\x _ { 3 } - x _ { 4 } = 1100\end{array}$

Question 64

Find the indicated sum or difference, if it is defined.
$$\left[ \begin{array} { l l } 5 & 4\end{array} \right] + \left[ \begin{array} { r } - 1 \\5\end{array} \right]$$

A) $\left[ \begin{array} { l l } 4 & 9 \end{array} \right]$
B) not defined
C) $\left[ \begin{array} { l } 4 \\ 9 \end{array} \right]$
D) $\left[ \begin{array} { r r } 5 & - 1 \\ 4 & 5 \end{array} \right]$

Question 65

Some people must eat a low-sodium diet with no more than 2000 mg of sodium per day. By eating 1 cracker, 1 pretzel, and 1 cookie, a person would ingest 149 mg of sodium. If a person ate 8 pretzels and 8 cookies, he or she
Would ingest 936 mg of sodium. By eating 6 crackers and 7 pretzels, a person would take in 535 mg of sodium.
Which of the following statements is true?

A) A pretzel contains 49 mg of sodium.
B) A cracker contains more sodium than a cookie
C) A cracker contains 30 mg of sodium.
D) A cookie contains 71 mg of sodium.

Question 66

The sum of a student's three scores is 224. If the first is 13 points more than the second, and the sum of the first two is 29 more than twice the third, then what was the first score?

A) 65
B) 86
C) 52
D) 73

Question 67

Find the solution or solutions, if any exist, to the system.
A man has $\$ 298,000$ invested in three rental properties. One property earns $8.5 \%$ per year on the investment, a second earns $11 \%$ , and the third earns $7 \%$ . The annual earnings from the the properties total $\$ 17,000$ . Write a system of two equations to represent the problem with $\mathrm { x } , \mathrm { y }$ , and $\mathrm { z }$ representing the $8.5 \% , 11 \%$ , and $7 \%$ investments, respectively.

A) $\left\{ \begin{array} { l } x + y + z = 17,000 \\ 0.085 x + 0.11 y + 0.07 z = 298,000 \end{array} \right.$
B) $\left\{ \begin{array} { l } x + y + z = 298,000 \\ 0.085 x + 0.07 y + 0.11 z = 17,000 \end{array} \right.$
C) $\left\{ \begin{array} { l } x + y + z = 298,000 \\ 0.11 x + 0.085 y + 0.07 z = 17,000 \end{array} \right.$
D) $\left\{ \begin{array} { l } x + y + z = 298,000 \\ 0.085 x + 0.11 y + 0.07 z = 17,000 \end{array} \right.$

Question 68

Find the indicated sum or difference, if it is defined.
$$\left[ \begin{array} { r } 5 x + 5 y \\- 6 k - 9 z \\2 w + 6 v \\- 2 m - 9 n\end{array} \right] - \left[ \begin{array} { c } 2 x + 2 y \\2 k - 4 z \\4 w + 9 v \\6 m - 3 n\end{array} \right]$$

A)
$\left[ \begin{array} { r } 3 x + 7 y \\ - 8 k - 13 z \\ - 2 w + 15 v \\ - 8 m - 12 n \end{array} \right]$
B) $\left[ \begin{array} { r } 3 x + 3 y \\ - 8 k + 5 z \\ - 2 w - 3 v \\ - 8 m + 6 n \end{array} \right]$
C) $\left[ \begin{array} { r } 3 x - 7 y \\ - 8 k - 13 z \\ - 2 w - 15 v \\ - 8 m - 12 n \end{array} \right]$
D) $\left[ \begin{array} { r } 3 x + 3 y \\ - 8 k - 5 z \\ - 2 w - 3 v \\ - 8 m - 6 n \end{array} \right]$

Question 69

A basketball fieldhouse seats 15,000. Courtside seats sell for $8, endzone for $7, and balcony for $5. A full house earns $88,000 in ticket revenue. If half the courtside and balcony seats and all the endzone seats are sold, the
Total ticket revenue is $51,000. How many of each type of seat are there?

A) 4000 courtside, 1500 endzone, 9500 balcony
B) 4000 courtside, 3000 endzone, 8000 balcony
C) 3000 courtside, 2000 endzone, 10,000 balcony
D) 3200 courtside, 1800 endzone, 10,000 balcony

Question 70

Find the indicated sum or difference, if it is defined.
$$\left[ \begin{array} { l l } - 1 & - 6 \\- 1 & - 8 \\- 6 & - 5\end{array} \right] + \left[ \begin{array} { r r } 6 & 3 \\- 2 & - 2 \\- 5 & - 6\end{array} \right]$$

A) $\left[ \begin{array} { r r } 5 & - 3 \\ - 3 & - 10 \\ - 11 & - 11 \end{array} \right]$
B) $\left[ \begin{array} { r r } - 7 & - 9 \\ 1 & - 6 \\ - 1 & - 12 \end{array} \right]$
C) $\left[ \begin{array} { r r } 5 & - 8 \\ - 3 & - 10 \\ - 11 & - 11 \end{array} \right]$
D) not defined

Question 71

Find the indicated matrix.
Let $B = \left[ \begin{array} { l l l l } - 1 & 6 & 4 & - 3 \end{array} \right]$ . Find -2B.

A) $\left[ \begin{array} { l l l l } - 2 & 12 & 8 & - 6 \end{array} \right]$
B) $\left[ \begin{array} { l l l l } 2 & 6 & 4 & - 3 \end{array} \right]$
C) $\left[ \begin{array} { l l l l } - 3 & 4 & 2 & - 5 \end{array} \right]$
D) $\left[ \begin{array} { l l l l } 2 & - 12 & - 8 & 6 \end{array} \right]$

Question 72

Find the indicated sum or difference, if it is defined.
$$\left[ \begin{array} { r r } - 1 & 5 \\0 & 4 \\8 & - 4\end{array} \right] - \left[ \begin{array} { l l } 2 & 1 \\7 & 4 \\4 & 2\end{array} \right]$$

A) $\left[ \begin{array} { r r } - 3 & 4 \\ - 7 & 0 \\ 4 & - 6 \end{array} \right]$
B) $\left[ \begin{array} { r r } 3 & - 4 \\ 7 & 0 \\ - 4 & 6 \end{array} \right]$
C) not defined
D) $\left[ \begin{array} { r r } 1 & 6 \\ 7 & 8 \\ 12 & - 2 \end{array} \right]$

Question 73

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?

A) 0, 1, 2, or 3
B) 0
C) 6
D) 0 or 2

Question 74

Find the indicated sum or difference, if it is defined.
$$\left[ \begin{array} { r } 3 \\- 1 \\- 4\end{array} \right] + \left[ \begin{array} { r } - 5 \\6 \\9\end{array} \right]$$

A) $\left[ \begin{array} { l } 2 \\ 5 \\ 6 \end{array} \right]$
B) $\left[ \begin{array} { r r } 3 & - 5 \\ - 1 & 6 \\ - 4 & 9 \end{array} \right]$
C) $\left[ \begin{array} { l l l } - 2 & 5 & 5 \end{array} \right]$
D) $\left[ \begin{array} { r } - 2 \\ 5 \\ 5 \end{array} \right]$

Question 75

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The total income from the 5% and 6% investments is equal to the income
From the 8% investment. Find the amount invested at each rate.

A) $10,000 at 5%, $10,000 at 6%, $5000 at 8%
B) $10,000 at 5%, $5000 at 6%, $10,000 at 8%
C) $8000 at 5%, $10,000 at 6%, $7000 at 8%
D) $5000 at 5%, $10,000 at 6%, $10,000 at 8%

Question 76

Find the indicated sum or difference, if it is defined.
Compute the sum of $A = \left[ \begin{array} { r r } - 7 & 1 \\ 2 & 5 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 6 & 2 \\ 6 & 3 \end{array} \right]$ .

A) $\left[ \begin{array} { r r } - 1 & 3 \\ 8 & 8 \end{array} \right]$
В) $[ 18 ]$
C) $\left[ \begin{array} { l l } 3 & 4 \\ 4 & 8 \end{array} \right]$
D) $\left[ \begin{array} { l l } - 1 & - 5 \\ - 1 & - 4 \end{array} \right]$

Question 77

Find the indicated matrix.
Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 3 \\ 0 & 2 \end{array} \right]$ . Find $3 \mathrm {~A}$ .

A) $\left[ \begin{array} { r r } - 9 & 9 \\ 0 & 2 \end{array} \right]$
B) $\left[ \begin{array} { r r } - 9 & 3 \\ 0 & 2 \end{array} \right]$
C) $\left[ \begin{array} { r r } - 9 & 9 \\ 0 & 6 \end{array} \right]$
D) $\left[ \begin{array} { l l } 0 & 6 \\ 3 & 5 \end{array} \right]$

Question 78

Find the indicated sum or difference, if it is defined.
Compute the difference $\mathrm { A } - \mathrm { B }$ if $\mathrm { A } = \left[ \begin{array} { r r } - 1 & 0 \\ 6 & 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 1 & 6 \\ 3 & 1 \end{array} \right]$ .

A) $\left[ \begin{array} { r r } 0 & - 6 \\ 3 & 2 \end{array} \right]$
B) $\left[ \begin{array} { r r } - 2 & 6 \\ 9 & 4 \end{array} \right]$
C) $[ - 1 ]$
D) $\left[ \begin{array} { r r } 0 & 6 \\ - 3 & - 2 \end{array} \right]$

Question 79

Find the solution or solutions, if any exist, to the system.
$$\left\{ \begin{array} { c } 2 x - 6 z - 6 w = - 4 \\x + y + z + 3 w = 2 \\4 x + 2 y - 4 z - 4 w = 0 \\6 x + 4 y - 2 z + 2 w = 4\end{array} \right.$$

A) infinitely many solutions of the form $x = 3 z - 2 , y = - 4 z + 4 , z = z , w = 0$
B) inconsistent system, no solution
C) $x = - 2 , y = 4 , z = 0 , w = 0$
D) infinitely many solutions of the form $x = - 3 z - 2 , y = 4 z + 4 , z = 0 , w = w$

Question 80

A $128,000 trust is to be invested in bonds paying 9%, CDs paying 6%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $10,900 annual income from the
Investments, how much should the bank invest in bonds?

A) $42,000
B) $64,000
C) $20,000
D) $22,000