Deck 10: Systems of Equations and Inequalities

Question 1

Solve the system.

- $$\left\{ \begin{array} { l } x - 4 y = - 10 \\2 x - 8 y = - 17\end{array} \right.$$

A) (2, 4)
B) (2, 3)
C) dependent (many solutions)
D) inconsistent (no solution)

Answer 1

inconsistent (no solution)

Question 2

Solve the problem.
A flat rectangular piece of aluminum has a perimeter of 68 inches. The length is 14 inches longer than the width. Find the width.

A) 28 inches
B) 26 inches
C) 8 inches
D) 18 inches

Answer 2

C

Question 3

Solve the system.

- $$\left\{ \begin{array} { l } x + y = 1 \\x + y = - 3\end{array} \right.$$

A) inconsistent (no solution)
B) (0, -2)
C) dependent (many solutions)
D) (1, -3)

Answer 3

inconsistent (no solution)

Question 4

Solve the system of equations by using substitution.

- $$\left\{ \begin{array} { l } 5 x + 3 y = 80 \\2 x + y = 30\end{array} \right.$$

A) x = 0, y = 0
B) x = 10, y = 10
C) x = 0, y = 10
D) x = 10, y = 0

Question 5

Use the elimination method to solve the system.

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

A) x = 5, y = 14
B) x = 8, y = -10
C) x = 2, y = -10
D) x = 9, y = 5

Question 6

Use the elimination method to solve the system.

- $$\left\{ \begin{array} { c } 2 x + 24 y = - 154 \\9 x + 4 y = 35\end{array} \right.$$

A) x = 9, y = -9
B) x = 7, y = -7
C) x = -4, y = 7
D) x = -7, y = 7

Question 7

Solve the problem.
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) 10 adults
B) 29 adults
C) 16 adults
D) 24 adults

Question 8

Verify that the values of the variables listed are solutions of the system of equations.

- $$\left\{ \begin{array} { r } x - y + 4 z = - 3 \\3 x + z = - 2 \\x + 2 y + z = - 12 \\x = 0 , y = - 5 , z = - 2\end{array} \right.$$

A) solution
B) not a solution

Question 9

Verify that the values of the variables listed are solutions of the system of equations.

- $$\begin{array} { l } \left\{ \begin{array} { l } x + y = 8 \\x - y = 4\end{array} \right. \\x = 6 , y = 2\end{array}$$

A) not a solution
B) solution

Question 10

Solve the system of equations by using substitution.

- $$\left\{ \begin{array} { r } x + 7 y = - 2 \\3 x + y = 34\end{array} \right.$$

A) x = -2, y = 3
B) x = 7, y = 12
C) x = 12, y = -2
D) x = 3, y = 7

Question 11

Solve the system of equations by using substitution.

- $$\left\{ \begin{array} { r } 5 x - 2 y = - 1 \\x + 4 y = 35\end{array} \right.$$

A) x = 3, y = 9
B) x = 2, y = 9
C) x = 3, y = 8
D) x = 2, y = 8

Question 12

Solve the system of equations by using substitution.

- $$\left\{ \begin{array} { l } 3 x + y = 13 \\2 x + 9 y = - 8\end{array} \right.$$

A) x = 5, y = 2
B) x = -5, y = 2
C) x = 5, y = -2
D) x = -5, y = -2

Question 13

Solve the problem.
A retired couple has $170,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 11% per year. How much
Should they invest in each to realize exactly $16,300 per year?

A) $120,000 at 5% and $70,000 at 12%
B) $130,000 at 5% and $60,000 at 12%
C) $130,000 at 12% and $60,000 at 5%
D) $140,000 at 12% and $50,000 at 5%

Question 14

Use the elimination method to solve the system.

- $$\left\{ \begin{array} { r } 3 x + 48 y = 48 \\3 x - 6 y = - 6\end{array} \right.$$

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

Question 15

Solve the system of equations by using substitution.

- $$\left\{ \begin{array} { c } x + 7 y = 7 \\3 x - 8 y = - 8\end{array} \right.$$

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

Question 16

Solve the problem.
The Family Fine Arts Center charges $20 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 560 people paid admission, the total receipts were $7,390. How many who paid were
Senior citizens?

A) 282 senior citizens
B) 115 senior citizens
C) 205 senior citizens
D) 372 senior citizens

Question 17

Solve the problem.
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) 168 miles
B) 153 miles
C) 147 miles
D) 132 miles

Question 18

Verify that the values of the variables listed are solutions of the system of equations.

- $$\begin{array} { l } \left\{ \begin{array} { l } 4 x + y = 13 \\2 x + 4 y = 24\end{array} \right. \\x = 2 , y = 5\end{array}$$

A) not a solution
B) solution

Question 19

Verify that the values of the variables listed are solutions of the system of equations.

- $$\begin{array} { l } \left\{ \begin{array} { c } x + y + z = 6 \\x - y + 4 z = 14 \\5 x + y + z = 26\end{array} \right. \\x = 5 , y = - 1 , z = 2\end{array}$$

A) not a solution
B) solution

Question 20

Solve the problem.
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 5 soft drinks for $40.29. The second group bought 6 slices of pizza and 7 soft drinks for $
35)71. How much does one slice of pizza cost?

A) $1.94 per slice of pizza
B) $2.44 per slice of pizza
C) $3.27 per slice of pizza
D) $2.77 per slice of pizza

Question 21

Write the augmented matrix for the system.

- $$\left\{ \begin{array} { c } - 2 x + 6 y = 8 \\5 y = 20\end{array} \right.$$

A) $\left[ \begin{array} { r r | r } 8 & 6 & - 2 \\ 20 & 0 & 5 \end{array} \right]$

B) $\left[ \begin{array} { r r | r } 5 & 0 & 20 \\ - 2 & 6 & 6 \end{array} \right]$

C) $\left[ \begin{array} { r r | r } - 2 & 6 & 8 \\ 0 & 5 & 20 \end{array} \right]$

D) $\left[ \begin{array} { r r | r } - 2 & 6 & 8 \\ 5 & 20 & 0 \end{array} \right]$

Question 22

Solve the problem.
The Family Arts Center charges $22 for adults, $12 for senior citizens, and $7 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $10,134 for 757 tickets sold. There
Were 41 more children than adults. How many children attended?

A) 200 children
B) 244 children
C) 234 children
D) 208 children

Question 23

Solve the system.

- $$\left\{ \begin{array} { l } 4 x - 7 y = 8 \\4 x - 7 y = - 5\end{array} \right.$$

A) $\left( \frac { 8 } { 7 } , - \frac { 5 } { 4 } \right)$
B) inconsistent (no solution)
C) $( 8 , - 5 )$
D) consistent (many solutions)

Question 24

Solve the system of equations.

- $\left\{\begin{array}{r}x+4 y-z=3 \\x+5 y-2 z=5 \\3 x+12 y-3 z=9\end{array}\right.$

A)
$\begin{array}{l}x=3 z+5 \\y=z-2 \\z=\text { any real number }\end{array}$

B)
$\begin{array}{l}x=-3 z-5 \\y=z+2 \\z=\text { any real number }\end{array}$

C)
$\begin{array}{l}x=z-2 \\y=-3 z-5 \\z=\text { any real number }\end{array}$

D) $\text { inconsistent (no solution) }$

Question 25

Solve the system of equations.

- $$\left\{ \begin{aligned}x - y + 3 z & = 10 \\2 x + z & = 4 \\x + 2 y + z & = 8\end{aligned} \right.$$

A) $x = - 4 , y = 0 , z = 5$
B) $x = - 4 , y = 5 , z = 0$
C) inconsistent (no solution)
D) $x = 0 , y = 5 , z = - 4$

Question 26

Solve the system of equations.

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

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

Question 27

Solve the system.

- $$\left\{ \begin{array} { c } x + 5 y = 4 \\3 x + 15 y = 12\end{array} \right.$$

A) $( 0,0 )$
B) $( - 3,0 )$
C) inconsistent (no solution)
D) $y = - \frac { x } { 5 } + \frac { 4 } { 5 }$ , where $x$ is any real number

Question 28

Write the word or phrase that best completes each statement or answers the question.
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?

Question 29

Write the word or phrase that best completes each statement or answers the question.

-A company has sales (measured in millions of dollars) of 50, 60, and 75 during the first three consecutive years. Find a quadratic function that fits these data, and use the result to predict the sales during the fourth year.
Assume that the quadratic function is of the form y = ax2 + bx + c

A) $$y = \frac { 5 } { 2 } x ^ { 2 } + \frac { 5 } { 2 } x + 45$$ 45; sales during the fourth year = $95 million
B) $$y = - 5 x ^ { 2 } + 40 x + 15$$ 15; sales during the fourth year = $95 million
C) $$y = 5 x ^ { 2 } + 5 x + 40$$ 40; sales during the fourth year = $180 million
D) $$y = \frac { 15 } { 2 } x ^ { 2 } - \frac { 25 } { 2 } x + \frac { 325 } { 4 }$$ ; sales during the fourth year = $151.25 million

Question 30

Solve the system of equations.

- $$\left\{ \begin{aligned}x + y + z & = 0 \\x - y + 4 z & = - 11 \\3 x + 3 y + 3 z & = 4\end{aligned} \right.$$

A) $x = - 3 , y = 1 , z = 2$
B) $x = - 3 , y = 2 , z = 1$
C) $x = 2 , y = 1 , z = - 3$
D) inconsistent (no solution)

Question 31

Solve the system.

- $$\left\{ \begin{array} { l } 7 x - 2 y = 5 \\28 x - 8 y = 15\end{array} \right.$$

A) inconsistent (no solution)
B) 6, - 9
C) (2, 5)
D) consistent (many solutions)

Question 32

Solve the system of equations.

- $$\left\{ \begin{aligned}x - y + 4 z & = 2 \\5 x + z & = 0 \\x + 2 y + z & = - 4\end{aligned} \right.$$

A) $x = 0 , y = 0 , z = - 2$
B) Inconsistent, $\varnothing$
C) $x = 0 , y = - 2 , z = 2$
D) $x = 0 , y = - 2 , z = 0$

Question 33

Solve the system of equations.

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

A) $x = 1 , y = - 2 , z = - 4$
B) $x = - 4 , y = - 2 , z = 1$
C) $x = 1 , y = - 4 , z = - 2$
D) inconsistent (no solution)

Question 34

Write the word or phrase that best completes each statement or answers the question.
Find real numbers a, b, and c such that the graph of the function c contains the points (1, 1), (2, 4),
and (-3, 29).

Question 35

Solve the system of equations.

- $$\left\{ \begin{array} { l } x - y + 3 z = - 4 \\3 x + z = 0 \\- x + y - 3 z = 16\end{array} \right.$$

A) $x = 4 , y = 2 , z = 0$
B) inconsistent (no solution)
C) $x = 0 , y = 0 , z = 2$
D) $x = 0 , y = 2 , z = 0$

Question 36

Write the augmented matrix for the system.

- $$\left\{ \begin{array} { l } 9 x + 8 y = 50 \\5 x + 6 y = 34\end{array} \right.$$

A) $\left[ \begin{array} { l l | l } 2 & 6 & 26 \\ 5 & 9 & 35 \end{array} \right]$

B) $\left[ \begin{array} { l l | l } 26 & 6 & 2 \\ 35 & 5 & 9 \end{array} \right]$

C) $\left[ \begin{array} { l | l } 2 & 6 \end{array} \right.$

D) $\left[ \begin{array} { l l | l } 2 & 5 & 26 \\ 6 & 9 & 35 \end{array} \right]$

Question 37

Solve the system of equations.

- $$\left\{ \begin{aligned}2 x - y + 5 z & = - 7 \\x + y - 2 z & = - 2 \\x - y + 4 z & = 8\end{aligned} \right.$$

A) $x = - 3 - z$
$y = 3 z + 1$
$z =$ any real number

B)
$$\begin{array}{l}x=3 z+1 \\y=z-3\end{array}$$
$z=$ any real number

C) $x = z + 3$
$y = 3 z + 1$
$z =$ any real number

D) inconsistent (no solution)

Question 38

Solve the system.

- $$\left\{ \begin{aligned}8 x + y & = 2 \\- 24 x - 3 y & = - 6\end{aligned} \right.$$

A) inconsistent (no solution)
B) y = 8x + 6, where x is any real number
C) y = -8x + 2, where x is any real number
D) x = -8y + 2, where y is any real number

Question 39

Solve the system of equations.

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

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 40

Solve the system of equations.

- $$\left\{ \begin{aligned}- x + y + 2 z & = 0 \\x + 2 y + z & = 6 \\- 2 x - y + z & = - 6\end{aligned} \right.$$

A) inconsistent (no solution)
B) $x = 2 - z$
$y = z + 2$
$z =$ any real number
C) $x = z + 2$
$y = 2 - z$
$z =$ any real number

D) $x = z + 2$
$y = z - 2$
$z =$ any real number

Question 41

Perform the row operation(s) on the given augmented matrix.

- $$\begin{array} { l } \mathrm { R } _ { 2 } = - 4 \mathrm { r } _ { 1 } \\{ \left[ \begin{array} { l l | l } 1 & 7 & 10 \\4 & 8 & - 4\end{array} \right] }\end{array}$$

A) $\left[ \begin{array} { r r | r } - 4 & - 28 & - 40 \\ 0 & - 20 & - 44 \end{array} \right]$

B) $\left[ \begin{array} { r r | r } 1 & 3 & 10 \\ 0 & 10 & 18 \end{array} \right]$

C) $\left[ \begin{array} { r r | r } - 4 & - 28 & - 40 \\ 4 & 8 & - 4 \end{array} \right]$

D) $\left[ \begin{array} { r r | r } 1 & 7 & 10 \\ 0 & - 20 & - 44 \end{array} \right]$

Question 42

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

- $$\left\{ \begin{aligned}3 x + 9 y - z & = 53 \\x - 9 y + 9 z & = 11 \\- 4 x + y + z & = 5\end{aligned} \right.$$

A) $x = 2 , y = 6 , z = 7$
B) $x = - 2 , y = 6 , z = 4$
C) $x = 2 , y = 7 , z = 6$
D) Inconsistent

Question 43

Perform the row operation(s) on the given augmented matrix.

- $$\begin{array} { l } \mathrm { R } _ { 1 } = \frac { 1 } { 2 } \mathrm { r } _ { 1 } \\{ \left[ \begin{array} { r r | r } 2 & - 2 & - 4 \\4 & 2 & - 5\end{array} \right] }\end{array}$$

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

B) $\left[ \begin{array} { r r | r } 1 & - 1 & - 4 \\ 4 & 2 & - 5 \end{array} \right]$

C) $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \\ 5 & 1 & - 7 \end{array} \right]$

D) $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \\ 4 & 2 & - 5 \end{array} \right]$

Question 44

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

- $$\left\{ \begin{array} { r } - 2 x - 8 y = - 8 \\4 x + 16 y = 3\end{array} \right.$$

A) $x = - 8 , y = 3$
B) $x = - 2 , y = 3$
C) $x = - 2 , y = - 2$
D) Inconsistent

Question 45

Perform the row operation(s) on the given augmented matrix.

- $$\begin{array} { l } R _ { 2 } = - 2 r _ { 2 } + r \\{ \left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\1 & 2 & 3 & 6 \\4 & 6 & 7 & 1\end{array} \right] }\end{array}$$

A) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ 0 & 0 & 0 & - 20 \\ 4 & 6 & 7 & 1 \end{array} \right]$

B) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ - 4 & - 8 & - 12 & - 20 \\ 4 & 6 & 7 & 1 \end{array} \right]$

C) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ 0 & 0 & 0 & 0 \\ 4 & 6 & 7 & 1 \end{array} \right]$

D) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ - 4 & - 8 & - 12 & 6 \\ 4 & 6 & 7 & 1 \end{array} \right]$

Question 46

Perform the row operation(s) on the given augmented matrix.

Question 47

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

- $$\begin{array} { l } x + 3 y - 2 z - w = 27 \\4 x + y + z + 2 w = 14 \\- 3 x - y - 3 z - 2 w = - 1 \\x - y - 3 z - 2 w = 11\end{array}$$

A) $\{ ( 27,14 , - 1,11 ) \}$
B) $\{ ( 3,5 , - 5,1 ) \}$
C) $\{ ( 2,7 , - 9,3 ) \}$
D) $\{ ( 3 + w , 5 - 2 w , - 5 + 2 w , w ) \}$

Question 48

Write the system of equations associated with the augmented matrix. Do not solve.

- $$\left[ \begin{array} { r r r | r } 3 & 6 & 2 & - 2 \\6 & 0 & 7 & 4 \\2 & 7 & 0 & 2\end{array} \right]$$

A) $\left\{ \begin{array} { l } 3 x - 6 y + 2 z = - 2 \\ 6 x + 7 z = - 4 \\ 2 x + 7 y = - 2 \end{array} \right.$

B) $\left\{ \begin{array} { l } 3 x + 6 y + 2 z = - 2 \\ 6 x + 7 z = 4 \\ 2 x + 7 y = 2 \end{array} \right.$

C) $\left\{ \begin{array} { l } 3 x + 6 y + 2 z = - 2 \\ 6 x + 7 z = 4 \\ 2 x + 7 z = 2 \end{array} \right.$

Question 49

Write the augmented matrix for the system.

- $$\left\{ \begin{aligned}4 x + 9 y + 2 z & = 30 \\- 2 x + 6 y + 2 z & = 0 \\9 x + 6 y + 5 z & = 38\end{aligned} \right.$$

A) $\left[ \begin{array} { r r | r } 4 & 9 & 2 \\ - 2 & 6 & 2 \\ 9 & 5 \end{array} \right]$

B) $\left[ \begin{array} { r r r | r } 30 & 2 & 9 & 4 \\ 0 & 2 & 6 & - 2 \\ 38 & 5 & 6 & 9 \end{array} \right]$

C) $\left[ \begin{array} { r r r | r } 4 & 9 & 2 & 30 \\ - 2 & 6 & 2 & 0 \\ 9 & 6 & 5 & 38 \end{array} \right]$

D) $\left[ \begin{array} { r r r | r } 4 & - 2 & 9 & 30 \\ 9 & 6 & 6 & 0 \\ 2 & 2 & 5 & 38 \end{array} \right]$

Question 50

Solve the problem using matrices.

-Find the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ for which $f ( 0 ) = - 2 , f ( 1 ) = 5 , f ( - 1 ) = 3$ , $f ( 2 ) = 4$ .

A) $f ( x ) = 10 x ^ { 3 } - 18 x ^ { 2 } - 13 x + 6$

B) $f ( x ) = - 8 x ^ { 3 } + 12 x ^ { 2 } + 5 x - 6$

C) $f ( x ) = - \frac { 10 } { 3 } x ^ { 3 } + 6 x ^ { 2 } + \frac { 13 } { 3 } x - 2$

D) $f ( x ) = \frac { 8 } { 3 } x ^ { 3 } + 4 x ^ { 2 } + \frac { 5 } { 3 } x - 2$

Question 51

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

- $$\left\{ \begin{aligned}- 3 x - y + 8 z & = 16 \\- 9 x + 6 z & = - 33 \\4 y + z & = 17\end{aligned} \right.$$

A) $x = 7 , y = 5 , z = 3$
B) $x = - 7 , y = 3 , z = 14$
C) $x = 7 , y = 3 , z = 5$
D) Inconsistent

Question 52

Perform the row operation(s) on the given augmented matrix.

-(a) $R _ { 2 } = - 2 r _ { 1 } + r _ { 2 }$
(b) $\mathrm { R } _ { 3 } = - 2 \mathrm { r } _ { 1 } + \mathrm { r } _ { 3 }$
(c) $\mathrm { R } _ { 3 } = 6 \mathrm { r } _ { 2 } + \mathrm { r } _ { 3 }$
$$\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\2 & - 5 & - 4 & 5 \\2 & 5 & 4 & 6\end{array} \right]$$

A) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & 4 & 11 & 11 \\ 0 & 21 & 35 & 25 \end{array} \right]$

B) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & 4 & 11 & 11 \\ 0 & 37 & 79 & 69 \end{array} \right]$

C) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & - 8 & - 9 & 3 \\ 0 & - 37 & - 40 & - 8 \end{array} \right]$

D) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & 14 & 19 & 4 \\ 0 & 87 & 119 & 34 \end{array} \right]$

Question 53

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

- $$\begin{array} { l } x - y + 2 z + w = 2 \\y + z = 2 \\z - w = 3\end{array}$$

A) $\varnothing$
B) $\{ ( - 9 , - 2,4,1 ) \}$
C) $\{(-5-4 \mathrm{w},-1-\mathrm{w}, 3+\mathrm{w}, \mathrm{w})\}$
D) $\{ ( - 5 , - 1,3 , \mathrm { w } ) \}$

Question 54

Write the system of equations associated with the augmented matrix. Do not solve.

- $$\left[ \begin{array} { r r r | r } 8 & 8 & - 8 & 7 \\6 & 6 & 0 & 9 \\3 & 0 & - 4 & - 6\end{array} \right]$$

A)
$$\left\{ \begin{array} { l } 8 x + 8 y - 8 z = 7 \\6 x + 6 y = 9 \\3 x - 4 y = - 6\end{array} \right.$$
B)
$$\left\{ \begin{array} { r } 8 x + 8 y - 8 z = 7 \\6 x + 6 y = 9 \\3 x - 4 z = - 6\end{array} \right.$$
C)
$$\left\{ \begin{array} { l } 8 x + 8 y - 8 z = 7 \\6 x + 6 y + 9 z = 0 \\3 x - 4 y - 6 z = 0\end{array} \right.$$
D)
$$\left\{ \begin{aligned}8 x + 8 y - 8 z & = 7 \\6 x + 6 z & = 9 \\3 x - 4 y & = - 6\end{aligned} \right.$$

Question 55

Write the augmented matrix for the system.

- $$\left\{ \begin{array} { r } 8 x - 2 z = 60 \\9 y + 7 z = 33 \\2 x - 2 y + 9 z = 74\end{array} \right.$$

A) $\left[ \begin{array} { r r r | r } 8 & - 2 & 0 & 60 \\ 9 & 7 & 0 & 33 \\ 2 & - 2 & 9 & 74 \end{array} \right]$

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

C) $\left[ \begin{array} { r r r | r } 8 & 0 & 2 & 60 \\ 0 & 9 & - 2 & 33 \\ - 2 & 7 & 9 & 74 \end{array} \right]$

D) $\left[ \begin{array} { r r r | r } 8 & 0 & - 2 & 60 \\ 0 & 9 & 7 & 33 \\ 2 & - 2 & 9 & 74 \end{array} \right]$

Question 56

Write the system of equations associated with the augmented matrix. Do not solve.

- $$\left[ \begin{array} { c c | c } 9 & 7 & 17 \\7 & 18 & 5\end{array} \right]$$

A) $\left\{ \begin{array} { l } 7 x + 9 y = 17 \\ 7 x + 18 y = 5 \end{array} \right.$

B) $\left\{ \begin{array} { l } 9 x + 7 y = 17 \\ 7 x + 18 y = 5 \end{array} \right.$

C) $\left\{ \begin{array} { l } 9 x + 7 y = - 17 \\ 7 x + 18 y = - 5 \end{array} \right.$

D) $\left\{ \begin{array} { l } 9 x + 7 y = 17 \\ 18 x + 7 y = 5 \end{array} \right.$

Question 57

Perform the row operation(s) on the given augmented matrix.

-(a) $R _ { 2 } = - 4 r _ { 1 } + r _ { 2 }$
(b) $R _ { 3 } = 4 r _ { 1 } + r _ { 3 }$
(c) $R _ { 3 } = 3 r _ { 2 } + r _ { 3 }$
$$\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & 2 \\4 & - 5 & 2 & 5 \\- 4 & - 5 & 4 & 6\end{array} \right]$$

A) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & 2 \\ 0 & 7 & 22 & - 3 \\ 0 & 28 & 90 & 5 \end{array} \right]$

B) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & 2 \\ 0 & 7 & 22 & - 3 \\ 0 & 4 & 50 & 5 \end{array} \right]$

C) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & 2 \\ 0 & 7 & 22 & - 3 \\ 0 & - 68 & - 70 & 5 \end{array} \right]$

D) $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & 2 \\ 0 & 17 & 22 & - 3 \\ 0 & 0 & 6 & 11 \end{array} \right]$

Question 58

Write the system of equations associated with the augmented matrix. Do not solve.

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

A) $\left\{ \begin{array} { l } - 5 x + 9 y = - 6 \\ - 7 x - 2 y = - 5 \end{array} \right.$

B) $\left\{ \begin{array} { c } 9 x - 5 y = - 6 \\ - 7 x - 2 y = - 5 \end{array} \right.$

C) $\left\{ \begin{array} { l } 9 x - 5 y = 0 \\ - 7 x - 2 y = 0 \end{array} \right.$

D) $\left\{ \begin{array} { c } 9 x - 5 y = - 6 \\ - 2 x - 7 y = - 5 \end{array} \right.$

Question 59

Write the system of equations associated with the augmented matrix. Do not solve.

- $$\left[ \begin{array} { r r r | r } 1 & 0 & 0 & - 8 \\0 & 1 & 0 & - 6 \\0 & 0 & 1 & - 10\end{array} \right]$$

A) $\left\{ \begin{array} { l } x = 0 \\ y = - 14 \\ z = - 18 \end{array} \right.$

B) $\left\{ \begin{array} { l } x = 15 \\ y = 1 \\ z = 0 \end{array} \right.$

C) $\left\{ \begin{array} { l } x = - 9 \\ y = 5 \\ z = 6 \end{array} \right.$

D) $\left\{ \begin{array} { l } x = - 8 \\ y = - 6 \\ z = - 10 \end{array} \right.$

Question 60

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

- $$\left\{ \begin{array} { l } 5 x + 6 y = 20 \\2 x + 8 y = 36\end{array} \right.$$

A) $x = - 2 , y = - 5$
B) $x = 5 , y = - 2$
C) $x = - 2 , y = 5$
D) Inconsistent

Question 61

Use Cramer's rule to solve the linear system.

- $$\left\{ \begin{array} { l } 9 x - 2 y - z = 64 \\x - 3 y - 3 z = - 27 \\- 6 x + y + z = - 42\end{array} \right.$$

A) x = 10, y = 3, z = 7
B) x = 9, y = -5, z = -7
C) x = 5, y = 7, z = 5
D) x = 9, y = 5, z = 7

Question 62

Use Cramer's rule to solve the linear system.

- $$\left\{ \begin{aligned}8 x - 3 z & = 29 \\2 x + 9 y + 5 z & = 104 \\2 x - 9 y & = - 31\end{aligned} \right.$$

A) x = 8, y = 3, z = 9
B) x = 7, y = 5, z = 9
C) x = 7, y = -5, z = -9
D) x = 5, y = 9, z = 5

Question 63

Use Cramer's rule to solve the linear system.

- $$\left\{ \begin{array} { c } 2 x + 5 y = 21 \\2 x + y = 1\end{array} \right.$$

A) x = 5, y = -2
B) x = -5, y = -2
C) x = -2, y = 5
D) x = 2, y = -5

Question 64

Solve for x.

Question 65

Find the value of the determinant.

- $$\left| \begin{array} { l l l } 3 & 5 & 5 \\6 & 3 & 2 \\1 & 1 & 5\end{array} \right|$$

A) -106
B) 256
C) -86
D) 86

Question 66

Find the value of the determinant.

- $$\left| \begin{array} { l l } 6 & 3 \\5 & 2\end{array} \right|$$

A) -3
B) 3
C) 27
D) 8

Question 67

Find the value of the determinant.

- $$\left| \begin{array} { r r r } - 2 & 5 & 4 \\3 & - 2 & 1 \\1 & 6 & - 3\end{array} \right|$$

A) -90
B) 130
C) 80
D) -12

Question 68

Solve the problem.

-The equation of the line passing through the distinct points $\left( x _ { 1 } , y _ { 1 } \right)$ and (x $\left. 2 , y _ { 2 } \right)$ is given by $\left| \begin{array} { l l l } x _ { 1 } & y _ { 1 } & 1 \\ x _ { 2 } & y _ { 2 } & 1 \end{array} \right| = 0 .$ Find the equation of the line passing through the points $( 3,5 )$ and $( - 1,4 )$ .

A) x + 4y + 17 = 0
B) -x + 4y - 17 = 0
C) x + 7y + 17 = 0
D) x - 4y + 17 = 0

Question 69

Solve the problem using matrices.
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?

Question 70

Use Cramer's rule to solve the linear system.

- $$\left\{ \begin{array} { l } - 2 x + 4 y = - 12 \\- 2 x + 5 y = - 11\end{array} \right.$$

A) x = 1, y = 8
B) x = -1, y = 8
C) x = -8, y = -1
D) x = 8, y = 1

Question 71

Use Cramer's rule to solve the linear system.

- $$\left\{ \begin{array} { l } 3 x + 3 y = 51 \\2 x - 2 y = 2\end{array} \right.$$

A) x = 8, y = 9
B) x = 9, y = 8
C) x = -9, y = -8
D) x = -8, y = 9

Question 72

Find the value of the determinant.

- $$\left| \begin{array} { r r } 12 & - 7 \\- 4 & 3\end{array} \right|$$

A) 4
B) 8
C) 64
D) -8

Question 73

Solve the problem using matrices.
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?

Question 74

Solve for x.

Question 75

Use Cramer's rule to solve the linear system.

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

A) $x = - \frac { 2 } { 17 } , y = \frac { 11 } { 17 }$

B) $x = \frac { 23 } { 3 } , y = - \frac { 11 } { 3 }$

C) $x = \frac { 2 } { 17 } , y = - \frac { 11 } { 17 }$

D) $x = \frac { 2 } { 3 } , y = \frac { 1 } { 3 }$

Question 76

Find the value of the determinant.

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

A) -1
B) 5
C) -5
D) 7

Question 77

Find the value of the determinant.

- $$\left| \begin{array} { r r } - 6 & - 4 \\- 8 & 8\end{array} \right|$$

A) 80
B) 88
C) -16
D) -80

Question 78

Find the value of the determinant.

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

A) 15
B) -15
C) 87
D) -39

Question 79

Use Cramer's rule to solve the linear system.

- $$\left\{ \begin{array} { l } - 4 x - 6 y - 8 z = - 84 \\- 4 x + 4 y - 3 z = 11 \\- 9 x - 5 y - 9 z = - 94\end{array} \right.$$

A) x = 3, y = 8, z = 3
B) x = 4, y = 6, z = 3
C) x = 8, y = 3, z = 8
D) x = 3, y = -8, z = -3

Question 80

Solve the problem using matrices.
Find real numbers a, b, and c such that the graph of the function c contains the points (-2, -4), (1,
-1), and (3, -19).