Deck 6: Systems of Equations and Inequalities

Question 1

Solve the problem.
The Family Arts Center charges $21 for adults, $13 for senior citizens, and $8 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $12,388 for 905 tickets sold. There
Were 50 more children than adults. How many children attended?

A) 291 children
B) 341 children
C) 273 children
D) 331 children

Answer 1

B

Question 2

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) 132 miles
B) 147 miles
C) 168 miles
D) 153 miles

Answer 2

D

Question 3

Verify that the values of the variables listed are solutions of the system of equations.
$$\left\{ \begin{array} { c } x + y + z = - 1 \\x - y + 3 z = - 13 \\4 x + y + z = 8 \\\end{array} \right.$$

x = 3 , y = 1 , z = - 5

A) not a solution
B) solution

Answer 3

B

Question 4

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

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

Question 5

Solve the problem.
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).

Question 6

Solve the problem.
The Family Fine Arts Center charges $24 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 538 people paid admission, the total receipts were $7592. How many who paid were
Senior citizens?

A) 158 senior citizens
B) 290 senior citizens
C) 248 senior citizens
D) 380 senior citizens

Question 7

Solve the problem.
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 8

Verify that the values of the variables listed are solutions of the system of equations.
$$\begin{array} { l } \left\{ \begin{array} { l } 2 x + y = 6 \\3 x + 2 y = 7\end{array} \right. \\x = 5 , y = - 4\end{array}$$

A) not a solution
B) solution

Question 9

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 7 soft drinks for $36.53. The second group bought 6 slices of pizza and 5 soft drinks for $
26)93. How much does one slice of pizza cost?

A) $1.87 per slice of pizza
B) $2.37 per slice of pizza
C) $2.43 per slice of pizza
D) $2.93 per slice of pizza

Question 10

Verify that the values of the variables listed are solutions of the system of equations.
$$\left\{ \begin{array} { l } x + y = - 1 \\x - y = 11 \\\end{array} \right.$$

x = 5 , y = - 6

A) solution
B) not a solution

Question 11

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

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

Verify that the values of the variables listed are solutions of the system of equations.
$$\left\{ \begin{array} { c } x - y + 3 z = - 20 \\4 x + z = - 5 \\x + 2 y + z = 5 \\\end{array} \right.$$

x = 0 , y = 5 , z = - 5

A) solution
B) not a solution

Question 14

Solve the problem.
A flat rectangular piece of aluminum has a perimeter of 64 inches. The length is 14 inches longer than the width. Find the width.

A) 32 inches
B) 9 inches
C) 37 inches
D) 23 inches

Question 15

Write the augmented matrix for the system.
$$\left\{ \begin{array} { l } 2 x + 8 y = 64 \\5 x + 7 y = 69\end{array} \right.$$

A) $\left[ \begin{array} { l l | l } 2 & 8 & 64 \\ 5 & 7 & 69 \end{array} \right]$
B) $\left[ \begin{array} { l l | l } 2 & 5 & 64 \\ 8 & 7 & 69 \end{array} \right]$
C) $\left[ \begin{array} { l l | l } 2 & 8 & 69 \\ 7 & 5 & 64 \end{array} \right]$
D) $\left[ \begin{array} { l l | l } 64 & 8 & 2 \\ 69 & 5 & 7 \end{array} \right]$

Question 16

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) 24 adults
C) 29 adults
D) 16 adults

Question 17

Write the augmented matrix for the system.
$$\left\{ \begin{array} { l } 9 x - 2 y + 6 z = 54 \\6 x + 2 y + 9 z = 51 \\5 x + 3 y + 2 z = 41\end{array} \right.$$

A) $\left[ \begin{array} { r r r | r } 54 & 6 & - 2 & 9 \\ 51 & 9 & 2 & 6 \\ 41 & 2 & 3 & 5 \end{array} \right]$
B) $\left[ \begin{array} { r r r | r } 9 & 6 & 5 & 54 \\ - 2 & 2 & 3 & 51 \\ 6 & 9 & 2 & 41 \end{array} \right]$
C) $\left[ \begin{array} { r r | r } 9 & - 2 & 6 \\ 6 & 2 & 9 \\ 5 & 3 & 2 \end{array} \right]$
D) $\left[ \begin{array} { r r r | r } 9 & - 2 & 6 & 54 \\ 6 & 2 & 9 & 51 \\ 5 & 3 & 2 & 41 \end{array} \right]$

Question 18

Solve the problem.
A retired couple has $140,000 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 12% per year. How much
Should they invest in each to realize exactly $13,300 per year?

A) $90,000 at 5% and $50,000 at 12%
B) $80,000 at 5% and $60,000 at 12%
C) $90,000 at 12% and $50,000 at 5%
D) $100,000 at 12% and $40,000 at 5%

Question 19

Solve the system of equations by using substitution.
$$\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.$$

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

Question 20

Write the augmented matrix for the system.
$$\left\{ \begin{array} { r } - 2 x + 8 y = 28 \\5 y = 20\end{array} \right.$$

A) $\left[ \begin{array} { r | r | r } 5 & 0 & 20 \\ - 2 & 8 & 8 \end{array} \right]$
B) $\left[ \begin{array} { r r | r } 28 & 8 & - 2 \\ 20 & 0 & 5 \end{array} \right]$
C) $\left[ \begin{array} { r r | r } - 2 & 8 & 28 \\ 0 & 5 & 20 \end{array} \right]$
D) $\left[ \begin{array} { r r | r } - 2 & 8 & 28 \\ 5 & 20 & 0 \end{array} \right]$

Question 21

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

Question 22

Perform the row operation(s) on the given augmented matrix.
$$\begin{array} { l } \mathrm { R } _ { 1 } = \frac { 1 } { 5 } \mathrm { r } _ { 1 } \\{ \left[ \begin{array} { r r | r } 5 & - 10 & 25 \\4 & 2 & - 5\end{array} \right] }\end{array}$$

A) $\left[ \begin{array} { r r | r } 1 & - 2 & 25 \\ 4 & 2 & - 5 \end{array} \right]$
B) $\left[ \begin{array} { r r | r } 1 & - 2 & 5 \\ \frac { 4 } { 5 } & \frac { 2 } { 5 } & - 1 \end{array} \right]$
C) $\left[ \begin{array} { r r | r } 1 & - 2 & 5 \\ 4 & 2 & - 5 \end{array} \right]$
D) $\left[ \begin{array} { r r | r } 1 & - 2 & 5 \\ 5 & 0 & 0 \end{array} \right]$

Question 23

Find the value of the determinant.
$$\left\{ \begin{aligned}6 x - 2 z & = 16 \\2 x + 9 y - 5 z & = 64 \\- 8 x - 5 y & = - 59\end{aligned} \right.$$

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

Question 24

Solve the problem.
$$\text { Given that } \left| \begin{array} { l l l } x & y & z \\a & b & c \\2 & 4 & 5\end{array} \right| = 3 \text {, find the value of the determinant } \left| \begin{array} { l l l } 2 & 4 & 5 \\3 a & 3 b & 3 c \\x - 2 & y - 4 & z - 5\end{array} \right| \text {. }$$

A) 0
B) 6
C) 9
D) -9

Question 25

Write the augmented matrix for the system.
$\left. \begin{array} { r r r | r } 4 & 8 & 8 & - 2 \\ 9 & 0 & 5 & 4 \\ 5 & 9 & 0 & 2 \end{array} \right]$

A) $\left\{ \begin{array} { l } 4 x + 8 y + 8 z = - 2 \\ 9 x + 5 z = 4 \\ 5 x + 9 y = 2 \end{array} \right.$
B) $\left\{ \begin{array} { l } 4 x + 8 y + 8 z = - 2 \\ 9 x + 5 z = 4 \\ 5 x + 9 z = 2 \end{array} \right.$
C) $\left\{ \begin{array} { l } 4 x - 8 y + 8 z = - 2 \\ 9 x + 5 z = - 4 \\ 5 x + 9 y = - 2 \end{array} \right.$

Question 26

Write the augmented matrix for the system.
$$\left\{ \begin{array} { r } 5 x + 5 z = 65 \\3 y + 9 z = 96 \\9 x - 2 y + 6 z = 77\end{array} \right.$$

A) $\left[ \begin{array} { r r | r } 5 & 0 & 5 \\ 0 & 3 & 9 \\ 9 & - 2 & 6 \end{array} \right]$
B) $\left[ \begin{array} { r r r | r } 5 & 0 & 9 & 65 \\ 0 & 3 & - 2 & 96 \\ 5 & 9 & 6 & 77 \end{array} \right]$
C) $\left[ \begin{array} { c c c | c } 5 & 5 & 0 & 65 \\ 3 & 9 & 0 & 96 \\ 9 & - 2 & 6 & 77 \end{array} \right]$
D) $\left[ \begin{array} { r r r | r } 5 & 0 & 5 & 65 \\ 0 & 3 & 9 & 96 \\ 9 & - 2 & 6 & 77 \end{array} \right]$

Question 27

Perform the row operation(s) on the given augmented matrix.
$$\begin{array} { l } \mathrm { R } _ { 2 } = - 2 \mathrm { r } _ { 2 } + \mathrm { r } 1 \\{ \left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\1 & 2 & 3 & 6 \\4 & 6 & 7 & 1\end{array} \right] }\end{array}$$ A) $\left[ \begin{array} { l l l | r } 2 & 4 & 6 & - 8 \\ 0 & 0 & 0 & 0 \\ 4 & 6 & 7 & 1 \end{array} \right]$
B) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ - 4 & - 8 & - 12 & - 20 \\ 4 & 6 & 7 & 1 \end{array} \right]$
C) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ 0 & 0 & 0 & - 20 \\ 4 & 6 & 7 & 1 \end{array} \right]$
D) $\left[ \begin{array} { r r r | r } 2 & 4 & 6 & - 8 \\ - 4 & - 8 & - 12 & 6 \\ 4 & 6 & 7 & 1 \end{array} \right]$

Question 28

Perform the row operation(s) on the given augmented matrix.
$$\begin{array} { l } R _ { 2 } = 5 r _ { 1 } + r _ { 2 } \\{ \left[ \begin{array} { c c | c } 1 & 6 & 11 \\- 5 & 9 & - 6\end{array} \right] }\end{array}$$

A) $\left[ \begin{array} { r r | r } 5 & 30 & 55 \\ - 5 & 9 & - 6 \end{array} \right]$
B) $\left[ \begin{array} { r r | r } 1 & 6 & 11 \\ 0 & 39 & 49 \end{array} \right]$
C) $\left[ \begin{array} { r r | r } 1 & 6 & 11 \\ 0 & - 21 & - 61 \end{array} \right]$
D) $\left[ \begin{array} { l l | l } 5 & 30 & 55 \\ 0 & 39 & 49 \end{array} \right]$

Question 29

Use the properties of determinants to find the value of the second determinant, given the value of the first.
$$\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & 3 & - 2\end{array} \right| = 14 \quad \left| \begin{array} { c c c } 1 & 3 & - 2 \\u & v & w \\x & y & z\end{array} \right| = ?$$

A) 14
B) -14
C) 0
D) Cannot determine

Question 30

Use the properties of determinants to find the value of the second determinant, given the value of the first.
$$\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & 2 & - 3\end{array} \right| = 118 \left| \begin{array} { c c c } x - 3 & y - 6 & z + 9 \\- 3 u - 1 & - 3 v - 2 & - 3 w + 3 \\1 & 2 & - 3\end{array} \right| = ?$$

A) 118
B) -118
C) 354
D) -354

Question 31

Find the value of the determinant.
$$\left| \begin{array} { r r r } x & - 4 & - 1 \\- 2 & 2 & 0 \\- 1 & - 2 & 8\end{array} \right| = 10$$

Question 32

Use the properties of determinants to find the value of the second determinant, given the value of the first.
$$\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & - 3 & - 4\end{array} \right| = 62 \left| \begin{array} { c c c } 1 & - 3 & - 4 \\3 u & 3 v & 3 w \\x - 1 & y + 3 & z + 4\end{array} \right| = ?$$

A) 186
B) -186
C) 62
D) -62

Question 33

Find the value of the determinant.
$$\left| \begin{array} { r r r } - 4 & - 3 & - 3 \\3 & - 5 & - 2 \\2 & - 3 & - 1\end{array} \right|$$

A) -20
B) -4
C) 4
D) 34

Question 34

Perform the row operation(s) on the given augmented matrix.
$$\begin{array} { l } \mathrm { R } _ { 3 } = 4 \mathrm { r } 1 + \mathrm { r } _ { 3 } \\{ \left[ \begin{array} { r r r | r } - 7 & - 5 & - 1 & - 10 \\6 & - 2 & 9 & 5 \\28 & - 6 & 6 & 18\end{array} \right] }\end{array}$$

Question 35

Use the properties of determinants to find the value of the second determinant, given the value of the first.
$$\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & - 3 & 4\end{array} \right| = - 138 \left| \begin{array} { c c c } x & y & z \\u & v & w \\2 & - 6 & 8\end{array} \right| = ?$$ $$\begin{array} { c c c } x & y & z \\a & v & w \\1 & - 3 & 4\end{array} | = - 138 | \begin{array} { c c c } x & y & z \\u & v & w \\2 & - 6 & 8\end{array} \mid = ?$$

A) -276
B) -138
C) 276
D) 138

Question 36

Solve the problem.
$$\text { Let } \mathrm { A } = \left[ \begin{array} { l } - 2 \\1 \\- 1\end{array} \right] \text { and } \mathrm { B } = \left[ \begin{array} { l l l } 1 & - 1 & 0\end{array} \right] \text {. Find } \mathrm { AB } \text {. }$$

Question 37

Use the properties of determinants to find the value of the second determinant, given the value of the first.
$$\text { Given } \left| \begin{array} { l l l } \mathrm { s } & \mathrm { t } & \mathrm { u } \\\mathrm { v } & \mathrm { w } & \mathrm { x } \\4 & 2 & 8\end{array} \right| = 3 \text {, find the value of } \mid \begin{array} { c c c } 32 - \mathrm { s } & 16 - \mathrm { t } & 64 - \mathrm { u } \\\mathrm { v } & \mathrm { w } & \mathrm { x } \\4 & 2 & 8\end{array} .$$

A) -24
B) 3
C) -3
D) 24

Question 38

Use the properties of determinants to find the value of the second determinant, given the value of the first.
$$\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & - 2 & - 1\end{array} \right| = 2 \quad \left| \begin{array} { c c c } u & v & w \\2 & - 4 & - 2 \\x & y & z\end{array} \right| = ?$$

A) -2
B) 4
C) -4
D) 2

Question 39

Find the value of the determinant.
$$\left| \begin{array} { l l l } 4 & 4 & 6 \\3 & 4 & 5 \\4 & 4 & 4\end{array} \right|$$

A) -8
B) -168
C) 8
D) 440

Question 40

Use the properties of determinants to find the value of the second determinant, given the value of the first.

A) -54
B) 0
C) 54
D) Cannot determine

Question 41

Find the inverse of the matrix.
$$\left[ \begin{array} { r r r } 1 & 0 & 0 \\- 1 & 1 & 0 \\1 & 1 & 1\end{array} \right]$$

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

Question 42

Find the inverse of the matrix.
$$\left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]$$

A) $\left[ \begin{array} { l l l } - 1 & - 1 & - 1 \\ - 2 & - 1 & - 1 \\ - 2 & - 2 & - 3 \end{array} \right]$
B) $\left[ \begin{array} { r r r } - 1 & 1 & 0 \\ 4 & - 1 & - 1 \\ - 2 & 0 & 1 \end{array} \right]$
C) $\left[ \begin{array} { c c c } 1 & 1 & 1 \\ \frac { 1 } { 2 } & 1 & 1 \\ \frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 } \end{array} \right]$
D) No inverse

Question 43

Find the inverse of the matrix.
$$\left[ \begin{array} { r r } 6 & 1 \\- 2 & 0\end{array} \right]$$

A) $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 2 } \\ 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { l r } 1 & 3 \\ 0 & - \frac { 1 } { 2 } \end{array} \right]$
C) $\left[ \begin{array} { r r } 0 & \frac { 1 } { 2 } \\ - 1 & 3 \end{array} \right]$
D) $\left[ \begin{array} { c c } 3 & - \frac { 1 } { 2 } \\ 1 & 0 \end{array} \right]$

Question 44

Solve the problem.
Let $A = \left[ \begin{array} { r r } - 2 & 3 \\ 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 0 \\ - 1 & 4 \end{array} \right]$ . Find $A B$ .

A) $\left[ \begin{array} { r r } 12 & 1 \\ 8 & - 8 \end{array} \right]$
B) $\left[ \begin{array} { r r } 4 & - 6 \\ - 4 & 5 \end{array} \right]$
C) $\left[ \begin{array} { r r } 4 & 0 \\ - 3 & 8 \end{array} \right]$
D) $\left[ \begin{array} { r r } 1 & 12 \\ - 8 & 8 \end{array} \right]$

Question 45

Find the inverse of the matrix.
$$A = \left[ \begin{array} { r r } 14 & - 6 \\- 7 & 3\end{array} \right]$$

Question 46

Find the inverse of the matrix.
$$\left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]$$

A) $\left[ \begin{array} { l l l } 1 & 1 & 2 \\ 0 & 2 & 5 \\ 8 & 3 & 3 \end{array} \right]$
B) $\left[ \begin{array} { r r r } - 1 & 0 & - 8 \\ - 1 & - 2 & - 3 \\ - 2 & - 5 & - 3 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 9 & - 40 & 16 \\ - 3 & 13 & - 5 \\ - 1 & 5 & - 2 \end{array} \right]$
D) No inverse

Question 47

Solve the problem.
Let $\mathrm { A } = \left[ \begin{array} { r r r } 6 & - 1 & - 2 \\ - 4 & 7 & - 2 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 7 & 6 & 2 \\ - 7 & 7 & - 9 \\ - 4 & - 9 & 7 \end{array} \right]$ . Find $\mathrm { AB }$ .

A) $\left[ \begin{array} { r r r } 6 & - 1 & - 2 \\ - 4 & 7 & - 2 \\ 7 & 7 & - 2 \\ - 7 & 7 & - 9 \\ - 4 & - 4 & 7 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 57 & 47 & 7 \\ - 69 & 43 & - 85 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 42 & - 6 & - 4 \\ 28 & 49 & 18 \\ 16 & - 63 & - 14 \end{array} \right]$
D) $\left[ \begin{array} { r r } 57 & - 69 \\ 47 & 43 \\ 7 & - 85 \end{array} \right]$

Question 48

Perform the indicated operations and simplify.
Let $\mathrm { A } = \left[ \begin{array} { r r } 3 & - 4 \\ - 2 & 5 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { r r r } 5 & - 2 & 8 \\ 1 & 0 & - 3 \end{array} \right]$ , and $\mathrm { C } = \left[ \begin{array} { r r r } 7 & - 9 & 0 \\ 3 & - 5 & 1 \\ - 1 & 6 & 2 \end{array} \right]$ . Find $\mathrm { AB } + \mathrm { BC }$ .

A) $\left[ \begin{array} { r r r } - 10 & - 19 & 12 \\ - 15 & 31 & - 25 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 32 & 19 & 40 \\ - 15 & 31 & - 37 \end{array} \right]$
C) $\left[ \begin{array} { l r r } 32 & 7 & 50 \\ 5 & - 23 & - 37 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 68 & 3 & 31 \\ 8 & - 2 & - 5 \end{array} \right]$

Question 49

Find the inverse of the matrix.
$\left[ \begin{array} { r r } 1 & - 2 \\ 6 & 2 \end{array} \right]$

A) $\left[ \begin{array} { c c } \frac { 1 } { 7 } & \frac { 1 } { 7 } \\ - \frac { 3 } { 7 } & \frac { 1 } { 14 } \end{array} \right]$
B) $\left[ \begin{array} { c c } - \frac { 3 } { 7 } & \frac { 1 } { 14 } \\ \frac { 1 } { 7 } & \frac { 1 } { 7 } \end{array} \right]$
C) $\left[ \begin{array} { c c } \frac { 1 } { 7 } & - \frac { 1 } { 7 } \\ \frac { 3 } { 7 } & \frac { 1 } { 14 } \end{array} \right]$
D) $\left[ \begin{array} { c c } \frac { 1 } { 14 } & \frac { 1 } { 7 } \\ - \frac { 3 } { 7 } & \frac { 1 } { 7 } \end{array} \right]$

Question 50

Find the inverse of the matrix.
$$\left[ \begin{array} { r r } 3 & 0 \\1 & - 1\end{array} \right]$$

A) $\left[ \begin{array} { c c } - 1 & 0 \\ \frac { 1 } { 3 } & \frac { 1 } { 3 } \end{array} \right]$
B) $\left[ \begin{array} { c c } \frac { 1 } { 3 } & 0 \\ - \frac { 1 } { 3 } & - 1 \end{array} \right]$
C) $\left[ \begin{array} { c c } \frac { 1 } { 3 } & 0 \\ \frac { 1 } { 3 } & - 1 \end{array} \right]$
D) No inverse

Question 51

Find the inverse of the matrix.
$$\left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$$

A) $\left[ \begin{array} { l l l } - 1 & - 3 & - 2 \\ - 1 & - 3 & - 3 \\ - 2 & - 7 & - 8 \end{array} \right]$
B) $\left[ \begin{array} { c c c } 1 & \frac { 1 } { 3 } & \frac { 1 } { 2 } \\ 1 & \frac { 1 } { 3 } & \frac { 1 } { 3 } \\ \frac { 1 } { 2 } & \frac { 1 } { 7 } & \frac { 1 } { 8 } \end{array} \right]$
C) $\left[ \begin{array} { r r r } - 3 & 10 & - 3 \\ 2 & - 4 & 1 \\ - 1 & 1 & 0 \end{array} \right]$
D) No inverse

Question 52

Find the inverse of the matrix.
$$\left[ \begin{array} { l l } 6 & - 3 \\3 & - 6\end{array} \right]$$

A) $\left[ \begin{array} { c } \frac { 2 } { 9 } - \frac { 1 } { 9 } \\ \frac { 1 } { 9 } - \frac { 2 } { 9 } \end{array} \right]$
B) $\left[ \begin{array} { r r } - \frac { 2 } { 9 } & - \frac { 1 } { 9 } \\ \frac { 1 } { 9 } & \frac { 2 } { 9 } \end{array} \right]$
C) $\left[ \begin{array} { r r } \frac { 2 } { 9 } & \frac { 1 } { 9 } \\ - \frac { 1 } { 9 } & - \frac { 2 } { 9 } \end{array} \right]$
D) $\left[ \begin{array} { l } \frac { 1 } { 9 } - \frac { 2 } { 9 } \\ \frac { 2 } { 9 } - \frac { 1 } { 9 } \end{array} \right]$

Question 53

Use a graphing utility to find the inverse of the matrix, if it exists. Round answers to two decimal places, if necessary.
$$\left[ \begin{array} { r r r r } 1 & - 4 & 0 & 0 \\0 & 1 & - 2 & 0 \\0 & 0 & 1 & - 1 \\0 & 0 & 0 & 1\end{array} \right]$$

A) $\left[ \begin{array} { r r r r } 1 & 1 & - 2 & 8 \\ 0 & 1 & 2 & - 8 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 1 \end{array} \right]$
B) $\left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 4 & 1 & 0 & 0 \\ 8 & 2 & 1 & 0 \\ 4 & 2 & 1 & 1 \end{array} \right]$
C) $\left[ \begin{array} { r r r r } 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ - 2 & 2 & 1 & 0 \\ 8 & - 8 & 4 & 1 \end{array} \right]$
D) $\left[ \begin{array} { l l l l } 1 & 4 & 8 & 8 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]$

Question 54

Find the inverse of the matrix.

A)
$\left[ \begin{array} { c c } 0 & - \frac { 1 } { 4 } \\ - \frac { 1 } { 6 } & - \frac { 5 } { 24 } \end{array} \right]$
B) $\left[ \begin{array} { c c } - \frac { 1 } { 6 } & 0 \\ - \frac { 5 } { 24 } & - \frac { 1 } { 4 } \end{array} \right]$
D) $\left[ \begin{array} { c c } - \frac { 5 } { 24 } & \frac { 1 } { 4 } \\ \frac { 1 } { 6 } & 0 \end{array} \right]$

Question 55

Find the inverse of the matrix.
$$\left[ \begin{array} { c c c } 2 & - 5 & 3 \\2 & - 4 & 8 \\4 & - 9 & 11\end{array} \right]$$

A) $\left[ \begin{array} { l l l } 1 & 0 & 14 \\ 0 & 1 & 5 \\ 0 & 0 & 0 \end{array} \right]$
B) No inverse
C) $\left[ \begin{array} { c c c } 1 & 0 & 14 \\ 0 & 1 & 5 \\ 0 & - 1 & 0 \end{array} \right]$
D) $\left[ \begin{array} { c c c } 2 & 2 & 4 \\ - 5 & - 4 & - 9 \\ 3 & 8 & 11 \end{array} \right]$

Question 56

Find the inverse of the matrix.
$$\left[ \begin{array} { c c } - 2 & - 2 \\- 6 & - 6\end{array} \right]$$

A) $\left[ \begin{array} { c c } \frac { 3 } { 16 } & \frac { 1 } { 16 } \\ \frac { 3 } { 16 } & \frac { 1 } { 16 } \end{array} \right]$
B) $\left[ \begin{array} { c c } - \frac { 3 } { 16 } & \frac { 1 } { 16 } \\ \frac { 3 } { 16 } & - \frac { 1 } { 16 } \end{array} \right]$
C) $\left[ \begin{array} { c c } \frac { 3 } { 16 } & - \frac { 1 } { 16 } \\ - \frac { 3 } { 16 } & \frac { 1 } { 16 } \end{array} \right]$
D) No inverse

Question 57

Use a graphing utility to find the inverse of the matrix, if it exists. Round answers to two decimal places, if necessary.
$$\left\{ \begin{array} { r } 2 x + 4 y - 5 z = - 8 \\x + 5 y + 2 z = - 1 \\3 x + 3 y + 3 z = 15\end{array} \right.$$

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

Question 58

Use a graphing utility to find the inverse of the matrix, if it exists. Round answers to two decimal places, if necessary.
$$\left[ \begin{array} { r r r } - 16 & 3 & 28 \\5 & - 14 & 15 \\34 & 25 & 2\end{array} \right]$$

A) $\left[ \begin{array} { r r r } - 0.01 & 0.02 & 0.01 \\ 0.02 & - 0.03 & 0.01 \\ 0.02 & 0.02 & 0 . \end{array} \right.$
B) $\left[ \begin{array} { r r r } - 0.02 & 0.03 & 0.02 \\ 0.02 & - 0.04 & 0.01 \\ 0.02 & 0.02 & 0.00 \end{array} \right]$
$C)$
$\left[\begin{array}{rrr}-0.01 & 0.03 & 0.02 \\ 0.02 & -0.03 & 0.01 \\ 0.02 & 0.02 & 0.01\end{array}\right]$
D)
$\left[\begin{array}{rrr}-0.02 & 0.03 & 0.02 \\ 0.02 & -0.04 & 0.02 \\ 0.02 & 0.02 & 0.01\end{array}\right]$

Question 59

Find the inverse of the matrix.
$$\left[ \begin{array} { r r r } 1 & 0 & 0 \\1 & 1 & 0 \\0 & - 9 & 1\end{array} \right]$$

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

Question 60

Find the inverse of the matrix.
$$\left[ \begin{array} { l l } 1 & - 2 \\0 & - 2\end{array} \right]$$

A) $\left[ \begin{array} { c c } - \frac { 1 } { 2 } & - 1 \\ 0 & 1 \end{array} \right]$
B) $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 2 } \\ 1 & - 1 \end{array} \right]$
C) $\left[ \begin{array} { r r } 1 & 1 \\ 0 & - \frac { 1 } { 2 } \end{array} \right]$
D) $\left[ \begin{array} { c c } 1 & - 1 \\ 0 & - \frac { 1 } { 2 } \end{array} \right]$

Question 61

Solve the system of equations using substitution.
$$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 169 \\x + y = 17\end{array} \right.$$

A) $x = 12 , y = 5 ; x = 5 , y = 12$
B) $x = 12 , y = - 5 ; x = 5 , y = - 12$
C) $x = - 12 , y = 5 ; x = - 5 , y = 12$
D) $x = - 12 , y = - 5 ; x = - 5 , y = - 12$

Question 62

Solve the system of equations using substitution.
$$\left\{ \begin{array} { l } y = - x ^ { 2 } + 3 \\x ^ { 2 } + y ^ { 2 } = 5\end{array} \right.$$

A) $x = 2 , y = - 1 ; x = 1 , y = 2 ; x = - 1 , y = 2 ; x = - 2 , y = - 1$
B) $x = 1 , y = 4 ; x = 4 , y = 19$
C) $x = 1 , y = 2 ; x = - 1 , y = 2$
D) $x = 2 , y = - 1 ; x = - 2 , y = - 1$

Question 63

Write the partial fraction decomposition of the rational expression.
$$\frac { 7 x - 26 } { ( x + 2 ) ( x - 3 ) }$$

A) $\frac { 1 } { x - 3 } - \frac { 8 } { x + 2 }$
B) $\frac { 8 } { x + 2 } - \frac { 1 } { x - 3 }$
C) $\frac { 8 } { x + 2 } + \frac { 1 } { x - 3 }$
D) $\frac { 7 } { x + 2 } - \frac { 26 } { x - 3 }$

Question 64

Write the partial fraction decomposition of the rational expression.
$$\frac { 12 x + 3 } { ( x - 1 ) \left( x ^ { 2 } + x + 1 \right) }$$

A) $\frac { 5 } { x - 1 } + \frac { - 5 x + 2 } { x ^ { 2 } + x + 1 }$
B) $\frac { 5 } { x - 1 } + \frac { 2 x - 5 } { x ^ { 2 } + x + 1 }$
C) $\frac { 5 } { x - 1 } + \frac { - 5 } { x + 1 } + \frac { 2 } { x - 1 }$
D) $\frac { - 5 } { x - 1 } + \frac { 5 x + 2 } { x ^ { 2 } + x + 1 }$

Question 65

Write the partial fraction decomposition of the rational expression.
$$\frac { 12 x ^ { 2 } + 162 x + 384 } { ( x + 8 ) ( x + 2 ) ( x + 11 ) }$$

A) $- \frac { 8 } { x + 8 } - \frac { 2 } { x + 2 } - \frac { 2 } { x + 11 }$
B) $\frac { 8 } { x + 8 } + \frac { 2 } { x + 2 } - \frac { 2 } { x + 11 }$
C) $- \frac { 8 } { x + 8 } + \frac { 2 } { x + 2 } + \frac { 2 } { x + 11 }$
D) $\frac { 8 } { x + 8 } + \frac { 2 } { x + 2 } + \frac { 2 } { x + 11 }$

Question 66

Tell whether the given rational expression is proper or improper.
$$\frac { x ^ { 3 } - 13 x + 42 } { x ^ { 2 } - 8 x + 15 }$$

A) Improper
B) Proper

Question 67

Tell whether the given rational expression is proper or improper.
$$\frac { x ^ { 2 } - 12 x + 36 } { \left( x ^ { 2 } - 8 x + 15 \right) ( x + 3 ) }$$

A) Proper
B) Improper

Question 68

Write the partial fraction decomposition of the rational expression.
$$\frac { 2 x - 5 } { x ^ { 2 } - 5 x - 6 }$$

A) $\frac { 1 } { x - 6 } + \frac { 1 } { x + 1 }$
B) $\frac { 1 } { x - 3 } + \frac { 1 } { x - 2 }$
C) $\frac { 17 } { x + 6 } - \frac { 3 } { x - 1 }$
D) $\frac { 9 } { x + 2 } + \frac { 1 } { x - 3 }$

Question 69

Write the partial fraction decomposition of the rational expression.
$\frac { 2 x ^ { 3 } + 4 x ^ { 2 } } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$

A) $\frac { 2 x + 4 } { x ^ { 2 } + 5 } + \frac { 10 x - 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
B) $\frac { 2 x - 4 } { x ^ { 2 } + 5 } + \frac { - 10 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
C) $\frac { 2 x + 4 } { x ^ { 2 } + 5 } + \frac { - 10 x - 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$
D) $\frac { 2 x + 4 } { x ^ { 2 } + 5 } + \frac { 10 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$

Question 70

Solve the system of equations using substitution.
$$\left\{ \begin{array} { r } x y = 20 \\x + y = 9\end{array} \right.$$

A) $x = 20 , y = 1 ; x = 1 , y = 20$
B) $x = 10 , y = 2 ; x = 2 , y = 10$
C) $x = 6 , y = 3 ; x = 3 , y = 6$
D) $x = 5 , y = 4 ; x = 4 , y = 5$

Question 71

Write the partial fraction decomposition of the rational expression.
$$\frac { 7 x ^ { 3 } - 2 } { x ^ { 2 } ( x + 1 ) ^ { 3 } }$$

Question 72

Write the partial fraction decomposition of the rational expression.
$\frac { x ^ { 2 } - 111 } { x ^ { 4 } - x ^ { 2 } - 72 }$

A) $\frac { 1 } { x + 3 } - \frac { 1 } { x - 3 } + \frac { 7 } { x ^ { 2 } + 8 }$
B) $\frac { 1 } { x + 3 } + \frac { 1 } { x - 3 } + \frac { 7 } { x ^ { 2 } + 8 }$
C) $\frac { 1 } { x + 3 } + \frac { 1 } { x - 3 } - \frac { 7 } { x ^ { 2 } + 8 }$
D) $\frac { 1 } { x + 3 } - \frac { 1 } { x - 3 } - \frac { 7 } { x ^ { 2 } + 8 }$

Question 73

Write the partial fraction decomposition of the rational expression.
$\frac { 2 x ^ { 2 } - x - 13 } { x ( x + 1 ) ( x - 1 ) }$

A) $\frac { 13 } { x } + \frac { 5 } { x + 1 } + \frac { - 6 } { x - 1 }$
B) $\frac { 13 } { x } + \frac { - 5 } { x + 1 } + \frac { 6 } { x - 1 }$
C) $\frac { 13 } { x } + \frac { - 6 } { x + 1 } + \frac { 5 } { x - 1 }$
D) $\frac { 13 } { x } + \frac { - 5 } { x + 1 } + \frac { - 6 } { x - 1 }$

Question 74

Write the partial fraction decomposition of the rational expression.
$$\frac { x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x }$$

A) $\frac { 2 } { x } + \frac { - 2 } { x - 1 } + \frac { 5 } { ( x - 1 ) ^ { 2 } }$
B) $\frac { 2 } { x } + \frac { 3 } { x - 1 } + \frac { - 2 } { ( x - 1 ) ^ { 2 } }$
C) $\frac { - 2 } { x } + \frac { 2 } { x - 1 } + \frac { 3 } { ( x - 1 ) ^ { 2 } }$
D) $\frac { 2 } { x } + \frac { - 2 } { x - 1 } + \frac { 3 } { ( x - 1 ) ^ { 2 } }$

Question 75

Write the partial fraction decomposition of the rational expression.
$$\frac { 2 x ^ { 3 } - 1 x ^ { 2 } + 13 x - 5 } { \left( x ^ { 2 } + 4 \right) ^ { 3 } }$$

A) $\frac { 2 x + 1 } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } + \frac { 5 x + 1 } { \left( x ^ { 2 } + 4 \right) ^ { 3 } }$
B) $\frac { x + 1 } { x ^ { 2 } + 4 } + \frac { 2 x - 1 } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } + \frac { 5 x - 1 } { \left( x ^ { 2 } + 4 \right) ^ { 3 } }$
C) $\frac { x } { x ^ { 2 } + 4 } + \frac { 2 x - 1 } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } + \frac { 5 x - 1 } { \left( x ^ { 2 } + 4 \right) ^ { 3 } }$
D) $\frac { 2 x - 1 } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } + \frac { 5 x - 1 } { \left( x ^ { 2 } + 4 \right) ^ { 3 } }$

Question 76

Write the partial fraction decomposition of the rational expression.
$$\frac { 9 x ^ { 2 } + 21 x + 11 } { ( x + 2 ) ( x + 1 ) ^ { 2 } }$$

A) $\frac { - 5 } { x + 2 } + \frac { 4 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }$
B) $\frac { 5 } { x + 2 } + \frac { - 4 } { x + 1 } + \frac { - 1 } { ( x + 1 ) ^ { 2 } }$
C) $\frac { - 1 } { x + 2 } + \frac { 4 } { x + 1 } + \frac { - 5 } { ( x + 1 ) ^ { 2 } }$
D) $\frac { 5 } { x + 2 } + \frac { 4 } { x + 1 } + \frac { - 1 } { ( x + 1 ) ^ { 2 } }$

Question 77

Write the partial fraction decomposition of the rational expression.
$\frac { x - 1 } { ( x - 3 ) ( x - 2 ) }$

A) $\frac { - 1 } { x - 3 } + \frac { 2 } { x - 2 }$
B) $\frac { 2 } { x - 3 } + \frac { - 1 } { x - 2 }$
C) $\frac { 1 } { x - 3 } + \frac { - 2 } { x - 2 }$
D) $\frac { 2 } { x - 3 } + \frac { 1 } { x - 2 }$

Question 78

Solve the system of equations using substitution.
$$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 4 \\x + y = 2\end{array} \right.$$

A) $x = 0 , y = 0 ; x = 2 , y = - 2$
B) $x = 0 , y = - 2 ; x = - 2 , y = 0$
C) $x = 0 , y = 2 ; x = 2 , y = 0$
D) $x = 2 , y = - 2 ; x = - 2 , y = - 2$

Question 79

Write the partial fraction decomposition of the rational expression.
$\frac { 3 x - 2 } { x ^ { 3 } - 1 }$

A) $\frac { 3 } { x - 1 } + \frac { - 3 ( x - 7 ) } { x ^ { 2 } + x + 1 }$
B) $\frac { \frac { 1 } { 2 } } { x - 1 } + \frac { \frac { 5 } { 2 } } { x + 1 }$
C) $\frac { \frac { 1 } { 3 } } { x - 1 } + \frac { - \frac { 1 } { 3 } x + \frac { 7 } { 3 } } { \left( x ^ { 2 } + x + 1 \right) }$
D) $\frac { 3 } { ( x - 1 ) ^ { 2 } } + \frac { 1 } { ( x - 1 ) ^ { 3 } }$

Question 80

Write the partial fraction decomposition of the rational expression.
$$\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$$

A) $\frac { - 1 } { x - 2 } + \frac { \frac { 1 } { 4 } x } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 4 } } { x + 4 }$
B) $\frac { 12 } { x - 2 } + \frac { 2 } { ( x - 2 ) ^ { 2 } } + \frac { - 12 } { x + 4 }$
C) $\frac { \frac { 1 } { 12 } } { x - 2 } + \frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$
D) $\frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$