Question 1

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form.
-

Accepted Answer

A)  $2 x + y \leq 4$
$3 x + 4 y \geq 8$ 
B)  $- 2 x + y \leq 4$
$- 3 x + 4 y \geq 8\ 
C)  \(- 2 x + y \geq 4$
$- 3 x + 4 y \leq 8$ 
D)  $2 x + y \geq 4$  )  
$3 x + 4 y \leq 8$ 
A)  $2 x + y \leq 4$
$3 x + 4 y \geq 8$ 
B)  $- 2 x + y \leq 4$
$- 3 x + 4 y \geq 8\ 
C)  \(- 2 x + y \geq 4$
$- 3 x + 4 y \leq 8$ 
D)  $2 x + y \geq 4$  )  
$3 x + 4 y \leq 8$

Question 2

A $74,000 trust is to be invested in bonds paying 8%, CDs paying 5%, and mortgages paying 9%. The bond and CD investment together must equal the mortgage investment. To earn a $5930 annual income from the
Investments, how much should the bank invest in bonds?

Accepted Answer

A)  $25,000 
B)  $37,000 
C)  $12,000 
D)  $23,000 
A)  $25,000 
B)  $37,000 
C)  $12,000 
D)  $23,000

Question 3

A bookstore is having a sale. All books included in the sale have a colored sticker on them to indicate the sale price. There are green stickers, red stickers, and orange stickers. Bob, Sue, and Fred each make purchases of
Books that are on sale. Each row of the table gives information about the numbers of book purchases and the
Total cost of the purchase (before taxes).

Use this information to set up a matrix equation of the form $A X = B$, which can be solved to determine the price each type of sale book. Solve this matrix equation to find the price of a book with an orange sticker.
Use the fact that for $A = \left[ \begin{array} { l l l } 1 & 2 & 2 \ 1 & 3 & 2 \ 1 & 2 & 3 \end{array} \right] , A ^ { - 1 } = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \ - 1 & 1 & 0 \ - 1 & 0 & 1 \end{array} \right]$.

Accepted Answer

A)  $5.32 
B)  $4.96 
C)  $5.21 
D)  $5.79 
A)  $5.32 
B)  $4.96 
C)  $5.21 
D)  $5.79

Question 4

Solve the system by elimination.
-$$\begin{array} { c } 
x + 9 y = - 36 \
7 x + 10 y = - 40
\end{array}$$

Accepted Answer

A)  $\{ ( 4,0 ) \}$ 
B)  $\{ ( 1 , - 5 ) \}$ 
C)  $\{ ( 0 , - 4 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 4,0 ) \}$ 
B)  $\{ ( 1 , - 5 ) \}$ 
C)  $\{ ( 0 , - 4 ) \}$ 
D)  $\varnothing$

Question 5

Perform the operation or operations when possible.
-$$\left[ \begin{array} { r } 
3 x + 3 y \
7 k - 7 z \
- 5 w + 9 v \
3 m - 6 n
\end{array} \right] - \left[ \begin{array} { r } 
- 5 x - 8 y \
- 5 k - 3 z \
- 8 w + 6 v \
8 m - 2 n
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { r } - 2 x + 11 y \ 2 k - 4 z \ - 13 w + 3 v \ 11 m - 4 n \end{array} \right]$ 
B) 
$\left[\begin{array}{r}
8 x+11 y \
12 k-4 z \
3 w+3 v \
-5 m-4 n
\end{array}\right]$ 
C) 
$\left[\begin{array}{r}
-2 x-5 y \
2 k-10 z \
-13 w+15 v \
11 m-8 n
\end{array}\right]$ 
D) 
$\left[\begin{array}{r}
8 x-5 y \
12 k-10 z \
3 w+15 v \
-5 m-8 n
\end{array}\right]$ 
A) 
$\left[ \begin{array} { r } - 2 x + 11 y \ 2 k - 4 z \ - 13 w + 3 v \ 11 m - 4 n \end{array} \right]$ 
B) 
$\left[\begin{array}{r}
8 x+11 y \
12 k-4 z \
3 w+3 v \
-5 m-4 n
\end{array}\right]$ 
C) 
$\left[\begin{array}{r}
-2 x-5 y \
2 k-10 z \
-13 w+15 v \
11 m-8 n
\end{array}\right]$ 
D) 
$\left[\begin{array}{r}
8 x-5 y \
12 k-10 z \
3 w+15 v \
-5 m-8 n
\end{array}\right]$

Question 6

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
$\begin{array} { r } 
- 5 x + 6 y - z = - 8 \
x + 8 y + 5 z = 86 \
- 8 x + y + z = - 35
\end{array}$

Accepted Answer

A)  {(7,3,8)} 
B)  {(6,5,8)} 
C)  {(6,-5,-8)} 
D)  {(5,8,5)} 
A)  {(7,3,8)} 
B)  {(6,5,8)} 
C)  {(6,-5,-8)} 
D)  {(5,8,5)}

Question 7

A basketball fieldhouse seats 15,000. Courtside seats sell for $10, endzone for $6, and balcony for $5. Total revenue for a sell-out is $92,000. If half the courtside and balcony and all the endzone seats are sold, the total
Revenue is $52,000. How many of each type of seat are there in the fieldhouse?

Accepted Answer

A)  3,200 courtside; 1,800 endzone; 10,000 balcony 
B)  3,000 courtside; 2,000 endzone; 10,000 balcony 
C)  4,000 courtside; 3,000 endzone; 8,000 balcony 
D)  2,000 courtside; 5,000 endzone; 8,000 balcony 
A)  3,200 courtside; 1,800 endzone; 10,000 balcony 
B)  3,000 courtside; 2,000 endzone; 10,000 balcony 
C)  4,000 courtside; 3,000 endzone; 8,000 balcony 
D)  2,000 courtside; 5,000 endzone; 8,000 balcony

Question 8

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of  b  ?
$\begin{array} { c | c } 
\text { Year } & \begin{array} { c } 
\text { Number of dog collars } \
\text { (in thousands) }
\end{array} \
\hline 0 & 20 \
\hline 6 & 14 \
\hline 18 & 23.6
\end{array}$

Accepted Answer

A)   -1.2 
B)   -1.6 
C)   1.9 
D)   2.3 
A)   -1.2 
B)   -1.6 
C)   1.9 
D)   2.3

Question 9

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = - \frac { 1 } { 3 } \
\frac { 1 } { x } - \frac { 1 } { y } - \frac { 4 } { z } = - \frac { 11 } { 6 } \
\frac { 4 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = - \frac { 4 } { 3 }
\end{array}$$

Accepted Answer

A)  $\left\{ \left( - \frac { 1 } { 3 } , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right) \right\}$ 
B)  $\{ ( - 3 , - 2 , - 2 ) \}$ 
C)  $\{ ( 3,2,2 ) \}$ 
D)  $\{ ( - 3 , - 2,2 ) \}$ 
A)  $\left\{ \left( - \frac { 1 } { 3 } , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right) \right\}$ 
B)  $\{ ( - 3 , - 2 , - 2 ) \}$ 
C)  $\{ ( 3,2,2 ) \}$ 
D)  $\{ ( - 3 , - 2,2 ) \}$

Question 10

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { l r r r } 
0 & 3 & 0 & - 3 \
4 & 9 & - 3 & - 6 \
8 & 6 & - 6 & - 3 \
12 & 0 & - 6 & 0
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c c c } 
- \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 2 } & \frac { 1 } { 4 } \
- 1 & \frac { 2 } { 3 } & - \frac { 1 } { 3 } & 0 \
- 1 & 1 & - 1 & \frac { 1 } { 3 } \
- \frac { 4 } { 3 } & \frac { 2 } { 3 } & - \frac { 1 } { 3 } & 0
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$$\left[ \begin{array} { r r r r } 
- \frac { 1 } { 2 } & - 1 & - 1 & - \frac { 4 } { 3 } \
\frac { 1 } { 2 } & \frac { 2 } { 3 } & 1 & \frac { 1 } { 3 } \
- \frac { 1 } { 2 } & - \frac { 1 } { 3 } & - 1 & - \frac { 1 } { 3 } \
\frac { 1 } { 4 } & 0 & \frac { 1 } { 3 } & 0
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r r } 
\frac { 1 } { 2 } & - \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 4 } \
1 & - \frac { 2 } { 3 } & \frac { 1 } { 3 } & 0 \
1 & - 1 & 1 & \frac { 1 } { 3 } \
- 1 & - \frac { 2 } { 3 } & \frac { 1 } { 3 } & 0
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c c c } 
- \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 2 } & \frac { 1 } { 4 } \
- 1 & \frac { 2 } { 3 } & - \frac { 1 } { 3 } & 0 \
- 1 & 1 & - 1 & \frac { 1 } { 3 } \
- \frac { 4 } { 3 } & \frac { 2 } { 3 } & - \frac { 1 } { 3 } & 0
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$$\left[ \begin{array} { r r r r } 
- \frac { 1 } { 2 } & - 1 & - 1 & - \frac { 4 } { 3 } \
\frac { 1 } { 2 } & \frac { 2 } { 3 } & 1 & \frac { 1 } { 3 } \
- \frac { 1 } { 2 } & - \frac { 1 } { 3 } & - 1 & - \frac { 1 } { 3 } \
\frac { 1 } { 4 } & 0 & \frac { 1 } { 3 } & 0
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r r } 
\frac { 1 } { 2 } & - \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 4 } \
1 & - \frac { 2 } { 3 } & \frac { 1 } { 3 } & 0 \
1 & - 1 & 1 & \frac { 1 } { 3 } \
- 1 & - \frac { 2 } { 3 } & \frac { 1 } { 3 } & 0
\end{array} \right]$$

Question 11

Perform the operation or operations when possible.
-$$\left[ \begin{array} { l l } 
4 & 4
\end{array} \right] + \left[ \begin{array} { l } 
1 \
9
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { r } 5 \ 13 \end{array} \right]$ 
B)

$\left[ \begin{array} { l l } 4 & 1 \ 4 & 9 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 5 & 13 \end{array} \right]$ 
D)  They cannot be added. 
A) 
$\left[ \begin{array} { r } 5 \ 13 \end{array} \right]$ 
B)

$\left[ \begin{array} { l l } 4 & 1 \ 4 & 9 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 5 & 13 \end{array} \right]$ 
D)  They cannot be added.

Question 12

Solve the linear programming problem.
-A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $2400, and the average monthly salary of an aide is $1100. The camp can
Accommodate up to 35 staff members and needs at least 20 to run properly. The camp must have at least 10
Aides and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the
Camp hire to minimize cost?

Accepted Answer

A)  13 counselors and 17 aides 
B)  13 counselors and 16 aides 
C)  19 counselors and 23 aides 
D)  8 counselors and 12 aides 
A)  13 counselors and 17 aides 
B)  13 counselors and 16 aides 
C)  19 counselors and 23 aides 
D)  8 counselors and 12 aides

Question 13

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
-$$\begin{array} { r } 
a x + b y = \frac { a } { b } \
x + y = \frac { 1 } { a }
\end{array}$$

Accepted Answer

A)  $\left\{ \left( \frac { a + b } { a b } , - \frac { 1 } { b } \right) \right\}$ 
B)  $\left\{ \left( \frac { \mathrm { a } - \mathrm { b } } { \mathrm { ab } } , \frac { 1 } { \mathrm {~b} } \right) \right\}$ 
C)  $\left\{ \left( \frac { a b } { a + b } , - b \right) \right\}$ 
D)  $\left\{ \left( \frac { a b } { a + b } , - \frac { 1 } { b } \right) \right\}$ 
A)  $\left\{ \left( \frac { a + b } { a b } , - \frac { 1 } { b } \right) \right\}$ 
B)  $\left\{ \left( \frac { \mathrm { a } - \mathrm { b } } { \mathrm { ab } } , \frac { 1 } { \mathrm {~b} } \right) \right\}$ 
C)  $\left\{ \left( \frac { a b } { a + b } , - b \right) \right\}$ 
D)  $\left\{ \left( \frac { a b } { a + b } , - \frac { 1 } { b } \right) \right\}$

Question 14

Write the augmented matrix for the system. Do not solve the system.
$\begin{array} { l } 
8 x + 5 y + 6 z = 40 \
9 x + 4 y + 3 z = 39 \
4 x - 2 y + 6 z = - 10
\end{array}$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 
8 & 5 & 6 \
9 & 4 & 3 \
4 & - 2 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { r r r | r } 
8 & 9 & 4 & 40 \
5 & 4 & - 2 & 39 \
6 & 3 & 6 & - 10
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r | r } 
40 & 6 & 4 & 8 \
39 & 3 & 4 & 9 \
- 10 & 6 & - 2 & 4
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r | r } 
8 & 5 & 6 & 40 \
9 & 4 & 3 & 39 \
4 & - 2 & 6 & - 10
\end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } 
8 & 5 & 6 \
9 & 4 & 3 \
4 & - 2 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { r r r | r } 
8 & 9 & 4 & 40 \
5 & 4 & - 2 & 39 \
6 & 3 & 6 & - 10
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r | r } 
40 & 6 & 4 & 8 \
39 & 3 & 4 & 9 \
- 10 & 6 & - 2 & 4
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r | r } 
8 & 5 & 6 & 40 \
9 & 4 & 3 & 39 \
4 & - 2 & 6 & - 10
\end{array} \right]$

Question 15

The sizes of two matrices are given. Find the size of the product AB and the size of the product BA, if the given product
can be calculated.
-$\mathrm { A }$ is $1 \times 4$; B is $1 \times 4$.

Accepted Answer

A)  $\mathrm { AB }$ is $1 \times 1 ; \mathrm { BA }$ is $4 \times 4$. 
B)  $\mathrm { AB }$ is $4 \times 1 ; \mathrm { BA }$ is $1 \times 4$. 
C)  AB cannot be calculated; BA cannot be calculated. 
D)  $\mathrm { AB }$ is $1 \times 4$; $\mathrm { BA }$ is $4 \times 1$. 
A)  $\mathrm { AB }$ is $1 \times 1 ; \mathrm { BA }$ is $4 \times 4$. 
B)  $\mathrm { AB }$ is $4 \times 1 ; \mathrm { BA }$ is $1 \times 4$. 
C)  AB cannot be calculated; BA cannot be calculated. 
D)  $\mathrm { AB }$ is $1 \times 4$; $\mathrm { BA }$ is $4 \times 1$.

Question 16

Find the matrix product when possible.
-$$\left[ \begin{array} { r r r } 
3 & - 3 & - 3 \
- 8 & 6 & - 2
\end{array} \right] \left[ \begin{array} { r } 
- 1 \
5 \
- 1
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { r } 40 \ - 15 \end{array} \right]$ 
B) 
[-1540] 
C) 
$\left[\begin{array}{rrr}
3 & -3 & -3 \
-8 & 6 & -2 \
-1 & 5 & -1
\end{array}\right]$ 
D) 
$\left[\begin{array}{r}
-15 \
40
\end{array}\right]$ 
A) 
$\left[ \begin{array} { r } 40 \ - 15 \end{array} \right]$ 
B) 
[-1540] 
C) 
$\left[\begin{array}{rrr}
3 & -3 & -3 \
-8 & 6 & -2 \
-1 & 5 & -1
\end{array}\right]$ 
D) 
$\left[\begin{array}{r}
-15 \
40
\end{array}\right]$

Question 17

Provide an appropriate response.
-Let $M$ be a $5 \times 5$ matrix. Which of the following statements are true?
(i) $\mathrm { I } 5 \mathrm { M } = \mathrm { M }$
(ii) $\mathrm { I } _ { 5 } + \mathrm { M } = \mathrm { M }$
(iii) $\mathrm { MI } _ { 5 } = \mathrm { I } _ { 5 }$
(iv) $\mathrm { MI } _ { 5 } = \mathrm { M }$

Accepted Answer

A)  (ii), (iii) 
B)  (i), (iii), (iv) 
C)  (i), (ii), (iii), (iv) 
D)  (i), (iv) 
A)  (ii), (iii) 
B)  (i), (iii), (iv) 
C)  (i), (ii), (iii), (iv) 
D)  (i), (iv)

Question 18

Solve the equation for x.
$\left| \begin{array} { l l } 
2 & 4 \
x & 3
\end{array} \right| = - 2$

Accepted Answer

A)  {-2} 
B)   {-1} 
C)   {2} 
D)  $\varnothing$ 
A)  {-2} 
B)   {-1} 
C)   {2} 
D)  $\varnothing$

Question 19

Find the cofactor of the indicated element.
-$\mathrm { a } _ { 13 }$
$\left| \begin{array} { r r r } - 2 & - 1 & 0 \ 2 & 1 & 2 \ 5 & 3 & 1 \end{array} \right|$

Accepted Answer

A)  1 
B)  $- 8$ 
C)  $- 5$ 
D)  $- 1$ 
A)  1 
B)  $- 8$ 
C)  $- 5$ 
D)  $- 1$

Question 20

Solve the system by elimination.
-$$\begin{array} { l } 
- 7 x - 6 y = - 13 \
- 3 x + 3 y = - 39
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 7 , - 6 ) \}$ 
C)  $\{ ( 6 , - 5 ) \}$ 
D)  $\{ ( 7 , - 5 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 7 , - 6 ) \}$ 
C)  $\{ ( 6 , - 5 ) \}$ 
D)  $\{ ( 7 , - 5 ) \}$

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form. -

A $74,000 trust is to be invested in bonds paying 8%, CDs paying 5%, and mortgages paying 9%. The bond and CD investment together must equal the mortgage investment. To earn a $5930 annual income from the Investments, how much should the bank invest in bonds?

Solve the system by elimination. - x+9y=-36 7x+10y=-40

Perform the operation or operations when possible. - $\left[ \begin{array} { r } 3 x + 3 y \\7 k - 7 z \\- 5 w + 9 v \\3 m - 6 n\end{array} \right] - \left[ \begin{array} { r } - 5 x - 8 y \\- 5 k - 3 z \\- 8 w + 6 v \\8 m - 2 n\end{array} \right]$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. -5x+6y-z=-8 x+8y+5z=86 -8x+y+z=-35

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of b ? Year Number of dog collars (in thousands) 0 20 6 14 18 23.6

If the system has infinitely many solutions, write the solution set with x arbitrary. - ++=- --=- ++=-

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l r r r } 0 & 3 & 0 & - 3 \\4 & 9 & - 3 & - 6 \\8 & 6 & - 6 & - 3 \\12 & 0 & - 6 & 0\end{array} \right]$

Perform the operation or operations when possible. - $\left[ \begin{array} { l l } 4 & 4\end{array} \right] + \left[ \begin{array} { l } 1 \\9\end{array} \right]$

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - ax+by= x+y=

Write the augmented matrix for the system. Do not solve the system. 8x+5y+6z=40 9x+4y+3z=39 4x-2y+6z=-10

The sizes of two matrices are given. Find the size of the product AB and the size of the product BA, if the given product can be calculated. - $\mathrm { A }$ is $1 \times 4$ ; B is $1 \times 4$ .

Find the matrix product when possible. - $\left[ \begin{array} { r r r } 3 & - 3 & - 3 \\- 8 & 6 & - 2\end{array} \right] \left[ \begin{array} { r } - 1 \\5 \\- 1\end{array} \right]$

Solve the equation for x. $\left| \begin{array} { l l } 2 & 4 \\x & 3\end{array} \right| = - 2$

Find the cofactor of the indicated element. - $\mathrm { a } _ { 13 }$ $\left| \begin{array} { r r r } - 2 & - 1 & 0 \\ 2 & 1 & 2 \\ 5 & 3 & 1 \end{array} \right|$

Solve the system by elimination. - -7x-6y=-13 -3x+3y=-39

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Arithmetic Sequence: Common Difference and First n Terms

Filters

Exam 6: Systems and Matrices