Question 1

Sketch the graph of the inequality. $5 x + 2 y \geq 1 y + 5$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)    A

Question 2

Find the maximum and minimum values of the objective function $C = 8 x + 4 y + 7$ on the region in the figure.

Accepted Answer

A)  The maximum is $19$ at $( 0,3 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
B)  The maximum is $51$ at $( 4,3 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
C)  The maximum is $3$ at $( 2,4 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
D)  The maximum is $20$ at $( 4,3 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
E)  The maximum is $67$ at $( 7,1 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
A)  The maximum is $19$ at $( 0,3 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
B)  The maximum is $51$ at $( 4,3 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
C)  The maximum is $3$ at $( 2,4 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
D)  The maximum is $20$ at $( 4,3 )$ ,
The minimum is $19$ at $( 0,3 )$ . 
E)  The maximum is $67$ at $( 7,1 )$ ,
The minimum is $19$ at $( 0,3 )$ . E

Question 3

Use the method of substitution to solve the system. $$\left\{ \begin{array} { l } 
y = x ^ { 2 } - 25 \
y = 3 x - 7
\end{array} \right.$$

Accepted Answer

A)  ( - 3, -16 ) 
B)  ( - 3, -16 ), ( 6, 11 ) 
C)  ( 6, 11 ) 
D)  ( - 3, -16 ), ( 5, 11 ) 
E)  no solution 
A)  ( - 3, -16 ) 
B)  ( - 3, -16 ), ( 6, 11 ) 
C)  ( 6, 11 ) 
D)  ( - 3, -16 ), ( 5, 11 ) 
E)  no solution B

Question 4

Use matrices to solve the system. $$\left\{ \begin{array} { l } 
5 x + y + z = 0 \
x - 2 y - 2 z = 0 \
x + y + z = 0
\end{array} \right.$$

Accepted Answer

A)  ( 0, 0, 0) 
B)  ( 0, z, z ) 
C)  ( 0, - z, z ) 
D)  ( 2, 8, 4 ) 
E)  The system is inconsistent 
A)  ( 0, 0, 0) 
B)  ( 0, z, z ) 
C)  ( 0, - z, z ) 
D)  ( 2, 8, 4 ) 
E)  The system is inconsistent

Question 5

Use matrices to solve the system. $$\left\{ \begin{aligned}
4 x + 3 y - z & = - 4 \
x - 4 y + 5 z & = 1 \
4 y + 5 z & = 36
\end{aligned} \right.$$

Accepted Answer

A)  $x = - 3 , y = 4 , z = 4$ 
B)  $x = - 3 , y = - 7 , z = 6$ 
C)  $x = 5 , y = - 4 , z = - 4$ 
D)  The system is inconsistent 
E)  The equations are dependent 
A)  $x = - 3 , y = 4 , z = 4$ 
B)  $x = - 3 , y = - 7 , z = 6$ 
C)  $x = 5 , y = - 4 , z = - 4$ 
D)  The system is inconsistent 
E)  The equations are dependent

Question 6

Find, if possible, $B A$ . $$A = \left[ \begin{array} { c c c } 
8 & - 7 & 4 \
- 1 & 1 & 9
\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 
2 & 1 \
0 & 1 \
- 4 & 7
\end{array} \right]$$

Accepted Answer

A)  $$A B = \left[ \begin{array} { l l } 
- 29 & 0 \
- 4 & 8
\end{array} \right]$$ 
B)  $$A B = \left[ \begin{array} { c c } 
- 63 & 38 \
- 0 & 29
\end{array} \right]$$ 
C)  $$A B = \left[ \begin{array} { c c } 
- 0 & 29 \
- 38 & 63
\end{array} \right]$$ 
D)  $$A B = \left[ \begin{array} { c c } 
- 63 & 8 \
- 1 & 1
\end{array} \right]$$ 
E)  $$A B = \left[ \begin{array} { c c } 
- 0 & 29 \
- 29 & 63
\end{array} \right]$$ 
A)  $$A B = \left[ \begin{array} { l l } 
- 29 & 0 \
- 4 & 8
\end{array} \right]$$ 
B)  $$A B = \left[ \begin{array} { c c } 
- 63 & 38 \
- 0 & 29
\end{array} \right]$$ 
C)  $$A B = \left[ \begin{array} { c c } 
- 0 & 29 \
- 38 & 63
\end{array} \right]$$ 
D)  $$A B = \left[ \begin{array} { c c } 
- 63 & 8 \
- 1 & 1
\end{array} \right]$$ 
E)  $$A B = \left[ \begin{array} { c c } 
- 0 & 29 \
- 29 & 63
\end{array} \right]$$

Question 7

Find the partial fraction decomposition. $\frac { 4 x ^ { 4 } - 4 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 }$

Accepted Answer

A)  $4 x + \frac { 1 } { x + 1 } + \frac { 3 x } { x ^ { 2 } - 1 }$ 
B)  $4 x + \frac { 1 } { x - 1 } + \frac { 3 x } { x ^ { 2 } + 1 }$ 
C)  $3 x + \frac { 1 } { x - 1 } + \frac { 4 x } { x ^ { 2 } + 1 }$ 
D)  $4 x - 4 + \frac { 1 } { x - 1 } + \frac { 3 x } { x ^ { 2 } + 1 }$ 
E)  $4 x + 4 - \frac { 1 } { x - 1 } + \frac { 3 x } { x ^ { 2 } + 1 }$ 
A)  $4 x + \frac { 1 } { x + 1 } + \frac { 3 x } { x ^ { 2 } - 1 }$ 
B)  $4 x + \frac { 1 } { x - 1 } + \frac { 3 x } { x ^ { 2 } + 1 }$ 
C)  $3 x + \frac { 1 } { x - 1 } + \frac { 4 x } { x ^ { 2 } + 1 }$ 
D)  $4 x - 4 + \frac { 1 } { x - 1 } + \frac { 3 x } { x ^ { 2 } + 1 }$ 
E)  $4 x + 4 - \frac { 1 } { x - 1 } + \frac { 3 x } { x ^ { 2 } + 1 }$

Question 8

Find the inverse of the matrix if it exists. $$\left[ \begin{array} { l l l } 
2 & 0 & 0 \
0 & 9 & 0 \
0 & 0 & 8
\end{array} \right]$$

Accepted Answer

A)  $$\left[ \begin{array} { c c c } 
1 & \frac { 1 } { 9 } & 1 \
\frac { 1 } { 8 } & 0 & - 2 \
0 & 1 & \frac { 1 } { 2 }
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c c } 
\frac { 1 } { 2 } & 0 & 0 \
0 & \frac { 1 } { 9 } & 0 \
0 & 0 & \frac { 1 } { 9 }
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c c } 
0 & 0 & \frac { 1 } { 2 } \
0 & \frac { 1 } { 8 } & 0 \
\frac { 1 } { 9 } & 0 & 0
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c c } 
\frac { 1 } { 2 } & 0 & 0 \
0 & \frac { 1 } { 9 } & 0 \
0 & 0 & \frac { 1 } { 8 }
\end{array} \right]$$ 
E)  Does not exist 
A)  $$\left[ \begin{array} { c c c } 
1 & \frac { 1 } { 9 } & 1 \
\frac { 1 } { 8 } & 0 & - 2 \
0 & 1 & \frac { 1 } { 2 }
\end{array} \right]$$ 
B)  $$\left[ \begin{array} { c c c } 
\frac { 1 } { 2 } & 0 & 0 \
0 & \frac { 1 } { 9 } & 0 \
0 & 0 & \frac { 1 } { 9 }
\end{array} \right]$$ 
C)  $$\left[ \begin{array} { c c c } 
0 & 0 & \frac { 1 } { 2 } \
0 & \frac { 1 } { 8 } & 0 \
\frac { 1 } { 9 } & 0 & 0
\end{array} \right]$$ 
D)  $$\left[ \begin{array} { c c c } 
\frac { 1 } { 2 } & 0 & 0 \
0 & \frac { 1 } { 9 } & 0 \
0 & 0 & \frac { 1 } { 8 }
\end{array} \right]$$ 
E)  Does not exist

Question 9

A shop specializes in preparing blends of gourmet coffees. From Colombian, Costa Rican, and Kenyan coffees, the owner wishes to prepare 3-pounds bags that will sell for $8.50. The cost per pound of these coffees is $10, $6, and $8, respectively. The amount of Colombian is to be three times the amount of Costa Rican. Find the amount of each type of coffee in the blend.

Accepted Answer

A)  1.275 lb Colombian , 0.425 lb Costa Rican , 1.3 lb Kenyan 
B)  1.575 lb Colombian , 0.525 lb Costa Rican , 0.9 lb Kenyan 
C)  1.125 lb Colombian , 0.375 lb Costa Rican , 1.5 lb Kenyan 
D)  1.125 lb Colombian , 0.375 lb Costa Rican , 3 lb Kenyan 
E)  1.425 lb Colombian , 0.475 lb Costa Rican , 1.1 lb Kenyan 
A)  1.275 lb Colombian , 0.425 lb Costa Rican , 1.3 lb Kenyan 
B)  1.575 lb Colombian , 0.525 lb Costa Rican , 0.9 lb Kenyan 
C)  1.125 lb Colombian , 0.375 lb Costa Rican , 1.5 lb Kenyan 
D)  1.125 lb Colombian , 0.375 lb Costa Rican , 3 lb Kenyan 
E)  1.425 lb Colombian , 0.475 lb Costa Rican , 1.1 lb Kenyan

Question 10

Find the partial fraction decomposition. $\frac { 11 x - 20 } { x ^ { 2 } - 4 x }$

Accepted Answer

A)  $\frac { 5 } { x } + \frac { 6 } { x - 4 }$ 
B)  $\frac { 5 } { x } - \frac { 6 } { x - 4 }$ 
C)  $\frac { 5 } { x } + \frac { 4 } { x - 6 }$ 
D)  $\frac { 6 } { x } - \frac { 5 } { x - 4 }$ 
E)  $\frac { 6 } { x } + \frac { 5 } { x - 4 }$ 
A)  $\frac { 5 } { x } + \frac { 6 } { x - 4 }$ 
B)  $\frac { 5 } { x } - \frac { 6 } { x - 4 }$ 
C)  $\frac { 5 } { x } + \frac { 4 } { x - 6 }$ 
D)  $\frac { 6 } { x } - \frac { 5 } { x - 4 }$ 
E)  $\frac { 6 } { x } + \frac { 5 } { x - 4 }$

Question 11

Find the determinant of the matrix. $$\left[ \begin{array} { c c c } 
- 3 & 4.5 & 9 \
- 0.3 & 5.5 & 1 \
5.3 & 6.7 & 13
\end{array} \right]$$

Accepted Answer

A)  -1083.6 
B)  -173 
C)  -433.44 
D)  -443.44 
A)  -1083.6 
B)  -173 
C)  -433.44 
D)  -443.44

Question 12

Use Cramer's rule, whenever possible, to solve the system. $$\left\{ \begin{array} { l } 
4 x + 6 y = 8 \
4 x - 9 y = 8
\end{array} \right.$$

Accepted Answer

A)  $( 0,3 )$ 
B)  $( 1,1 )$ 
C)  $( 2,0 )$ 
D)  The equations are dependent. 
E)  The system is inconsistent. 
A)  $( 0,3 )$ 
B)  $( 1,1 )$ 
C)  $( 2,0 )$ 
D)  The equations are dependent. 
E)  The system is inconsistent.

Question 13

Let $I = I _ { 2 }$ be the identity matrix of order 2, and let $f ( x ) = | A - x I |$ . Find the polynomial $f ( x )$ for the given matrix A in order to find the zeros of $f ( x )$ .
(In the study of matrices, $f ( x )$ is the characteristic polynomial of A, and the zeros of $f ( x )$ are the characteristic values (eigenvalues) of A.) $$A = \left[ \begin{array} { r r } 
37 & - 90 \
12 & - 29
\end{array} \right]$$

Accepted Answer

A)  $x = 7 , x = 1$ 
B)  $x = - 7 , x = 1$ 
C)  $x = 4 , x = 5$ 
D)  $x = - 10 , x = 6$ 
E)  $x = 7 , x = 4$ 
A)  $x = 7 , x = 1$ 
B)  $x = - 7 , x = 1$ 
C)  $x = 4 , x = 5$ 
D)  $x = - 10 , x = 6$ 
E)  $x = 7 , x = 4$

Question 14

Find the determinant of the matrix after introducing zeros. $$\left[ \begin{array} { c c c } 
2 & 1 & 0 \
- 6 & 0 & 1 \
1 & 2 & - 1
\end{array} \right]$$

Accepted Answer

A)  - 9 
B)  - 8 
C)  - 10 
D)  7 
E)  0 
A)  - 9 
B)  - 8 
C)  - 10 
D)  7 
E)  0

Question 15

Find the partial fraction decomposition. $\frac { 7 x + 1 } { ( x - 1 ) ^ { 2 } }$

Accepted Answer

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

Question 16

A hospital dietician wishes to prepare a corn-squash vegetable dish that will provide at least $3$ grams of protein and cost no more than $30$ cents per serving. An ounce of creamed corn provides $\frac { 1 } { 2 }$ gram of protein and costs $4$ cents. An ounce of squash supplies $\frac { 1 } { 4 }$ gram of protein and costs $3$ cents. For taste, there must be at least $2$ ounces of corn and at least as much squash as corn. It is important to keep the total number of ounces in a serving as small as possible. Find the combination of corn and squash that will minimize the amount of ingredients used per serving.

Accepted Answer

A)  $2$ ounces of corn and $6$ ounces of squash 
B)  $0$ ounces of corn and $12$ ounces of squash 
C)  $4$ ounces of corn and $4$ ounces of squash 
D)  $5$ ounces of corn and $2$ ounces of squash 
E)  $3$ ounces of corn and $3$ ounces of squash 
A)  $2$ ounces of corn and $6$ ounces of squash 
B)  $0$ ounces of corn and $12$ ounces of squash 
C)  $4$ ounces of corn and $4$ ounces of squash 
D)  $5$ ounces of corn and $2$ ounces of squash 
E)  $3$ ounces of corn and $3$ ounces of squash

Question 17

The data in the table are generated by the function $f ( x ) = a e ^ { - b x }$ . Approximate the unknown constants a and b to four decimal places. x
1 2 3 4
F ( x )
0)71939 0.41687 0.24157 0.13998

Accepted Answer

A)  $a = 1.2414 , a = 0.5456$ 
B)  $a = 0.0000 , a = 0.5456$ 
C)  $$a = 0.5456 , a = 0.5456$$ 
D)  $a = 1.2414 , a = 0.0000$ 
E)  $a = 0.5456 , a = 1.2414$ 
A)  $a = 1.2414 , a = 0.5456$ 
B)  $a = 0.0000 , a = 0.5456$ 
C)  $$a = 0.5456 , a = 0.5456$$ 
D)  $a = 1.2414 , a = 0.0000$ 
E)  $a = 0.5456 , a = 1.2414$

Question 18

Find, if possible, $B A$ . $$A = \left[ \begin{array} { l l l } 
8 & - 1 & 5
\end{array} \right] , B = \left[ \begin{array} { c } 
- 2 \
5
\end{array} \right]$$

Accepted Answer

A)  $$B A = \left[ \begin{array} { c c c } 
- 10 & 2 & 8 \
- 5 & 40 & 25
\end{array} \right]$$ 
B)  $$B A = \left[ \begin{array} { c c c } 
8 & 1 & 2 \
25 & - 5 & 40
\end{array} \right]$$ 
C)  $$B A = \left[ \begin{array} { c c c } 
- 16 & 2 & 5 \
40 & 25 & - 5
\end{array} \right]$$ 
D)  $$B A = \left[ \begin{array} { c c c } 
- 16 & 2 & - 10 \
40 & - 5 & 25
\end{array} \right]$$ 
E)  $$B A = \left[ \begin{array} { c c c } 
- 10 & 40 & - 16 \
40 & - 5 & 25
\end{array} \right]$$ 
A)  $$B A = \left[ \begin{array} { c c c } 
- 10 & 2 & 8 \
- 5 & 40 & 25
\end{array} \right]$$ 
B)  $$B A = \left[ \begin{array} { c c c } 
8 & 1 & 2 \
25 & - 5 & 40
\end{array} \right]$$ 
C)  $$B A = \left[ \begin{array} { c c c } 
- 16 & 2 & 5 \
40 & 25 & - 5
\end{array} \right]$$ 
D)  $$B A = \left[ \begin{array} { c c c } 
- 16 & 2 & - 10 \
40 & - 5 & 25
\end{array} \right]$$ 
E)  $$B A = \left[ \begin{array} { c c c } 
- 10 & 40 & - 16 \
40 & - 5 & 25
\end{array} \right]$$

Question 19

A manufacturer of tennis rackets makes a profit of $15 on each oversized racket and $10 on each standard racket. To meet dealer demand, daily production of standard rackets should be between 25 and 75, and production of oversized rackets should be between 8 and 27. To maintain high quality, the total number of rackets produced should not exceed 75 per day. How many of each type should be manufactured daily to maximize the profit?

Accepted Answer

A)  25 standard and 8 oversized 
B)  25 standard and 27 oversized 
C)  27 standard and 48 oversized 
D)  48 standard and 27 oversized 
E)  48 standard and 8 oversized 
A)  25 standard and 8 oversized 
B)  25 standard and 27 oversized 
C)  27 standard and 48 oversized 
D)  48 standard and 27 oversized 
E)  48 standard and 8 oversized

Question 20

Find the inverse of the matrix if it exists. $$\left[ \begin{array} { l l } 
5 & 2 \
4 & 9
\end{array} \right]$$

Accepted Answer

A)  $$9 \left[ \begin{array} { l l } 
5 & 4 \
2 & 1
\end{array} \right]$$ 
B)  $$\frac { 1 } { 5 } \left[ \begin{array} { c c } 
9 & 4 \
- 2 & 37
\end{array} \right]$$ 
C)  $$\frac { 1 } { 37 } \left[ \begin{array} { c c } 
9 & - 2 \
- 4 & 5
\end{array} \right]$$ 
D)  $$\frac { 1 } { 37 } \left[ \begin{array} { c c } 
63 & - 14 \
- 28 & 35
\end{array} \right]$$ 
E)  Does not exist 
A)  $$9 \left[ \begin{array} { l l } 
5 & 4 \
2 & 1
\end{array} \right]$$ 
B)  $$\frac { 1 } { 5 } \left[ \begin{array} { c c } 
9 & 4 \
- 2 & 37
\end{array} \right]$$ 
C)  $$\frac { 1 } { 37 } \left[ \begin{array} { c c } 
9 & - 2 \
- 4 & 5
\end{array} \right]$$ 
D)  $$\frac { 1 } { 37 } \left[ \begin{array} { c c } 
63 & - 14 \
- 28 & 35
\end{array} \right]$$ 
E)  Does not exist

Sketch the graph of the inequality. $5 x + 2 y \geq 1 y + 5$

Find the maximum and minimum values of the objective function $C = 8 x + 4 y + 7$ on the region in the figure.

Use the method of substitution to solve the system. $\left\{ \begin{array} { l } y = x ^ { 2 } - 25 \\y = 3 x - 7\end{array} \right.$

Use matrices to solve the system. $\left\{ \begin{array} { l } 5 x + y + z = 0 \\x - 2 y - 2 z = 0 \\x + y + z = 0\end{array} \right.$

Use matrices to solve the system. $\left\{ \begin{aligned}4 x + 3 y - z & = - 4 \\x - 4 y + 5 z & = 1 \\4 y + 5 z & = 36\end{aligned} \right.$

Find, if possible, $B A$ . $A = \left[ \begin{array} { c c c } 8 & - 7 & 4 \\- 1 & 1 & 9\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 2 & 1 \\0 & 1 \\- 4 & 7\end{array} \right]$

Find the partial fraction decomposition. $\frac { 4 x ^ { 4 } - 4 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 }$

Find the inverse of the matrix if it exists. $\left[ \begin{array} { l l l } 2 & 0 & 0 \\0 & 9 & 0 \\0 & 0 & 8\end{array} \right]$

Find the partial fraction decomposition. $\frac { 11 x - 20 } { x ^ { 2 } - 4 x }$

Find the determinant of the matrix. $\left[ \begin{array} { c c c } - 3 & 4.5 & 9 \\- 0.3 & 5.5 & 1 \\5.3 & 6.7 & 13\end{array} \right]$

Use Cramer's rule, whenever possible, to solve the system. $\left\{ \begin{array} { l } 4 x + 6 y = 8 \\4 x - 9 y = 8\end{array} \right.$

Find the determinant of the matrix after introducing zeros. $\left[ \begin{array} { c c c } 2 & 1 & 0 \\- 6 & 0 & 1 \\1 & 2 & - 1\end{array} \right]$

Find the partial fraction decomposition. $\frac { 7 x + 1 } { ( x - 1 ) ^ { 2 } }$

The data in the table are generated by the function $f ( x ) = a e ^ { - b x }$ . Approximate the unknown constants a and b to four decimal places. x 1 2 3 4 F ( x ) 0)71939 0.41687 0.24157 0.13998

Find, if possible, $B A$ . $A = \left[ \begin{array} { l l l } 8 & - 1 & 5\end{array} \right] , B = \left[ \begin{array} { c } - 2 \\5\end{array} \right]$

Find the inverse of the matrix if it exists. $\left[ \begin{array} { l l } 5 & 2 \\4 & 9\end{array} \right]$

Topics From Algebra

Functions and Graphs

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Inverse, Exponential, and Logarithmic Functions

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Exam 8: Systems of Equations and Inequalities

Sketch the graph of the inequality. 5x+2y≥1y+55 x + 2 y \geq 1 y + 55x+2y≥1y+5

Find the maximum and minimum values of the objective function C=8x+4y+7C = 8 x + 4 y + 7C=8x+4y+7 on the region in the figure.

Use the method of substitution to solve the system. {y=x2−25y=3x−7\left\{ \begin{array} { l } y = x ^ { 2 } - 25 \\y = 3 x - 7\end{array} \right.{y=x2−25y=3x−7​

Use matrices to solve the system. {5x+y+z=0x−2y−2z=0x+y+z=0\left\{ \begin{array} { l } 5 x + y + z = 0 \\x - 2 y - 2 z = 0 \\x + y + z = 0\end{array} \right.⎩⎨⎧​5x+y+z=0x−2y−2z=0x+y+z=0​

Use matrices to solve the system. {4x+3y−z=−4x−4y+5z=14y+5z=36\left\{ \begin{aligned}4 x + 3 y - z & = - 4 \\x - 4 y + 5 z & = 1 \\4 y + 5 z & = 36\end{aligned} \right.⎩⎨⎧​4x+3y−zx−4y+5z4y+5z​=−4=1=36​

Find, if possible, BAB ABA . A=[8−74−119] and B=[2101−47]A = \left[ \begin{array} { c c c } 8 & - 7 & 4 \\- 1 & 1 & 9\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 2 & 1 \\0 & 1 \\- 4 & 7\end{array} \right]A=[8−1​−71​49​] and B=​20−4​117​​

Find the partial fraction decomposition. 4x4−4x3+8x2−7x+1x3−x2+x−1\frac { 4 x ^ { 4 } - 4 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 }x3−x2+x−14x4−4x3+8x2−7x+1​

Find the inverse of the matrix if it exists. [200090008]\left[ \begin{array} { l l l } 2 & 0 & 0 \\0 & 9 & 0 \\0 & 0 & 8\end{array} \right]​200​090​008​​

Find the partial fraction decomposition. 11x−20x2−4x\frac { 11 x - 20 } { x ^ { 2 } - 4 x }x2−4x11x−20​

Find the determinant of the matrix. [−34.59−0.35.515.36.713]\left[ \begin{array} { c c c } - 3 & 4.5 & 9 \\- 0.3 & 5.5 & 1 \\5.3 & 6.7 & 13\end{array} \right]​−3−0.35.3​4.55.56.7​9113​​

Use Cramer's rule, whenever possible, to solve the system. {4x+6y=84x−9y=8\left\{ \begin{array} { l } 4 x + 6 y = 8 \\4 x - 9 y = 8\end{array} \right.{4x+6y=84x−9y=8​

Find the determinant of the matrix after introducing zeros. [210−60112−1]\left[ \begin{array} { c c c } 2 & 1 & 0 \\- 6 & 0 & 1 \\1 & 2 & - 1\end{array} \right]​2−61​102​01−1​​

Find the partial fraction decomposition. 7x+1(x−1)2\frac { 7 x + 1 } { ( x - 1 ) ^ { 2 } }(x−1)27x+1​

The data in the table are generated by the function f(x)=ae−bxf ( x ) = a e ^ { - b x }f(x)=ae−bx . Approximate the unknown constants a and b to four decimal places. x 1 2 3 4 F ( x ) 0)71939 0.41687 0.24157 0.13998

Find, if possible, BAB ABA . A=[8−15],B=[−25]A = \left[ \begin{array} { l l l } 8 & - 1 & 5\end{array} \right] , B = \left[ \begin{array} { c } - 2 \\5\end{array} \right]A=[8​−1​5​],B=[−25​]

Find the inverse of the matrix if it exists. [5249]\left[ \begin{array} { l l } 5 & 2 \\4 & 9\end{array} \right][54​29​]

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Sketch the graph of the inequality. $5 x + 2 y \geq 1 y + 5$

Find the maximum and minimum values of the objective function $C = 8 x + 4 y + 7$ on the region in the figure.

Use the method of substitution to solve the system. $\left\{ \begin{array} { l } y = x ^ { 2 } - 25 \\y = 3 x - 7\end{array} \right.$

Use matrices to solve the system. $\left\{ \begin{array} { l } 5 x + y + z = 0 \\x - 2 y - 2 z = 0 \\x + y + z = 0\end{array} \right.$

Use matrices to solve the system. $\left\{ \begin{aligned}4 x + 3 y - z & = - 4 \\x - 4 y + 5 z & = 1 \\4 y + 5 z & = 36\end{aligned} \right.$

Find, if possible, $B A$ . $A = \left[ \begin{array} { c c c } 8 & - 7 & 4 \\- 1 & 1 & 9\end{array} \right] \text { and } B = \left[ \begin{array} { c c } 2 & 1 \\0 & 1 \\- 4 & 7\end{array} \right]$

Find the partial fraction decomposition. $\frac { 4 x ^ { 4 } - 4 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 }$

Find the inverse of the matrix if it exists. $\left[ \begin{array} { l l l } 2 & 0 & 0 \\0 & 9 & 0 \\0 & 0 & 8\end{array} \right]$

Find the partial fraction decomposition. $\frac { 11 x - 20 } { x ^ { 2 } - 4 x }$

Find the determinant of the matrix. $\left[ \begin{array} { c c c } - 3 & 4.5 & 9 \\- 0.3 & 5.5 & 1 \\5.3 & 6.7 & 13\end{array} \right]$

Use Cramer's rule, whenever possible, to solve the system. $\left\{ \begin{array} { l } 4 x + 6 y = 8 \\4 x - 9 y = 8\end{array} \right.$

Find the determinant of the matrix after introducing zeros. $\left[ \begin{array} { c c c } 2 & 1 & 0 \\- 6 & 0 & 1 \\1 & 2 & - 1\end{array} \right]$

Find the partial fraction decomposition. $\frac { 7 x + 1 } { ( x - 1 ) ^ { 2 } }$

The data in the table are generated by the function $f ( x ) = a e ^ { - b x }$ . Approximate the unknown constants a and b to four decimal places. x 1 2 3 4 F ( x ) 0)71939 0.41687 0.24157 0.13998

Find, if possible, $B A$ . $A = \left[ \begin{array} { l l l } 8 & - 1 & 5\end{array} \right] , B = \left[ \begin{array} { c } - 2 \\5\end{array} \right]$

Find the inverse of the matrix if it exists. $\left[ \begin{array} { l l } 5 & 2 \\4 & 9\end{array} \right]$