Question 1

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

Accepted Answer

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

Question 2

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$\begin{aligned}
x ^ { 2 } + y ^ { 2 } & = 49 \
- 6 x - 4 y & = 7
\end{aligned}$$

Accepted Answer

Substitution should be used, b

Question 3

Solve the system using the inverse method.
-$$\left\{ \begin{aligned}
x + 2 y + 3 z & = - 8 \
x + y + z & = - 4 \
- x + y + 2 z & = - 1
\end{aligned} \right.$$

Accepted Answer

A)  x = 2, y = -9, z = 1 
B)  x = -13, y = -46, z = -29 
C)  x = -8, y = -16, z = 2 
D)  x = -3, y = 2, z = -3 
A)  x = 2, y = -9, z = 1 
B)  x = -13, y = -46, z = -29 
C)  x = -8, y = -16, z = 2 
D)  x = -3, y = 2, z = -3

Question 4

Find the inverse of the matrix.
-$$\left[ \begin{array} { r r } 
- 6 & 5 \
3 & 0
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { l } 0 \frac { 1 } { 3 } \ \frac { 1 } { 5 } \frac { 2 } { 5 } \end{array} \right]$

В) $\left[ \begin{array} { l l } \frac { 1 } { 5 } & \frac { 2 } { 5 } \ 0 & \frac { 1 } { 3 } \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } \frac { 2 } { 5 } & \frac { 1 } { 3 } \ \frac { 1 } { 5 } & 0 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 3 } \ - \frac { 1 } { 5 } & \frac { 2 } { 5 } \end{array} \right]$ 
A)  $\left[ \begin{array} { l } 0 \frac { 1 } { 3 } \ \frac { 1 } { 5 } \frac { 2 } { 5 } \end{array} \right]$

В) $\left[ \begin{array} { l l } \frac { 1 } { 5 } & \frac { 2 } { 5 } \ 0 & \frac { 1 } { 3 } \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } \frac { 2 } { 5 } & \frac { 1 } { 3 } \ \frac { 1 } { 5 } & 0 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 3 } \ - \frac { 1 } { 5 } & \frac { 2 } { 5 } \end{array} \right]$

Question 5

Graph the system of inequalities.
-$\begin{array}{l}
x^{2}+y^{2} \leq 100 \
6 x+7 y \leq 42
\end{array}$

A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 6

Write a system of linear inequalities that has the given graph.
-  A)
$\left\{\begin{array}{r}
y \geq 0 \
x \geq 0 \
x \leq 6 \
y \leq 9 \
x+y \geq 3
\end{array}\right.$

Accepted Answer

B) 
$\left\{\begin{array}{r}
y \geq 0 \
x \geq 0 \
x \leq 9 \
x+y \geq 3
\end{array}\right.$ 
C) 
$\left\{\begin{array}{r}
x \leq 6 \
y \leq 9 \
x+y \geq 3
\end{array}\right.$ 
D) 
$\left\{\begin{array}{r}
y \geq 0 \
x \geq 0 \
y \leq 9 \
x+y \geq 3
\end{array}\right.$ 
B) 
$\left\{\begin{array}{r}
y \geq 0 \
x \geq 0 \
x \leq 9 \
x+y \geq 3
\end{array}\right.$ 
C) 
$\left\{\begin{array}{r}
x \leq 6 \
y \leq 9 \
x+y \geq 3
\end{array}\right.$ 
D) 
$\left\{\begin{array}{r}
y \geq 0 \
x \geq 0 \
y \leq 9 \
x+y \geq 3
\end{array}\right.$

Question 7

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

Accepted Answer

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) 
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 8

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

Accepted Answer

The answer of Write the partial fraction decomposition of the...

Question 9

Solve the 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 45 staff members and needs at least 30 to run properly. They 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)  35 counselors and 10 aides 
B)  12 counselors and 18 aides 
C)  27 counselors and 18 aides 
D)  18 counselors and 12 aides 
A)  35 counselors and 10 aides 
B)  12 counselors and 18 aides 
C)  27 counselors and 18 aides 
D)  18 counselors and 12 aides

Question 10

Write the partial fraction decomposition of the rational expression.
-$$\frac { - 4 x ^ { 2 } - 6 x + 2 } { ( x + 2 ) ( x + 1 ) ^ { 2 } }$$

Accepted Answer

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

Question 11

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$\begin{array} { c } 
x ^ { 2 } + 8 y ^ { 2 } = 25 \
- 7 x ^ { 2 } - 5 y ^ { 2 } = 100
\end{array}$$

Accepted Answer

Either method may be used. Elimination m

Question 12

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

Accepted Answer

A)  $\left[ \begin{array} { l l } \frac { 3 } { 4 } & - \frac { 1 } { 4 } \ \frac { 1 } { 4 } & \frac { 1 } { 4 } \end{array} \right]$

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

В) $\left[ \begin{array} { r } - \frac { 1 } { 4 } \frac { 1 } { 4 } \ \frac { 3 } { 4 } \frac { 1 } { 4 } \end{array} \right]$ 
C)  $\left[ \begin{array} { c c } \frac { 1 } { 4 } & \frac { 1 } { 4 } \ \frac { 3 } { 4 } & - \frac { 1 } { 4 } \end{array} \right]$ 
D)  $\left[ \begin{array} { r } \frac { 1 } { 4 } - \frac { 1 } { 4 } \ - \frac { 3 } { 4 } - \frac { 1 } { 4 } \end{array} \right]$

Question 13

Write a system of linear inequalities that has the given graph.
-  A) $y \geq 0 , x \geq 0 , x \leq 9$, and $y + x \leq 7$

Accepted Answer

B)  $y \geq 0 , x \geq 0 , x \leq 7$, and $y + x \geq 9$ 
C)  $x \leq 7$, and $y + x \leq 9$ 
D)  $y \geq 0 , x \geq 0 , x \leq 7$, and $y + x \leq 9$ 
B)  $y \geq 0 , x \geq 0 , x \leq 7$, and $y + x \geq 9$ 
C)  $x \leq 7$, and $y + x \leq 9$ 
D)  $y \geq 0 , x \geq 0 , x \leq 7$, and $y + x \leq 9$

Question 14

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

Accepted Answer

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)  $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & 2 \ 0 & 17 & 22 & - 3 \ 0 & 0 & 6 & 11 \end{array} \right]$ 
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 15

The graph of two equations along with the points of intersection are given. Substitute the points of intersection into the
systems of equations. Are the points of intersection solutions to the system of equations (Y/N)?
-  $$\begin{array} { l } 
x ^ { 2 } = y - 1 \
y = - 3 x + 11
\end{array}$$

Accepted Answer

A)  No 
B)  Yes 
A)  No 
B)  Yes

Question 16

Graph the system of inequalities.
-$$\begin{array}{l}
x ^ { 2 } + y \leq 1\
x ^ { 2 } - y \leq 3
\end{array}$$

A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 17

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { l } 
3 x + y = 13 \
2 x + 9 y = - 8
\end{array} \right.$$

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Solve the system of equations using substitution.
-$$\left\{ \begin{array} { c } 
x y = 72 \
x + y = 17
\end{array} \right.$$

Accepted Answer

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

Question 20

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?

Accepted Answer

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

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

Which method should be used to solve the system? Explain your answer, including a description of the first step. - + =49 -6x-4y =7

Solve the system using the inverse method. - $\left\{ \begin{aligned}x + 2 y + 3 z & = - 8 \\x + y + z & = - 4 \\- x + y + 2 z & = - 1\end{aligned} \right.$

Find the inverse of the matrix. - $\left[ \begin{array} { r r } - 6 & 5 \\3 & 0\end{array} \right]$

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

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

Write the partial fraction decomposition of the rational expression. - $\frac { - 4 x ^ { 2 } - 6 x + 2 } { ( x + 2 ) ( x + 1 ) ^ { 2 } }$

Which method should be used to solve the system? Explain your answer, including a description of the first step. - +8=25 -7-5=100

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

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

Solve the system of equations by using substitution. - $\left\{ \begin{array} { l } 3 x + y = 13 \\2 x + 9 y = - 8\end{array} \right.$

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

Solve the system of equations using substitution. - $\left\{ \begin{array} { c } x y = 72 \\x + y = 17\end{array} \right.$

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?

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Exam 10: Systems of Equations and Inequalities

Find the value of the determinant. - ∣3−1−12∣\left| \begin{array} { r r } 3 & - 1 \\- 1 & 2\end{array} \right|​3−1​−12​​

Which method should be used to solve the system? Explain your answer, including a description of the first step. - + =49 -6x-4y =7

Solve the system using the inverse method. - {x+2y+3z=−8x+y+z=−4−x+y+2z=−1\left\{ \begin{aligned}x + 2 y + 3 z & = - 8 \\x + y + z & = - 4 \\- x + y + 2 z & = - 1\end{aligned} \right.⎩⎨⎧​x+2y+3zx+y+z−x+y+2z​=−8=−4=−1​

Find the inverse of the matrix. - [−6530]\left[ \begin{array} { r r } - 6 & 5 \\3 & 0\end{array} \right][−63​50​]

Graph the system of inequalities. - +\leq100 6x+7y\leq42 A) B) C) D)

Solve the system of equations. - {2x−y+5z=−7x+y−2z=−2x−y+4z=8\left\{ \begin{aligned}2 x - y + 5 z & = - 7 \\x + y - 2 z & = - 2 \\x - y + 4 z & = 8\end{aligned} \right.⎩⎨⎧​2x−y+5zx+y−2zx−y+4z​=−7=−2=8​

Write the partial fraction decomposition of the rational expression. - 7x3−2x2(x+1)3\frac { 7 x ^ { 3 } - 2 } { x ^ { 2 } ( x + 1 ) ^ { 3 } }x2(x+1)37x3−2​

Write the partial fraction decomposition of the rational expression. - −4x2−6x+2(x+2)(x+1)2\frac { - 4 x ^ { 2 } - 6 x + 2 } { ( x + 2 ) ( x + 1 ) ^ { 2 } }(x+2)(x+1)2−4x2−6x+2​