Question 1

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
x + y = 6 \
3 x + 3 y = 18
\end{array}$$

Accepted Answer

A)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
B)  Cramer's rule does not apply since $D = 0 ; \{ ( 6 - y , y ) \}$ 
C)  $\{ ( 2,4 ) \}$ 
D)  $\{ ( 0,0 ) \}$ 
A)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
B)  Cramer's rule does not apply since $D = 0 ; \{ ( 6 - y , y ) \}$ 
C)  $\{ ( 2,4 ) \}$ 
D)  $\{ ( 0,0 ) \}$

Question 2

Find the partial fraction decomposition for the rational expression.
-$$\frac { 4 x ^ { 3 } - 22 x ^ { 2 } + 33 x - 6 } { ( x - 2 ) ^ { 3 } ( x + 2 ) }$$

Accepted Answer

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

Question 3

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
- 7 x - 7 y = 14 \
- 4 x - 4 y = 8
\end{array}$$

Accepted Answer

A)  $\{ ( - 2,0 ) \}$ 
B)  $\{ ( - 2,1 ) \}$ 
C)  $\{ ( - 3,1 ) \}$ 
D)  Cramer's rule does not apply since $D = 0$; $\varnothing$ 
A)  $\{ ( - 2,0 ) \}$ 
B)  $\{ ( - 2,1 ) \}$ 
C)  $\{ ( - 3,1 ) \}$ 
D)  Cramer's rule does not apply since $D = 0$; $\varnothing$

Question 4

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { r } 
5 x ^ { 2 } + 3 x y - 5 y ^ { 2 } = - 9 \
x ^ { 2 } - x y - y ^ { 2 } = - 5
\end{array}$$

Accepted Answer

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

Question 5

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

Accepted Answer

A)  $- 32$ 
B)  $- 92$ 
C)  $- 132$ 
D)  92 
A)  $- 32$ 
B)  $- 92$ 
C)  $- 132$ 
D)  92

Question 6

Find the value of the determinant.
-$$\left| \begin{array} { l l } 
2 & 9 \
5 & 6
\end{array} \right|$$

Accepted Answer

A)  57 
B)  $- 33$ 
C)  33 
D)  $- 12$ 
A)  57 
B)  $- 33$ 
C)  33 
D)  $- 12$

Question 7

Find the partial fraction decomposition for the rational expression.
-$$\frac { 26 x ^ { 2 } - 139 x + 189 } { ( x - 5 ) ( 2 x - 4 ) ^ { 2 } }$$

Accepted Answer

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

Question 8

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
7 x - 34 = 8 y \
- 3 x + 6 y = - 12
\end{array}$$

Accepted Answer

A)  $\{ ( 6,2 ) \}$ 
B)  $\{ ( 5,2 ) \}$ 
C)  $\{ ( 6,1 ) \}$ 
D)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
A)  $\{ ( 6,2 ) \}$ 
B)  $\{ ( 5,2 ) \}$ 
C)  $\{ ( 6,1 ) \}$ 
D)  Cramer's rule does not apply since $D = 0 ; \varnothing$

Question 9

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$y \geq - 5 x ^ { 2 } + 1$$

Accepted Answer

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

Question 10

Provide an appropriate response.
-Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & - 6 \ 3 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 3 & - 6 \ 4 & 5 \end{array} \right]$.
Does the matrix A - B have an inverse?

Accepted Answer

A)  $\mathrm { Yes }$ 
B)  $\mathrm { No }$ 
A)  $\mathrm { Yes }$ 
B)  $\mathrm { No }$

Question 11

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-$$\begin{array} { l } 
x - \sqrt { 2 } y = 3.7 \
2.85 x + y = - 8.23
\end{array}$$

Accepted Answer

A)  $\{ ( 3.43297,0.18882 ) \}$ 
B)  $\{ ( - 1.57817 , - 3.73223 ) \}$ 
C)  $\{ ( 2.61630 , - 2.88772 ) \}$ 
D)  $\{ ( 0,0 ) \}$ 
A)  $\{ ( 3.43297,0.18882 ) \}$ 
B)  $\{ ( - 1.57817 , - 3.73223 ) \}$ 
C)  $\{ ( 2.61630 , - 2.88772 ) \}$ 
D)  $\{ ( 0,0 ) \}$

Question 12

Find the partial fraction decomposition for the rational expression.
-$$\frac { 13 x - 34 } { x ^ { 2 } - 6 x + 8 }$$

Accepted Answer

A)  $\frac { 1 } { x - 4 } + \frac { 1 } { x - 2 }$ 
B)  $\frac { 9 } { x - 4 } + \frac { 4 } { x - 2 }$ 
C)  $\frac { 9 } { x + 4 } + \frac { 4 } { x + 2 }$ 
D)  $\frac { 4 } { x - 4 } + \frac { 9 } { x - 2 }$ 
A)  $\frac { 1 } { x - 4 } + \frac { 1 } { x - 2 }$ 
B)  $\frac { 9 } { x - 4 } + \frac { 4 } { x - 2 }$ 
C)  $\frac { 9 } { x + 4 } + \frac { 4 } { x + 2 }$ 
D)  $\frac { 4 } { x - 4 } + \frac { 9 } { x - 2 }$

Question 13

Graph the solution set of the system of inequalities.
-$$\begin{array} { c } 
x + 2 y \leq 2 \
x + y \geq 0
\end{array}$$

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $\left\{ \frac { 4 } { 3 } \right\}$ 
B)  $\{ 16 \}$ 
C)  $\{ - 4,4 \}$ 
D)  $\varnothing$ 
A)  $\left\{ \frac { 4 } { 3 } \right\}$ 
B)  $\{ 16 \}$ 
C)  $\{ - 4,4 \}$ 
D)  $\varnothing$

Question 15

Find the value of the determinant.
-$$\left| \begin{array} { r r r } 
0 & 0 & 1 \
6 & - 6 & 1 \
0 & 3 & 5
\end{array} \right|$$

Accepted Answer

A)  $- 48$ 
B)  $- 12$ 
C)  18 
D)  $- 18$ 
A)  $- 48$ 
B)  $- 12$ 
C)  18 
D)  $- 18$

Question 16

Provide an appropriate response.
-Fill in the blank to complete the statement. A matrix having the same number of rows as columns is called a $\underline { ? }$ matrix.

Accepted Answer

A)  column 
B)  square 
C)  regular 
D)  rectangular 
A)  column 
B)  square 
C)  regular 
D)  rectangular

Question 17

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

Accepted Answer

A)  $- 3$ 
B)  3 
C)  5 
D)  0 
A)  $- 3$ 
B)  3 
C)  5 
D)  0

Question 18

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-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).
$\begin{array}{c|c|c|c|r}
\text { Person } & \text { Green } & \text { Red } & \text { Orange } & \text { Total Cost } \
\hline \text { Bob } & 1 & 2 & 2 & \$ 28.98 \
\text { Sue } & 1 & 3 & 2 & \$ 35.85 \
\text { Fred } & 1 & 2 & 3 & \$ 34.12
\end{array}$

Use this information to set up a matrix equation of the form $A X = B$, which can be solved to determ: the price for each type of sale book. Solve this matrix equation to find the price of a book with a gree 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)  $\$ 4.96$ 
B)  $\$ 4.91$ 
C)  $\$ 4.99$ 
D)  $\$ 4.89$ 
A)  $\$ 4.96$ 
B)  $\$ 4.91$ 
C)  $\$ 4.99$ 
D)  $\$ 4.89$

Question 19

Use a graphing calculator to find the value of the determinant.
-$$\left| \begin{array} { l l l } 
4 & 2 & 3 \
1 & 4 & 2 \
3 & 2 & 4
\end{array} \right|$$

Accepted Answer

A)  $- 2$ 
B)  22 
C)  $- 22$ 
D)  142 
A)  $- 2$ 
B)  22 
C)  $- 22$ 
D)  142

Question 20

Solve the problem.
-The perimeter of a rectangle is $60 \mathrm {~m}$. If the width were doubled and the length were increased by 19 $\mathrm { m }$, the perimeter would be $116 \mathrm {~m}$. What are the length and width of the rectangle?

Accepted Answer

A)  Length: $15 \mathrm {~m}$; width: $15 \mathrm {~m}$ 
B)  Length: $21 \mathrm {~m}$; width: $9 \mathrm {~m}$ 
C)  Length: $15 \mathrm {~m}$; width: $10 \mathrm {~m}$ 
D)  Length: $9 \mathrm {~m}$; width: $21 \mathrm {~m}$ 
A)  Length: $15 \mathrm {~m}$; width: $15 \mathrm {~m}$ 
B)  Length: $21 \mathrm {~m}$; width: $9 \mathrm {~m}$ 
C)  Length: $15 \mathrm {~m}$; width: $10 \mathrm {~m}$ 
D)  Length: $9 \mathrm {~m}$; width: $21 \mathrm {~m}$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+y=6 3x+3y=18

Find the partial fraction decomposition for the rational expression. - $\frac { 4 x ^ { 3 } - 22 x ^ { 2 } + 33 x - 6 } { ( x - 2 ) ^ { 3 } ( x + 2 ) }$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - -7x-7y=14 -4x-4y=8

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - 5+3xy-5=-9 -xy-=-5

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

Find the value of the determinant. - $\left| \begin{array} { l l } 2 & 9 \\5 & 6\end{array} \right|$

Find the partial fraction decomposition for the rational expression. - $\frac { 26 x ^ { 2 } - 139 x + 189 } { ( x - 5 ) ( 2 x - 4 ) ^ { 2 } }$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 7x-34=8y -3x+6y=-12

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $y \geq - 5 x ^ { 2 } + 1$ $Which method should be used to solve the system? Explain your answer, including a description of the first step. - y \geq - 5 x ^ { 2 } + 1$

Provide an appropriate response. -Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & - 6 \\ 3 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 3 & - 6 \\ 4 & 5 \end{array} \right]$ . Does the matrix A - B have an inverse?

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - x-y=3.7 2.85x+y=-8.23

Find the partial fraction decomposition for the rational expression. - $\frac { 13 x - 34 } { x ^ { 2 } - 6 x + 8 }$

Graph the solution set of the system of inequalities. - x+2y\leq2 x+y\geq0 $Graph the solution set of the system of inequalities. - \begin{array} { c } x + 2 y \leq 2 \\ x + y \geq 0 \end{array}$

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

Find the value of the determinant. - $\left| \begin{array} { r r r } 0 & 0 & 1 \\6 & - 6 & 1 \\0 & 3 & 5\end{array} \right|$

Provide an appropriate response. -Fill in the blank to complete the statement. A matrix having the same number of rows as columns is called a $\underline { ? }$ matrix.

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

Use a graphing calculator to find the value of the determinant. - $\left| \begin{array} { l l l } 4 & 2 & 3 \\1 & 4 & 2 \\3 & 2 & 4\end{array} \right|$

Solve the problem. -The perimeter of a rectangle is $60 \mathrm {~m}$ . If the width were doubled and the length were increased by 19 $\mathrm { m }$ , the perimeter would be $116 \mathrm {~m}$ . What are the length and width of the rectangle?

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Analytic Geometry

Further Topics in Algebra

Filters

Exam 10: Systems and Matrices

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+y=6 3x+3y=18

Find the partial fraction decomposition for the rational expression. - 4x3−22x2+33x−6(x−2)3(x+2)\frac { 4 x ^ { 3 } - 22 x ^ { 2 } + 33 x - 6 } { ( x - 2 ) ^ { 3 } ( x + 2 ) }(x−2)3(x+2)4x3−22x2+33x−6​