Question 1

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
2 x + 8 y + 6 z = 20 \
4 x + 2 y - 2 z = - 2 \
3 x - y + z = 11
\end{array}$$

Accepted Answer

A)  $\{ ( 2,1,6 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 2 , - 1,4 ) \}$ 
D)  $\{ ( - 1,2,1 ) \}$ 
A)  $\{ ( 2,1,6 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 2 , - 1,4 ) \}$ 
D)  $\{ ( - 1,2,1 ) \}$

Question 2

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective
function.
-$$\text { objective function } = 5 x + 3 y ; \text { minimum }$$

Accepted Answer

A)  0 
B)  14 
C)  18 
D)  30 
A)  0 
B)  14 
C)  18 
D)  30

Question 3

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of
the concepts involved in graphing inequalities.
-

Accepted Answer

A)  $ y \geq x+4 $ 
B)  $ y \geq-x+4 $ 
C)  $ y \leq x+4 $ 
D)  $ y \leq-x+4 $ 
A)  $ y \geq x+4 $ 
B)  $ y \geq-x+4 $ 
C)  $ y \leq x+4 $ 
D)  $ y \leq-x+4 $

Question 4

Choose the one alternative that best completes the statement or answers the question.
Graph the inequality.
$y \leq 5$

Accepted Answer

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

Question 5

Choose the one alternative that best completes the statement or answers the question.
Graph the inequality.
$y \geq 6$

Accepted Answer

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

Question 6

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$5 x - 8 y = 1$$
$- 10 x + 16 y = 1$

Accepted Answer

A)  $\left\{ \left( - \frac { 3 } { 32 } , \frac { 47 } { 256 } \right) \right\}$ 
B)  $\left\{ \left( x , \frac { 1 } { 4 } + \frac { 5 } { 8 } x \right) \right\}$ 
C)  $\left\{ \left( x , \frac { 3 } { 32 } x \right) \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \left( - \frac { 3 } { 32 } , \frac { 47 } { 256 } \right) \right\}$ 
B)  $\left\{ \left( x , \frac { 1 } { 4 } + \frac { 5 } { 8 } x \right) \right\}$ 
C)  $\left\{ \left( x , \frac { 3 } { 32 } x \right) \right\}$ 
D)  $\varnothing$

Question 7

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-Use the terms increasing, decreasing, and/or constant to describe the trends for the population of the City X metropolitan area.

Accepted Answer

A)  The population of the City X metropolitan area was increasing from Year 1 to Year 6. 
B)  The population of the City X metropolitan area was decreasing from Year 1 to Year 5 and was constant from Year 5 to Year 6. 
C)  The population of the City X metropolitan area was decreasing from Year 1 to Year 6. 
D)  The population of the City X metropolitan area was increasing from Year 1 to Year 5 and was constant from Year 5 to Year 6. 
A)  The population of the City X metropolitan area was increasing from Year 1 to Year 6. 
B)  The population of the City X metropolitan area was decreasing from Year 1 to Year 5 and was constant from Year 5 to Year 6. 
C)  The population of the City X metropolitan area was decreasing from Year 1 to Year 6. 
D)  The population of the City X metropolitan area was increasing from Year 1 to Year 5 and was constant from Year 5 to Year 6.

Question 8

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { c } 
x - 3 y = 6 \
8 x - 2 y = 4
\end{array}$$

Accepted Answer

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

Question 9

Find the partial fraction decomposition for the rational expression.
$\frac { 3 x - 20 } { ( x - 4 ) ^ { 2 } }$

Accepted Answer

A)  $\frac { 8 } { x - 4 } + \frac { - 3 } { ( x - 4 ) ^ { 2 } }$ 
B)  $\frac { 3 } { x - 4 } + \frac { 8 } { ( x - 4 ) ^ { 2 } }$ 
C)  $\frac { 3 } { x - 4 } + \frac { - 8 } { ( x - 4 ) ^ { 2 } }$ 
D)  $\frac { 6 } { x - 4 } + \frac { 16 } { ( x - 4 ) ^ { 2 } }$ 
A)  $\frac { 8 } { x - 4 } + \frac { - 3 } { ( x - 4 ) ^ { 2 } }$ 
B)  $\frac { 3 } { x - 4 } + \frac { 8 } { ( x - 4 ) ^ { 2 } }$ 
C)  $\frac { 3 } { x - 4 } + \frac { - 8 } { ( x - 4 ) ^ { 2 } }$ 
D)  $\frac { 6 } { x - 4 } + \frac { 16 } { ( x - 4 ) ^ { 2 } }$

Question 10

Provide an appropriate response.
-Describe the elements of a $5 \times 5$ zero matrix.

Accepted Answer

A)  The matrix has ones on the main diagonal, and all other elements are zeros. 
B)  The matrix has zeros on the main diagonal, and all other elements are ones. 
C)  Every element of the matrix is zero. 
D)  The matrix has zeros on the main diagonal, and there are no restrictions on the other elements. 
A)  The matrix has ones on the main diagonal, and all other elements are zeros. 
B)  The matrix has zeros on the main diagonal, and all other elements are ones. 
C)  Every element of the matrix is zero. 
D)  The matrix has zeros on the main diagonal, and there are no restrictions on the other elements.

Question 11

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary.
-$$\begin{array} { l } 
4 x + y = 22 \
2 x + 5 y = 2
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 2 , - 6 ) \}$ 
C)  $\{ ( - 2,6 ) \}$ 
D)  $\{ ( 6 , - 2 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 2 , - 6 ) \}$ 
C)  $\{ ( - 2,6 ) \}$ 
D)  $\{ ( 6 , - 2 ) \}$

Question 12

Let $A = \left[ \begin{array} { l l } 
e & 0 \
0 & a
\end{array} \right]$ where e and a are nonzero constants. Find $\mathrm { A } ^ { - 1 }$

Accepted Answer

A)  $\left[ \begin{array} { c } 
\frac { 1 } { e } {1} \
1 \frac { 1 } { a }
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
0 & e \
a & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
e & 0 \
0 & a
\end{array} \right]$ 
D)    
A)  $\left[ \begin{array} { c } 
\frac { 1 } { e } {1} \
1 \frac { 1 } { a }
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 
0 & e \
a & 0
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 
e & 0 \
0 & a
\end{array} \right]$ 
D)

Question 13

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-Express the solution of the system as an ordered pair.

Accepted Answer

A)  approximately (3.1, 0.95) 
B)  approximately (3.9, 0.92) 
C)  approximately (4.6, 0.87) 
D)  approximately (1.2, 1.03) 
A)  approximately (3.1, 0.95) 
B)  approximately (3.9, 0.92) 
C)  approximately (4.6, 0.87) 
D)  approximately (1.2, 1.03)

Question 14

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

Accepted Answer

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

Question 15

Solve the system by using the inverse of the coefficient matrix.
-$$- 5 x + 3 y = 8$$
$3 x - 6 y = - 30$

Accepted Answer

A)  $\{ ( - 2 , - 6 ) \}$ 
B)  $\{ ( - 6 , - 2 ) \}$ 
C)  $\{ ( 6,2 ) \}$ 
D)  $\{ ( 2,6 ) \}$ 
A)  $\{ ( - 2 , - 6 ) \}$ 
B)  $\{ ( - 6 , - 2 ) \}$ 
C)  $\{ ( 6,2 ) \}$ 
D)  $\{ ( 2,6 ) \}$

Question 16

A nonlinear system is given, along with the graphs of both equations in the system. Determine if the points of
intersection specified on the graph are solutions of the system by substituting directly into both equations.
-$$\begin{array} { l } 
x ^ { 2 } = y - 1 \
y = - 2 x + 16
\end{array}$$

Accepted Answer

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

Question 17

Find the partial fraction decomposition for the rational expression.
$\frac { 546 x ^ { 2 } + 78 x } { \left( x ^ { 2 } + 3 \right) ( x + 6 ) }$

Accepted Answer

A)  $\frac { 54 x + 492 } { x ^ { 2 } + 3 } + \frac { - 246 } { x + 6 }$ 
B)  $\frac { 54 x } { x + 6 } + \frac { 246 } { x ^ { 2 } + 3 }$ 
C)  $\frac { 54 x - 246 } { x ^ { 2 } + 3 } + \frac { 492 } { x + 6 }$ 
D)  $\frac { 54 x - 246 } { x ^ { 2 } + 3 } + \frac { - 492 } { x + 6 }$ 
A)  $\frac { 54 x + 492 } { x ^ { 2 } + 3 } + \frac { - 246 } { x + 6 }$ 
B)  $\frac { 54 x } { x + 6 } + \frac { 246 } { x ^ { 2 } + 3 }$ 
C)  $\frac { 54 x - 246 } { x ^ { 2 } + 3 } + \frac { 492 } { x + 6 }$ 
D)  $\frac { 54 x - 246 } { x ^ { 2 } + 3 } + \frac { - 492 } { x + 6 }$

Question 18

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
$\begin{array} { c } 
- 3 x - 7 y - 6 z = - 65 \
5 x + 9 y - 5 z = 43 \
9 x - 9 y + 8 z = 95
\end{array}$

Accepted Answer

A)  {(9,-2,-4)} 
B)  {(9,2,4)} 
C)  {(10,0,4)} 
D)  {(2,4,2)} 
A)  {(9,-2,-4)} 
B)  {(9,2,4)} 
C)  {(10,0,4)} 
D)  {(2,4,2)}

Question 19

Suppose that you are solving a system of three linear equations by the Gauss-Jordan method and obtain the following augmented matrix. $$\left[ \begin{array} { r r r | r } 
1 & 5 & - 6 & - 9 \
0 & 1 & 2 & - 12 \
0 & 0 & 0 & 6
\end{array} \right]$$ What conclusion can you draw about the solutions of the system?

Accepted Answer

A)  The system has infinitely many solutions. 
B)  The system has exactly one solution. 
C)  The system has no solutions. 
D)  There is not enough information to draw any conclusion about the solutions of the system. 
A)  The system has infinitely many solutions. 
B)  The system has exactly one solution. 
C)  The system has no solutions. 
D)  There is not enough information to draw any conclusion about the solutions of the system.

Question 20

A nonlinear system is given, along with the graphs of both equations in the system. Determine if the points of
intersection specified on the graph are solutions of the system by substituting directly into both equations.
-$\begin{array}{l}
x^{2}+y^{2}=25 \
2 y+x=5
\end{array}$

Accepted Answer

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

Solve the system by using the inverse of the coefficient matrix. - 2x+8y+6z=20 4x+2y-2z=-2 3x-y+z=11

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $y \leq 5$ $Choose the one alternative that best completes the statement or answers the question. Graph the inequality. y \leq 5$

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $y \geq 6$ $Choose the one alternative that best completes the statement or answers the question. Graph the inequality. y \geq 6$

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - $5 x - 8 y = 1$ $- 10 x + 16 y = 1$

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -Use the terms increasing, decreasing, and/or constant to describe the trends for the population of the City X metropolitan area.

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-3y=6 8x-2y=4

Find the partial fraction decomposition for the rational expression. $\frac { 3 x - 20 } { ( x - 4 ) ^ { 2 } }$

Provide an appropriate response. -Describe the elements of a $5 \times 5$ zero matrix.

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - 4x+y=22 2x+5y=2

Let $A = \left[ \begin{array} { l l } e & 0 \\0 & a\end{array} \right]$ where e and a are nonzero constants. Find $\mathrm { A } ^ { - 1 }$

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -Express the solution of the system as an ordered pair.

Graph the solution set of the system of inequalities. - x+2y\geq2 x-y\leq0 $Graph the solution set of the system of inequalities. - \begin{array}{c} x+2 y \geq 2 \\ x-y \leq 0 \end{array}$

Solve the system by using the inverse of the coefficient matrix. - $- 5 x + 3 y = 8$ $3 x - 6 y = - 30$

Find the partial fraction decomposition for the rational expression. $\frac { 546 x ^ { 2 } + 78 x } { \left( x ^ { 2 } + 3 \right) ( x + 6 ) }$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. -3x-7y-6z=-65 5x+9y-5z=43 9x-9y+8z=95

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

Solve the system by using the inverse of the coefficient matrix. - 2x+8y+6z=20 4x+2y-2z=-2 3x-y+z=11

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. - objective function =5x+3y; minimum \text { objective function } = 5 x + 3 y ; \text { minimum } objective function =5x+3y; minimum

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. y≤5y \leq 5y≤5

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. y≥6y \geq 6y≥6

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - 5x−8y=15 x - 8 y = 15x−8y=1 −10x+16y=1- 10 x + 16 y = 1−10x+16y=1