Question 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.

Accepted Answer

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

Question 2

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { l l } 
- 5 & 1 \
- 7 & 1
\end{array} \right] \text { and } \left[ \begin{array} { l l } 
\frac { 1 } { 2 } & - \frac { 1 } { 2 } \
\frac { 7 } { 2 } & - \frac { 5 } { 2 }
\end{array} \right]$$

Accepted Answer

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

Question 3

Solve the problem.
-Find the equation of the line $y = a x + b$ that passes through the points $( - 1,4 )$ and $( - 5 , - 6 )$

Accepted Answer

A)  $y = \frac { 13 } { 2 } x + \frac { 5 } { 2 }$ 
B)  $y = \frac { 1 } { 2 } x + \frac { 7 } { 2 }$ 
C)  $y = \frac { 5 } { 2 } x + \frac { 13 } { 2 }$ 
D)  $y = \frac { 1 } { 2 } x + \frac { 4 } { 9 }$ 
A)  $y = \frac { 13 } { 2 } x + \frac { 5 } { 2 }$ 
B)  $y = \frac { 1 } { 2 } x + \frac { 7 } { 2 }$ 
C)  $y = \frac { 5 } { 2 } x + \frac { 13 } { 2 }$ 
D)  $y = \frac { 1 } { 2 } x + \frac { 4 } { 9 }$

Question 4

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { l l l } 
1 & 0 & 8 \
1 & 2 & 3 \
2 & 5 & 3
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { l l l } 
1 & 1 & 2 \
0 & 2 & 5 \
8 & 3 & 3
\end{array} \right]$$ 
B)

$$\left[ \begin{array} { r r r } 
- 1 & 0 & - 8 \
- 1 & - 2 & - 3 \
- 2 & - 5 & - 3
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
9 & - 40 & 16 \
- 3 & 13 & - 5 \
- 1 & 5 & - 2
\end{array} \right]$$ 
D)  The inverse does not exist. 
A) 
$$\left[ \begin{array} { l l l } 
1 & 1 & 2 \
0 & 2 & 5 \
8 & 3 & 3
\end{array} \right]$$ 
B)

$$\left[ \begin{array} { r r r } 
- 1 & 0 & - 8 \
- 1 & - 2 & - 3 \
- 2 & - 5 & - 3
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
9 & - 40 & 16 \
- 3 & 13 & - 5 \
- 1 & 5 & - 2
\end{array} \right]$$ 
D)  The inverse does not exist.

Question 5

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last
variable be the arbitrary variable.
-$$\begin{array} { l } 
2 x - 2 y + 3 z = - 7 \
x + 2 y - z = 10 \
2 y + z = 1
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 3,11 , - 11 ) \}$ 
C)  $\{ ( - 3,2,9 ) \}$ 
D)  $\{ ( 3,2 , - 3 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 3,11 , - 11 ) \}$ 
C)  $\{ ( - 3,2,9 ) \}$ 
D)  $\{ ( 3,2 , - 3 ) \}$

Question 6

Solve the problem using matrices.
-John has a jarful of quarters and nickels. There are 101 coins in the jar. The value of the coins is $\$ 16.05$. How many of each type of coin are there?

Accepted Answer

A)  55 quarters; 46 nickels 
B)  60 quarters ; 41 nickels 
C)  46 quarters; 55 nickels 
D)  96 quarters; 5 nickels 
A)  55 quarters; 46 nickels 
B)  60 quarters ; 41 nickels 
C)  46 quarters; 55 nickels 
D)  96 quarters; 5 nickels

Question 7

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
x y = 18 \
x ^ { 2 } + y ^ { 2 } = 85
\end{array}$$

Accepted Answer

A)  $\{ ( 9,2 ) , ( - 9 , - 2 ) , ( 9 , - 2 ) , ( - 9,2 ) \}$ 
B)  $\{ ( 9,2 ) , ( 2,9 ) , ( 9 , - 2 ) , ( 2 , - 9 ) \}$ 
C)  $\{ ( - 9 , - 2 ) , ( - 2 , - 9 ) , ( - 9,2 ) , ( - 2,9 ) \}$ 
D)  $\{ ( 9,2 ) , ( - 9 , - 2 ) , ( 2,9 ) , ( - 2 , - 9 ) \}$ 
A)  $\{ ( 9,2 ) , ( - 9 , - 2 ) , ( 9 , - 2 ) , ( - 9,2 ) \}$ 
B)  $\{ ( 9,2 ) , ( 2,9 ) , ( 9 , - 2 ) , ( 2 , - 9 ) \}$ 
C)  $\{ ( - 9 , - 2 ) , ( - 2 , - 9 ) , ( - 9,2 ) , ( - 2,9 ) \}$ 
D)  $\{ ( 9,2 ) , ( - 9 , - 2 ) , ( 2,9 ) , ( - 2 , - 9 ) \}$

Question 8

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$y \geq|x+6|$

Accepted Answer

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

Question 9

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { r r r } 
0 & 4 & 4 \
- 3 & 0 & 5 \
0 & 4 & 0
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
\frac { 5 } { 12 } & - \frac { 1 } { 3 } & - \frac { 5 } { 12 } \
0 & 0 & \frac { 1 } { 4 } \
\frac { 1 } { 4 } & 0 & - \frac { 1 } { 4 }
\end{array} \right]$$ 
B)

$$\left[ \begin{array} { r r r } 
- \frac { 5 } { 12 } & - \frac { 1 } { 3 } & - \frac { 5 } { 12 } \
- \frac { 1 } { 4 } & 0 & \frac { 1 } { 4 } \
\frac { 1 } { 4 } & 0 & 0
\end{array} \right]$$ 
C)  The inverse does not exist. 
D) 
$$\left[ \begin{array} { r r r } 
\frac { 5 } { 12 } & 0 & \frac { 1 } { 4 } \
- \frac { 1 } { 3 } & 0 & 0 \
- \frac { 5 } { 12 } & \frac { 1 } { 4 } & - \frac { 1 } { 4 }
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
\frac { 5 } { 12 } & - \frac { 1 } { 3 } & - \frac { 5 } { 12 } \
0 & 0 & \frac { 1 } { 4 } \
\frac { 1 } { 4 } & 0 & - \frac { 1 } { 4 }
\end{array} \right]$$ 
B)

$$\left[ \begin{array} { r r r } 
- \frac { 5 } { 12 } & - \frac { 1 } { 3 } & - \frac { 5 } { 12 } \
- \frac { 1 } { 4 } & 0 & \frac { 1 } { 4 } \
\frac { 1 } { 4 } & 0 & 0
\end{array} \right]$$ 
C)  The inverse does not exist. 
D) 
$$\left[ \begin{array} { r r r } 
\frac { 5 } { 12 } & 0 & \frac { 1 } { 4 } \
- \frac { 1 } { 3 } & 0 & 0 \
- \frac { 5 } { 12 } & \frac { 1 } { 4 } & - \frac { 1 } { 4 }
\end{array} \right]$$

Question 10

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective
function.
-$$\text { objective function } = 4 x + 3 y ^ { * } \text { maximnun }$$

Accepted Answer

A)  27 
B)  13 
C)  100 
D)  29 
A)  27 
B)  13 
C)  100 
D)  29

Question 11

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-$$\begin{array} { l } 
x ^ { 2 } - y ^ { 2 } = 7.82 \
x = 2 y - 5
\end{array}$$

Accepted Answer

A)  $\{ ( - 6.31,5.65 ) , ( 2.97,1.01 ) \}$ 
B)  $\{ ( - 2.97 , - 1.01 ) , ( 6.31 , - 5.65 ) \}$ 
C)  $\{ ( - 6.31 , - 5.65 ) , ( 2.97 , - 1.01 ) \}$ 
D)  $\{ ( - 2.97,1.01 ) , ( 6.31,5.65 ) \}$ 
A)  $\{ ( - 6.31,5.65 ) , ( 2.97,1.01 ) \}$ 
B)  $\{ ( - 2.97 , - 1.01 ) , ( 6.31 , - 5.65 ) \}$ 
C)  $\{ ( - 6.31 , - 5.65 ) , ( 2.97 , - 1.01 ) \}$ 
D)  $\{ ( - 2.97,1.01 ) , ( 6.31,5.65 ) \}$

Question 12

Find the partial fraction decomposition for the rational expression.
-$$\frac { - 5 x ^ { 2 } - 5 x + 6 } { ( x + 3 ) ^ { 2 } ( 3 x + 1 ) }$$

Accepted Answer

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

Question 13

Provide an appropriate response.
-Fill in the blanks to complete the statement. For a system of 6 equations and 6 unknowns, the corresponding augmented matrix will have rows and columns.

Accepted Answer

A)  7; 7 
B)  6; 7 
C)  6; 6 
D)  7; 6 
A)  7; 7 
B)  6; 7 
C)  6; 6 
D)  7; 6

Question 14

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
x + y = - 1 \
x + y = - 6
\end{array}$$

Accepted Answer

A)  $\{ ( 0 , - 7 ) \}$ 
B)  $\{ ( - 1 , - 6 ) \}$ 
C)  Cramer's rule does not apply since $D = 0 ; \{ ( - 6 - y , y ) \}$ 
D)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
A)  $\{ ( 0 , - 7 ) \}$ 
B)  $\{ ( - 1 , - 6 ) \}$ 
C)  Cramer's rule does not apply since $D = 0 ; \{ ( - 6 - y , y ) \}$ 
D)  Cramer's rule does not apply since $D = 0 ; \varnothing$

Question 15

Solve the system by substitution.
-$$\begin{aligned}
x - 8 y & = 5 \
x & = 9 y
\end{aligned}$$

Accepted Answer

A)  $\{ ( 5,45 ) \}$ 
B)  $\left\{ \left( \frac { 8 } { 17 } , \frac { 5 } { 17 } \right) \right\}$ 
C)  $\{ ( 45,5 ) \}$ 
D)  $\{ ( - 45 , - 5 ) \}$ 
A)  $\{ ( 5,45 ) \}$ 
B)  $\left\{ \left( \frac { 8 } { 17 } , \frac { 5 } { 17 } \right) \right\}$ 
C)  $\{ ( 45,5 ) \}$ 
D)  $\{ ( - 45 , - 5 ) \}$

Question 16

Use a graphing calculator to Express solutions with approximations to the nearest thousandth.
-$$\begin{array} { l } 
\frac { 9 } { 7 } x + y = 5 \\
0.4 x - y = 3
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 2.258,2.097 ) \}$ 
C)  $\{ ( 4.746 , - 1.102 ) \}$ 
D)  $\{ ( - 2.947,8.789 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 2.258,2.097 ) \}$ 
C)  $\{ ( 4.746 , - 1.102 ) \}$ 
D)  $\{ ( - 2.947,8.789 ) \}$

Question 17

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

Accepted Answer

A)  $\{ ( 1,4,3 ) \}$ 
B)  $\{ ( 4,1,3 ) \}$ 
C)  Cramer's rule does not apply since $D = 0$; $\varnothing$ 
D)  $\{ ( 1,3,4 ) \}$ 
A)  $\{ ( 1,4,3 ) \}$ 
B)  $\{ ( 4,1,3 ) \}$ 
C)  Cramer's rule does not apply since $D = 0$; $\varnothing$ 
D)  $\{ ( 1,3,4 ) \}$

Question 18

Evaluate the determinant.
-$$\left| \begin{array} { r r r } 
9 & - 8 & 6 \
- 3 & - 5 & - 6 \
0 & 0 & 0
\end{array} \right|$$

Accepted Answer

A)  135 
B)  87 
C)  $- 183$ 
D)  0 
A)  135 
B)  87 
C)  $- 183$ 
D)  0

Question 19

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { l l l } 
2 & - 5 & 4 \
2 & - 4 & 7 \
4 & - 9 & 11
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c c } 
1 & 0 & \frac { 19 } { 2 } \
0 & 1 & 3 \
0 & - \frac { 1 } { 2 } & 0
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$\left[\begin{array}{lll}
2 & 2 & 4 \
-5 & -4 & -9 \
4 & 7 & 11
\end{array}\right]$ 
D) 
$\left[\begin{array}{lll}
1 & 0 & \frac{19}{2} \
0 & 1 & 3 \
0 & 0 & 0
\end{array}\right]$ 
A) 
$$\left[ \begin{array} { c c c } 
1 & 0 & \frac { 19 } { 2 } \
0 & 1 & 3 \
0 & - \frac { 1 } { 2 } & 0
\end{array} \right]$$ 
B)  The inverse does not exist. 
C) 
$\left[\begin{array}{lll}
2 & 2 & 4 \
-5 & -4 & -9 \
4 & 7 & 11
\end{array}\right]$ 
D) 
$\left[\begin{array}{lll}
1 & 0 & \frac{19}{2} \
0 & 1 & 3 \
0 & 0 & 0
\end{array}\right]$

Question 20

Evaluate the determinant.
-$$\left| \begin{array} { l l l } 
5 & 4 & 6 \
3 & 7 & 3 \
4 & 4 & 9
\end{array} \right|$$

Accepted Answer

A)  $- 99$ 
B)  3 
C)  99 
D)  771 
A)  $- 99$ 
B)  3 
C)  99 
D)  771

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.

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } - 5 & 1 \\- 7 & 1\end{array} \right] \text { and } \left[ \begin{array} { l l } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \\\frac { 7 } { 2 } & - \frac { 5 } { 2 }\end{array} \right]$

Solve the problem. -Find the equation of the line $y = a x + b$ that passes through the points $( - 1,4 )$ and $( - 5 , - 6 )$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]$

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. - 2x-2y+3z=-7 x+2y-z=10 2y+z=1

Solve the problem using matrices. -John has a jarful of quarters and nickels. There are 101 coins in the jar. The value of the coins is $\$ 16.05$ . How many of each type of coin are there?

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - xy=18 +=85

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $y \geq|x+6|$ $Which method should be used to solve the system? Explain your answer, including a description of the first step. - y \geq|x+6|$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r r } 0 & 4 & 4 \\- 3 & 0 & 5 \\0 & 4 & 0\end{array} \right]$

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. - -=7.82 x=2y-5

Find the partial fraction decomposition for the rational expression. - $\frac { - 5 x ^ { 2 } - 5 x + 6 } { ( x + 3 ) ^ { 2 } ( 3 x + 1 ) }$

Provide an appropriate response. -Fill in the blanks to complete the statement. For a system of 6 equations and 6 unknowns, the corresponding augmented matrix will have rows and columns.

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+y=-1 x+y=-6

Solve the system by substitution. - x-8y =5 x =9y

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - x+y=5 0.4x-y=3

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

Evaluate the determinant. - $\left| \begin{array} { r r r } 9 & - 8 & 6 \\- 3 & - 5 & - 6 \\0 & 0 & 0\end{array} \right|$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 2 & - 5 & 4 \\2 & - 4 & 7 \\4 & - 9 & 11\end{array} \right]$

Evaluate the determinant. - $\left| \begin{array} { l l l } 5 & 4 & 6 \\3 & 7 & 3 \\4 & 4 & 9\end{array} \right|$

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