Question 1

Use a graphing calculator to Express solutions with approximations to the nearest thousandth.
-$$\begin{array} { l } 
0.4 \mathrm { x } + \sqrt { 2 } \mathrm { y } = 1 \
\sqrt { 5 } \mathrm { x } + 0.8 \mathrm { y } = 1
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 0.216,0.646 ) \}$ 
C)  $\{ ( 0.097,0.980 ) \}$ 
D)  $\{ ( 0.176,0.757 )$ 
A)  $\varnothing$ 
B)  $\{ ( 0.216,0.646 ) \}$ 
C)  $\{ ( 0.097,0.980 ) \}$ 
D)  $\{ ( 0.176,0.757 )$

Question 2

Provide an appropriate response.
-You are graphing $2 x ^ { 2 } + 3 y ^ { 2 } > 5$. You have drawn the graph of $2 x ^ { 2 } + 3 y ^ { 2 } = 5$. What single point will allow you to determine the correct region?

Accepted Answer

A)  $( 2,5 )$ 
B)  $( 2,3 )$ 
C)  The origin 
D)  $( 3,5 )$ 
A)  $( 2,5 )$ 
B)  $( 2,3 )$ 
C)  The origin 
D)  $( 3,5 )$

Question 3

Find the matrix product when possible.
-Mike's Bait Shop sells three types of lures: discount, normal, and professional. Location I sells 38 discount lures, 100 regular lures, and 30 professional lures each day. Location II sells 35 discount lures and Location III sells 60 discount lures each day. Daily sales of regular lures are 90 at Location II and 120 at Location III. At Location II, 35 expert lures are sold each day, and 40 expert lures are sold each day at Location III. Write a $3 \times 3$ matrix that shows the sales figures for the three locations, with the rows representing the three locations.

Accepted Answer

A) 
$\left[\begin{array}{rrr}
38 & 100 & 30 \
35 & 90 & 35 \
60 & 120 & 40
\end{array}\right]$ 
B) 
$\left[\begin{array}{ccc}
38 & 100 & 30 \
35 & 35 & 90 \
60 & 120 & 40
\end{array}\right]$ 
C) 
$\left[\begin{array}{crr}
100 & 38 & 30 \
35 & 90 & 35 \
60 & 120 & 40
\end{array}\right]$ 
D) 
$\left[\begin{array}{rrr}
38 & 100 & 35 \
35 & 90 & 30 \
60 & 120 & 40
\end{array}\right]$ 
A) 
$\left[\begin{array}{rrr}
38 & 100 & 30 \
35 & 90 & 35 \
60 & 120 & 40
\end{array}\right]$ 
B) 
$\left[\begin{array}{ccc}
38 & 100 & 30 \
35 & 35 & 90 \
60 & 120 & 40
\end{array}\right]$ 
C) 
$\left[\begin{array}{crr}
100 & 38 & 30 \
35 & 90 & 35 \
60 & 120 & 40
\end{array}\right]$ 
D) 
$\left[\begin{array}{rrr}
38 & 100 & 35 \
35 & 90 & 30 \
60 & 120 & 40
\end{array}\right]$

Question 4

Write the augmented matrix for the system. Do not
-$$\begin{array} { r } 
6 x + 5 z - 1 = 0 \
4 y + 8 z - 16 = 0 \
- 2 x + 9 y + 7 z - 45 = 0
\end{array}$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
6 & 0 & 5 \
0 & 4 & 8 \
- 2 & 9 & 7
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r | r } 
6 & 5 & 0 & 1 \
4 & 8 & 0 & 16 \
- 2 & 9 & 7 & 45
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r | r } 
6 & 0 & - 2 & 1 \
0 & 4 & 9 & 16 \
5 & 8 & 7 & 45
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r | r } 
6 & 0 & 5 & 1 \
0 & 4 & 8 & 16 \
- 2 & 9 & 7 & 45
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
6 & 0 & 5 \
0 & 4 & 8 \
- 2 & 9 & 7
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r | r } 
6 & 5 & 0 & 1 \
4 & 8 & 0 & 16 \
- 2 & 9 & 7 & 45
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r | r } 
6 & 0 & - 2 & 1 \
0 & 4 & 9 & 16 \
5 & 8 & 7 & 45
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r | r } 
6 & 0 & 5 & 1 \
0 & 4 & 8 & 16 \
- 2 & 9 & 7 & 45
\end{array} \right]$$

Question 5

Perform the operation or operations when possible.
-$$\left[ \begin{array} { r r } 
6 & - 5 \
5 & 6
\end{array} \right] + \left[ \begin{array} { l l } 
- 5 & - 6 \
- 3 & - 2
\end{array} \right] - \left[ \begin{array} { l l } 
- 5 & 6 \
- 4 & 3
\end{array} \right]$$

Accepted Answer

A) 
$\left[\begin{array}{rr}
6 & 7 \
4 & 11
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr}
16 & -5 \
12 & 5
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
-4 & -5 \
-2 & 7
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
6 & -17 \
6 & 1
\end{array}\right]$ 
A) 
$\left[\begin{array}{rr}
6 & 7 \
4 & 11
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr}
16 & -5 \
12 & 5
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
-4 & -5 \
-2 & 7
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
6 & -17 \
6 & 1
\end{array}\right]$

Question 6

Solve the linear programming problem.
-An airline with two types of airplanes, $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$, has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip, airplane $P _ { 1 }$ costs $\$ 10,000$ to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane $P _ { 2 }$ costs $\$ 8500$ to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

Accepted Answer

A)  $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes 
B)  $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes 
C)  $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes 
D)  $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes 
A)  $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes 
B)  $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes 
C)  $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes 
D)  $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes

Question 7

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
\frac { 9 } { x } + \frac { 1 } { y } = \frac { 29 } { 28 } \
\frac { - 6 } { x } - \frac { 1 } { y } = - \frac { 17 } { 28 }
\end{array}$$

Accepted Answer

A)  $\{ ( - 42 , - 4 ) \}$ 
B)  $\{ ( - 4,7 ) \}$ 
C)  $\{ ( 7 , - 4 ) \}$ 
D)  $\{ ( - 7,4 ) \}$ 
A)  $\{ ( - 42 , - 4 ) \}$ 
B)  $\{ ( - 4,7 ) \}$ 
C)  $\{ ( 7 , - 4 ) \}$ 
D)  $\{ ( - 7,4 ) \}$

Question 8

Graph the solution set of the system of inequalities.
-$$\begin{array} { l } 
y - x \leq 5 \
x + y \geq 3 \
y - 3 x \geq - 1
\end{array}$$

Accepted Answer

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

Question 9

Find the partial fraction decomposition for the rational expression.
-$$\frac { 19 x - 16 } { 6 x ^ { 2 } - 19 x + 8 }$$

Accepted Answer

A)  $\frac { 1 } { 2 x + 1 } + \frac { 8 } { 3 x + 8 }$ 
B)  $\frac { 8 } { 3 x - 8 } + \frac { - 1 } { 2 x - 1 }$ 
C)  $\frac { 1 } { 2 x - 1 } + \frac { 8 } { 3 x - 8 }$ 
D)  $\frac { 1 } { 2 x - 1 } + \frac { - 8 } { 3 x - 8 }$ 
A)  $\frac { 1 } { 2 x + 1 } + \frac { 8 } { 3 x + 8 }$ 
B)  $\frac { 8 } { 3 x - 8 } + \frac { - 1 } { 2 x - 1 }$ 
C)  $\frac { 1 } { 2 x - 1 } + \frac { 8 } { 3 x - 8 }$ 
D)  $\frac { 1 } { 2 x - 1 } + \frac { - 8 } { 3 x - 8 }$

Question 10

Solve the problem.
-Carole's car averages $15.5 \mathrm { mi } / \mathrm { gal }$ in city driving and $25.8 \mathrm { mi } / \mathrm { gal }$ in highway driving. If she drove a total of $593.6 \mathrm { mi }$ on $27 \mathrm { gal }$ of gas, how many of the gallons were used for city driving?

Accepted Answer

A)  $10 \mathrm { gal }$ 
B)  $22 \mathrm { gal }$ 
C)  12 gal 
D)  $17 \mathrm { gal }$ 
A)  $10 \mathrm { gal }$ 
B)  $22 \mathrm { gal }$ 
C)  12 gal 
D)  $17 \mathrm { gal }$

Question 11

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 5 x + 3 y = 8 \
3 x - 6 y = - 30
\end{array}$$

Accepted Answer

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

Question 12

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.
-$$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 52 \
2 y + 3 x = 0
\end{array}$$

Accepted Answer

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

Question 13

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-In what years was the population of the City X metropolitan area less than that of the City Y metropolitan area?

Accepted Answer

A)  Years 1-3 
B)  Years 1-5 
C)  Years 5-6 
D)  Years 3-5 
A)  Years 1-3 
B)  Years 1-5 
C)  Years 5-6 
D)  Years 3-5

Question 14

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.
-$$\begin{aligned}
3 x - 7 y - z & = - 28 \
x - 9 y + 9 z & = - 28 \
3 x + y + z & = 34
\end{aligned}$$

Accepted Answer

A)  $\{ ( - 8,7,16 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 8,7,3 ) \}$ 
D)  $\{ ( 8,3,7 ) \}$ 
A)  $\{ ( - 8,7,16 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 8,7,3 ) \}$ 
D)  $\{ ( 8,3,7 ) \}$

Question 15

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$3 x + 5 y \leq 15$$

Accepted Answer

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

Question 16

Find the matrix product when possible.
-$$\left[ \begin{array} { r r } 
- 2 & 3 \
2 & 2
\end{array} \right] \left[ \begin{array} { l l } 
- 2 & 0 \
- 1 & 3
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
1 & 9 \
- 6 & 6
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
9 & 1 \
6 & - 6
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
4 & - 6 \
- 2 & 3
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
4 & 0 \
- 2 & 6
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r } 
1 & 9 \
- 6 & 6
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
9 & 1 \
6 & - 6
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
4 & - 6 \
- 2 & 3
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
4 & 0 \
- 2 & 6
\end{array} \right]$$

Question 17

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

Accepted Answer

A)  13 
B)  32 
C)  50 
D)  35 
A)  13 
B)  32 
C)  50 
D)  35

Question 18

Use the given row transformation to change the matrix as indicated.
-$\left[ \begin{array} { r r } 2 & 5 \ 4 & 11 \end{array} \right] ; - 2$ times row 1 added to row 2

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
2 & 5 \
0 & - 1
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 4 & - 10 \
0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l } 
2 & 5 \
0 & 1
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { l l } 
2 & 5 \
1 & 1
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r } 
2 & 5 \
0 & - 1
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 4 & - 10 \
0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l } 
2 & 5 \
0 & 1
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { l l } 
2 & 5 \
1 & 1
\end{array} \right]$$

Question 19

Evaluate the determinant.
-$$\left| \begin{array} { r r r } 
6 & 0 & 0 \
8 & - 4 & 0 \
- 4 & 6 & 9
\end{array} \right|$$

Accepted Answer

A)  168 
B)  $- 264$ 
C)  216 
D)  $- 216$ 
A)  168 
B)  $- 264$ 
C)  216 
D)  $- 216$

Question 20

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-$$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 7 \
y = \sqrt [ 3 ] { x + 6 }
\end{array}$$

Accepted Answer

A)  $\{ ( - 2.13 , - 1.57 ) , ( 1.76 , - 1.98 ) \}$ 
B)  $\{ ( - 2.13,1.57 ) , ( 1.76,1.98 ) \}$ 
C)  $\{ ( - 2.29 , - 1.32 ) , ( 2.29,1.32 ) \}$ 
D)  $\{ ( - 1.76 , - 1.98 ) , ( 2.13 , - 1.57 ) \}$ 
A)  $\{ ( - 2.13 , - 1.57 ) , ( 1.76 , - 1.98 ) \}$ 
B)  $\{ ( - 2.13,1.57 ) , ( 1.76,1.98 ) \}$ 
C)  $\{ ( - 2.29 , - 1.32 ) , ( 2.29,1.32 ) \}$ 
D)  $\{ ( - 1.76 , - 1.98 ) , ( 2.13 , - 1.57 ) \}$

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - 0.4+=1 +0.8=1

Provide an appropriate response. -You are graphing $2 x ^ { 2 } + 3 y ^ { 2 } > 5$ . You have drawn the graph of $2 x ^ { 2 } + 3 y ^ { 2 } = 5$ . What single point will allow you to determine the correct region?

Write the augmented matrix for the system. Do not - 6x+5z-1=0 4y+8z-16=0 -2x+9y+7z-45=0

Perform the operation or operations when possible. - $\left[ \begin{array} { r r } 6 & - 5 \\5 & 6\end{array} \right] + \left[ \begin{array} { l l } - 5 & - 6 \\- 3 & - 2\end{array} \right] - \left[ \begin{array} { l l } - 5 & 6 \\- 4 & 3\end{array} \right]$

If the system has infinitely many solutions, write the solution set with x arbitrary. - += -=-

Graph the solution set of the system of inequalities. - y-x\leq5 x+y\geq3 y-3x\geq-1 $Graph the solution set of the system of inequalities. - \begin{array} { l } y - x \leq 5 \\ x + y \geq 3 \\ y - 3 x \geq - 1 \end{array}$

Find the partial fraction decomposition for the rational expression. - $\frac { 19 x - 16 } { 6 x ^ { 2 } - 19 x + 8 }$

Solve the problem. -Carole's car averages $15.5 \mathrm { mi } / \mathrm { gal }$ in city driving and $25.8 \mathrm { mi } / \mathrm { gal }$ in highway driving. If she drove a total of $593.6 \mathrm { mi }$ on $27 \mathrm { gal }$ of gas, how many of the gallons were used for city driving?

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

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -In what years was the population of the City X metropolitan area less than that of the City Y metropolitan area?

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. - 3x-7y-z =-28 x-9y+9z =-28 3x+y+z =34

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $3 x + 5 y \leq 15$ $Which method should be used to solve the system? Explain your answer, including a description of the first step. - 3 x + 5 y \leq 15$

Find the matrix product when possible. - $\left[ \begin{array} { r r } - 2 & 3 \\2 & 2\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 3\end{array} \right]$

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r } 2 & 5 \\ 4 & 11 \end{array} \right] ; - 2$ times row 1 added to row 2

Evaluate the determinant. - $\left| \begin{array} { r r r } 6 & 0 & 0 \\8 & - 4 & 0 \\- 4 & 6 & 9\end{array} \right|$

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. - +=7 y=

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