Question 1

Solve the system.
-$$\begin{array} { l } 
\frac { 3 } { x } + \frac { 6 } { y } = - 1 \
\frac { 6 } { x } - \frac { 5 } { y } = - \frac { 23 } { 3 }
\end{array}$$

Accepted Answer

A)  $\{ ( - 1,3 ) \}$ 
B)  $\{ ( 1 , - 3 ) \}$ 
C)  $\left\{ \left( - \frac { 33 } { 31 } , 3 \right) \right\}$ 
D)  $\{ ( 3 , - 1 ) \}$ 
A)  $\{ ( - 1,3 ) \}$ 
B)  $\{ ( 1 , - 3 ) \}$ 
C)  $\left\{ \left( - \frac { 33 } { 31 } , 3 \right) \right\}$ 
D)  $\{ ( 3 , - 1 ) \}$

Question 2

Use a graphing calculator and the method of matrix inverses to solve the system. Give five decimal places, if necessary.
-$$\begin{array} { l } 
( \log 2 ) x + ( \ln 3 ) y + ( \ln 5 ) z = 4 \
( \ln 3 ) x + ( \log 8 ) y + ( \ln 10 ) z = 11 \
( \log 23 ) x + ( \ln 7 ) y + ( \ln 4 ) z = 15
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 4.12321,3.91519,1.27440 ) \}$ 
C)  $\{ ( 0.30103,9.0309,1.38629 ) \}$ 
D)  $\{ ( 9.25474,1.35236 , - 0.16880 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 4.12321,3.91519,1.27440 ) \}$ 
C)  $\{ ( 0.30103,9.0309,1.38629 ) \}$ 
D)  $\{ ( 9.25474,1.35236 , - 0.16880 ) \}$

Question 3

Find the partial fraction decomposition for the rational expression.
-$$\frac { - 2 x ^ { 2 } + 7 x + 24 } { ( x + 3 ) ^ { 2 } ( 3 x + 4 ) }$$

Accepted Answer

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

Question 4

Decide whether or not the matrices are inverses of each other.
-$\left[ \begin{array} { l l } 5 & 3 \ 3 & 2 \end{array} \right]$ and $\left[ \begin{array} { r r } 2 & - 3 \ - 3 & 5 \end{array} \right]$

Accepted Answer

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

Question 5

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

Accepted Answer

A)  about 0.95 million 
B)  about 0.87 million 
C)  about 0.92 million 
D)  about 1.03 million 
A)  about 0.95 million 
B)  about 0.87 million 
C)  about 0.92 million 
D)  about 1.03 million

Question 6

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { r r r } 
1 & - 1 & 2 \
- 3 & - 2 & 4 \
- 1 & 1 & 1
\end{array} \right] \text { and } \left[ \begin{array} { r r r } 
2 & - 1 & 0 \
- 1 & 1 & - 2 \
1 & 0 & - 1
\end{array} \right]$$

Accepted Answer

A)  $Y e s$ 
B)  No 
A)  $Y e s$ 
B)  No

Question 7

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary.
-$$A = \left[ \begin{array} { c c } 
\frac { 4 } { 3 } & 52 \
\sqrt { 21 } & 8.754
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c } 
0.75 & 0.01923 \
0.21822 & 0.11423
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 0.03863 & 0.22946 \
0.02022 & - 0.00588
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
- 0.70513 & 0.41667 \
0.36859 & - 0.10417
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c } 
0.75 & 0.01923 \
0.21822 & 0.11423
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 0.03863 & 0.22946 \
0.02022 & - 0.00588
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
- 0.70513 & 0.41667 \
0.36859 & - 0.10417
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { l l } 
1 & 0 \
0 & 1
\end{array} \right]$$

Question 8

Graph the solution set of the system of inequalities.
-

Accepted Answer

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

Question 9

Find the value of the determinant.
-$$\left| \begin{array} { l l l } 
1 & 7 & 2 \
2 & 6 & 4 \
6 & 3 & 1
\end{array} \right|$$

Accepted Answer

A)  284 
B)  88 
C)  $- 248$ 
D)  $- 88$ 
A)  284 
B)  88 
C)  $- 248$ 
D)  $- 88$

Question 10

Find the partial fraction decomposition for the rational expression.
-$$\frac { 10 x ^ { 2 } - 124 x + 12 } { \left( x ^ { 2 } + 2 x - 6 \right) ( x - 10 ) }$$

Accepted Answer

A)  $\frac { 12 x } { x ^ { 2 } + 2 x - 6 } + \frac { 2 } { x - 10 }$ 
B)  $\frac { 12 x } { x ^ { 2 } + 2 x - 6 } + \frac { - 2 } { x - 10 }$ 
C)  $\frac { 2 x } { x ^ { 2 } + 2 x - 6 } + \frac { - 12 } { x - 10 }$ 
D)  $\frac { 12 x } { x - 10 } + \frac { - 2 } { x ^ { 2 } + 2 x - 6 }$ 
A)  $\frac { 12 x } { x ^ { 2 } + 2 x - 6 } + \frac { 2 } { x - 10 }$ 
B)  $\frac { 12 x } { x ^ { 2 } + 2 x - 6 } + \frac { - 2 } { x - 10 }$ 
C)  $\frac { 2 x } { x ^ { 2 } + 2 x - 6 } + \frac { - 12 } { x - 10 }$ 
D)  $\frac { 12 x } { x - 10 } + \frac { - 2 } { x ^ { 2 } + 2 x - 6 }$

Question 11

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Consider the matrix
$$A = \left[ \begin{array} { r r r } 
10 & - 3 & 2 \
8 & - 4 & 0 \
- 8 & - 10 & 0
\end{array} \right]$$
Without using a calculator, determine whether $A ^ { - 1 }$ exists. Explain your reasoning.

Accepted Answer

Expanding

Question 12

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 13

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
x - y + z = 5 \
x + y + z = 7 \
x + y - z = 3
\end{array}$$

Accepted Answer

A)  $\{ ( 4,2,1 ) \}$ 
B)  $\{ ( 2,4,1 ) \}$ 
C)  Cramer's rule does not apply since $\mathrm { D } = 0$; $\varnothing$ 
D)  $\{ ( 4,1,2 ) \}$ 
A)  $\{ ( 4,2,1 ) \}$ 
B)  $\{ ( 2,4,1 ) \}$ 
C)  Cramer's rule does not apply since $\mathrm { D } = 0$; $\varnothing$ 
D)  $\{ ( 4,1,2 ) \}$

Question 14

Evaluate the determinant.
-$\left| \begin{array} { r r r } 1 & 0 & 0 \ 8 & - 1 & 0 \ - 1 & 4 & 8 \end{array} \right|$

Accepted Answer

A)  $- 40$ 
B)  $- 24$ 
C)  $- 8$ 
D)  8 
A)  $- 40$ 
B)  $- 24$ 
C)  $- 8$ 
D)  8

Question 15

Graph the inequality.
-$x^{2}+(y+3)^{2} \leq 16$

Accepted Answer

A) 
  
B)

C) 
  
D) 
  
A) 
  
B)

C) 
  
D)

Question 16

Solve the problem.
-The sum of the squares of the digits of a positive two-digit number is 20 , and the tens digit is 2 more than the units digit. Find the number.

Accepted Answer

A)  42 
B)  35 
C)  53 
D)  24 
A)  42 
B)  35 
C)  53 
D)  24

Question 17

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$x ^ { 2 } + y ^ { 2 } = 25$ $x - y = 1$

Accepted Answer

A)  $\{ ( - 4 , - 3 ) , ( - 3 , - 4 ) \}$ 
B)  $\{ ( 4,3 ) , ( - 3 , - 4 ) \}$ 
C)  $\{ ( 4 , - 3 ) , ( 3 , - 4 ) \}$ 
D)  $\{ ( - 4,3 ) , ( - 3,4 ) \}$ 
A)  $\{ ( - 4 , - 3 ) , ( - 3 , - 4 ) \}$ 
B)  $\{ ( 4,3 ) , ( - 3 , - 4 ) \}$ 
C)  $\{ ( 4 , - 3 ) , ( 3 , - 4 ) \}$ 
D)  $\{ ( - 4,3 ) , ( - 3,4 ) \}$

Question 18

Graph the solution set of the system of inequalities.
-$$\begin{array} { l } 
\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } \leq 1 \
x ^ { 2 } + y ^ { 2 } \leq 36
\end{array}$$

Accepted Answer

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

Question 19

Graph the solution set of the system of inequalities.
-

Accepted Answer

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

Question 20

Solve each problem.
-Mikeʹs Bait Shop sells three types of lures: discount, normal, and professional. Location I sells 31 discount lures, 100 regular lures, and 30 professional lures each day. Location II sells 18 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, 18 expert lures are sold each day, and 40 expert
Lures are sold each day at Location III. Write a 3 × 3 matrix that shows the sales figures for the three
Locations, with the rows representing the three locations.

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
31 & 100 & 30 \
18 & 18 & 90 \
60 & 120 & 40
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r } 
100 & 31 & 30 \
18 & 90 & 18 \
60 & 120 & 40
\end{array} \right]$$ 
C) 
$\left[\begin{array}{rrr}
31 & 100 & 18 \
18 & 90 & 30 \
60 & 120 & 40
\end{array}\right]$ 
D) 
$$\left[ \begin{array} { r r r } 
31 & 100 & 30 \
18 & 90 & 18 \
60 & 120 & 40
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
31 & 100 & 30 \
18 & 18 & 90 \
60 & 120 & 40
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r } 
100 & 31 & 30 \
18 & 90 & 18 \
60 & 120 & 40
\end{array} \right]$$ 
C) 
$\left[\begin{array}{rrr}
31 & 100 & 18 \
18 & 90 & 30 \
60 & 120 & 40
\end{array}\right]$ 
D) 
$$\left[ \begin{array} { r r r } 
31 & 100 & 30 \
18 & 90 & 18 \
60 & 120 & 40
\end{array} \right]$$

Solve the system. - +=-1 -=-

Use a graphing calculator and the method of matrix inverses to solve the system. Give five decimal places, if necessary. - (2)x+(3)y+(5)z=4 (3)x+(8)y+(10)z=11 (23)x+(7)y+(4)z=15

Find the partial fraction decomposition for the rational expression. - $\frac { - 2 x ^ { 2 } + 7 x + 24 } { ( x + 3 ) ^ { 2 } ( 3 x + 4 ) }$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 5 & 3 \\ 3 & 2 \end{array} \right]$ and $\left[ \begin{array} { r r } 2 & - 3 \\ - 3 & 5 \end{array} \right]$

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r r } 1 & - 1 & 2 \\- 3 & - 2 & 4 \\- 1 & 1 & 1\end{array} \right] \text { and } \left[ \begin{array} { r r r } 2 & - 1 & 0 \\- 1 & 1 & - 2 \\1 & 0 & - 1\end{array} \right]$

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary. - $A = \left[ \begin{array} { c c } \frac { 4 } { 3 } & 52 \\\sqrt { 21 } & 8.754\end{array} \right]$

Graph the solution set of the system of inequalities. -

Find the value of the determinant. - $\left| \begin{array} { l l l } 1 & 7 & 2 \\2 & 6 & 4 \\6 & 3 & 1\end{array} \right|$

Find the partial fraction decomposition for the rational expression. - $\frac { 10 x ^ { 2 } - 124 x + 12 } { \left( x ^ { 2 } + 2 x - 6 \right) ( x - 10 ) }$

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 Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-y+z=5 x+y+z=7 x+y-z=3

Evaluate the determinant. - $\left| \begin{array} { r r r } 1 & 0 & 0 \\ 8 & - 1 & 0 \\ - 1 & 4 & 8 \end{array} \right|$

Graph the inequality. - $x^{2}+(y+3)^{2} \leq 16$ $Graph the inequality. - x^{2}+(y+3)^{2} \leq 16$

Solve the problem. -The sum of the squares of the digits of a positive two-digit number is 20 , and the tens digit is 2 more than the units digit. Find the number.

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - $x ^ { 2 } + y ^ { 2 } = 25$ $x - y = 1$

Graph the solution set of the system of inequalities. - -\leq1 +\leq36 $Graph the solution set of the system of inequalities. - \begin{array} { l } \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } \leq 1 \\ x ^ { 2 } + y ^ { 2 } \leq 36 \end{array}$

Graph the solution set of the system of inequalities. -

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

Review of Basic Concepts

Filters

Exam 9: Systems and Matrices

Solve the system. - +=-1 -=-

Use a graphing calculator and the method of matrix inverses to solve the system. Give five decimal places, if necessary. - (2)x+(3)y+(5)z=4 (3)x+(8)y+(10)z=11 (23)x+(7)y+(4)z=15

Find the partial fraction decomposition for the rational expression. - −2x2+7x+24(x+3)2(3x+4)\frac { - 2 x ^ { 2 } + 7 x + 24 } { ( x + 3 ) ^ { 2 } ( 3 x + 4 ) }(x+3)2(3x+4)−2x2+7x+24​

Decide whether or not the matrices are inverses of each other. - [5332]\left[ \begin{array} { l l } 5 & 3 \\ 3 & 2 \end{array} \right][53​32​] and [2−3−35]\left[ \begin{array} { r r } 2 & - 3 \\ - 3 & 5 \end{array} \right][2−3​−35​]

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary. - A=[4352218.754]A = \left[ \begin{array} { c c } \frac { 4 } { 3 } & 52 \\\sqrt { 21 } & 8.754\end{array} \right]A=[34​21​​528.754​]

Graph the solution set of the system of inequalities. -

Find the value of the determinant. - ∣172264631∣\left| \begin{array} { l l l } 1 & 7 & 2 \\2 & 6 & 4 \\6 & 3 & 1\end{array} \right|​126​763​241​​

Find the partial fraction decomposition for the rational expression. - 10x2−124x+12(x2+2x−6)(x−10)\frac { 10 x ^ { 2 } - 124 x + 12 } { \left( x ^ { 2 } + 2 x - 6 \right) ( x - 10 ) }(x2+2x−6)(x−10)10x2−124x+12​

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 Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-y+z=5 x+y+z=7 x+y-z=3

Evaluate the determinant. - ∣1008−10−148∣\left| \begin{array} { r r r } 1 & 0 & 0 \\ 8 & - 1 & 0 \\ - 1 & 4 & 8 \end{array} \right|​18−1​0−14​008​​

Graph the inequality. - x2+(y+3)2≤16x^{2}+(y+3)^{2} \leq 16x2+(y+3)2≤16

Solve the problem. -The sum of the squares of the digits of a positive two-digit number is 20 , and the tens digit is 2 more than the units digit. Find the number.

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25 x−y=1x - y = 1x−y=1

Graph the solution set of the system of inequalities. - -\leq1 +\leq36

Graph the solution set of the system of inequalities. -

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

Review of Basic Concepts

Filters

Find the partial fraction decomposition for the rational expression. - $\frac { - 2 x ^ { 2 } + 7 x + 24 } { ( x + 3 ) ^ { 2 } ( 3 x + 4 ) }$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 5 & 3 \\ 3 & 2 \end{array} \right]$ and $\left[ \begin{array} { r r } 2 & - 3 \\ - 3 & 5 \end{array} \right]$

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary. - $A = \left[ \begin{array} { c c } \frac { 4 } { 3 } & 52 \\\sqrt { 21 } & 8.754\end{array} \right]$

Find the value of the determinant. - $\left| \begin{array} { l l l } 1 & 7 & 2 \\2 & 6 & 4 \\6 & 3 & 1\end{array} \right|$

Find the partial fraction decomposition for the rational expression. - $\frac { 10 x ^ { 2 } - 124 x + 12 } { \left( x ^ { 2 } + 2 x - 6 \right) ( x - 10 ) }$

Evaluate the determinant. - $\left| \begin{array} { r r r } 1 & 0 & 0 \\ 8 & - 1 & 0 \\ - 1 & 4 & 8 \end{array} \right|$

Graph the inequality. - $x^{2}+(y+3)^{2} \leq 16$ $Graph the inequality. - x^{2}+(y+3)^{2} \leq 16$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - $x ^ { 2 } + y ^ { 2 } = 25$ $x - y = 1$

Graph the solution set of the system of inequalities. - -\leq1 +\leq36 $Graph the solution set of the system of inequalities. - \begin{array} { l } \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } \leq 1 \\ x ^ { 2 } + y ^ { 2 } \leq 36 \end{array}$