Question 1

Solve the problem.
-Find the equation of the line $y = a x + b$ that passes through the points $( - 3,5 )$ and $( - 4 , - 7 )$.

Accepted Answer

A)  $y = 2 x + - 1$ 
B)  $\mathrm { y } = 12 \mathrm { x } + 41$ 
C)  $\mathrm { y } = 41 \mathrm { x } + 12$ 
D)  $y = 2 x + \frac { 1 } { 9 }$ 
A)  $y = 2 x + - 1$ 
B)  $\mathrm { y } = 12 \mathrm { x } + 41$ 
C)  $\mathrm { y } = 41 \mathrm { x } + 12$ 
D)  $y = 2 x + \frac { 1 } { 9 }$

Question 2

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { r r } 
10 & 1 \
- 1 & 0
\end{array} \right] \text { and } \left[ \begin{array} { r r } 
0 & 1 \
- 1 & 10
\end{array} \right]$$

Accepted Answer

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

Question 3

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

Accepted Answer

A)  $\{ ( - 3,1 ) \}$ 
B)  $\{ ( 3 , - 1 ) \}$ 
C)  $\{ ( - 1,3 ) \}$ 
D)  $\{ ( 1 , - 3 ) \}$ 
A)  $\{ ( - 3,1 ) \}$ 
B)  $\{ ( 3 , - 1 ) \}$ 
C)  $\{ ( - 1,3 ) \}$ 
D)  $\{ ( 1 , - 3 ) \}$

Question 4

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

Accepted Answer

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

Question 5

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the
solution with y arbitrary.
-$\begin{array}{l}
x+y=1 \
x-y=-4
\end{array}$

Accepted Answer

A)  $\left\{ \left( \frac { 5 } { 2 } , - 2 \right) \right\}$ 
B)  $\left\{ \left( - \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right) \right\}$ 
C)  $\left\{ \left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \left( \frac { 5 } { 2 } , - 2 \right) \right\}$ 
B)  $\left\{ \left( - \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right) \right\}$ 
C)  $\left\{ \left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right) \right\}$ 
D)  $\varnothing$

Question 6

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-$$\begin{array} { l } 
y = \log ( x + 7 ) \
y = x ^ { 3 }
\end{array}$$

Accepted Answer

A)  $\{ ( - 1.83,1.64 ) , ( 1.68,2.16 ) \}$ 
B)  $\{ ( - 2.50,1.49 ) \}$ 
C)  $\{ ( 2.07,0.69 ) \}$ 
D)  $\{ ( 0.97,0.90 ) \}$ 
A)  $\{ ( - 1.83,1.64 ) , ( 1.68,2.16 ) \}$ 
B)  $\{ ( - 2.50,1.49 ) \}$ 
C)  $\{ ( 2.07,0.69 ) \}$ 
D)  $\{ ( 0.97,0.90 ) \}$

Question 7

Graph the solution set of the system of inequalities.
-

Accepted Answer

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

Question 8

Solve the problem.
-Find the equation of the line $y = a x + b$ that passes through the points $( - 4,3 )$ and $( 2 , - 8 )$.

Accepted Answer

A)  $y = - \frac { 11 } { 6 } x - \frac { 13 } { 3 }$ 
B)  $y = - \frac { 5 } { 6 } x - \frac { 6 } { 35 }$ 
C)  $y = - \frac { 13 } { 3 } x - \frac { 11 } { 6 }$ 
D)  $y = - \frac { 11 } { 6 } x + \frac { 19 } { 3 }$ 
A)  $y = - \frac { 11 } { 6 } x - \frac { 13 } { 3 }$ 
B)  $y = - \frac { 5 } { 6 } x - \frac { 6 } { 35 }$ 
C)  $y = - \frac { 13 } { 3 } x - \frac { 11 } { 6 }$ 
D)  $y = - \frac { 11 } { 6 } x + \frac { 19 } { 3 }$

Question 9

Write the augmented matrix for the system. Do not solve the system.
-$$\begin{array} { r } 
9 x + 7 y = 93 \
2 y = 6
\end{array}$$

Accepted Answer

A) 
$\left[\begin{array}{rr|r}
93 & 7 & 9 \
6 & 0 & 2
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr|r}
9 & 7 & 93 \
2 & 6 & 0
\end{array}\right]$ 
C) 
$\left[\begin{array}{ll|l}
2 & 0 & 6 \
9 & 7 & 7
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr|r}
9 & 7 & 93 \
0 & 2 & 6
\end{array}\right]$ 
A) 
$\left[\begin{array}{rr|r}
93 & 7 & 9 \
6 & 0 & 2
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr|r}
9 & 7 & 93 \
2 & 6 & 0
\end{array}\right]$ 
C) 
$\left[\begin{array}{ll|l}
2 & 0 & 6 \
9 & 7 & 7
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr|r}
9 & 7 & 93 \
0 & 2 & 6
\end{array}\right]$

Question 10

Solve the system.
-$$\begin{array} { l } 
\frac { 2 } { x } + \frac { 1 } { y } = \frac { 8 } { 9 } \
\frac { - 2 } { x } - \frac { 7 } { y } = - \frac { 2 } { 9 }
\end{array}$$

Accepted Answer

A)  $\{ ( - 9,2 ) \}$ 
B)  $\left\{ \left( \frac { 27 } { 10 } , - 9 \right) \right\}$ 
C)  $\{ ( - 2,9 ) \}$ 
D)  $\{ ( 2 , - 9 ) \}$ 
A)  $\{ ( - 9,2 ) \}$ 
B)  $\left\{ \left( \frac { 27 } { 10 } , - 9 \right) \right\}$ 
C)  $\{ ( - 2,9 ) \}$ 
D)  $\{ ( 2 , - 9 ) \}$

Question 11

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { r r } 
- 5 & - 1 \
6 & 0
\end{array} \right] \text { and } \left[ \begin{array} { r } 
0 \frac { 1 } { 6 } \
- 1 \frac { 5 } { 6 }
\end{array} \right]$$

Accepted Answer

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

Question 12

Solve the system.
-$$\begin{array} { l } 
- 3 x - y + 9 z = 57 \
9 x + 7 y - 7 z = 13 \
- 5 x - 5 y + z = - 37
\end{array}$$

Accepted Answer

A)  $\{ ( - 3,6,6 ) \}$ 
B)  $\{ ( 3,8,6 ) \}$ 
C)  $\{ ( 3,6,8 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( - 3,6,6 ) \}$ 
B)  $\{ ( 3,8,6 ) \}$ 
C)  $\{ ( 3,6,8 ) \}$ 
D)  $\varnothing$

Question 13

Find the partial fraction decomposition for the rational expression.
-$$\frac { 4 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 7 } { 2 x ^ { 2 } - x - 1 }$$

Accepted Answer

A)  $2 x - 5 + \frac { - 5 } { 6 x + 3 } + \frac { 1 } { 3 x - 3 }$ 
B)  $2 x + 5 + \frac { 16 } { 6 x + 3 } + \frac { 10 } { 3 x - 3 }$ 
C)  $2 x - 5 + \frac { 13 } { 2 x + 1 } + \frac { 7 } { x - 1 }$ 
D)  $2 x + 5 + \frac { - 7 } { 6 x + 3 } + \frac { 13 } { 3 x - 3 }$ 
A)  $2 x - 5 + \frac { - 5 } { 6 x + 3 } + \frac { 1 } { 3 x - 3 }$ 
B)  $2 x + 5 + \frac { 16 } { 6 x + 3 } + \frac { 10 } { 3 x - 3 }$ 
C)  $2 x - 5 + \frac { 13 } { 2 x + 1 } + \frac { 7 } { x - 1 }$ 
D)  $2 x + 5 + \frac { - 7 } { 6 x + 3 } + \frac { 13 } { 3 x - 3 }$

Question 14

Find the values of the variables for which the statement is true, if possible.
-$\left[ \begin{array} { l l l } 5 & p + 3 & q - 7 \end{array} \right] = \left[ \begin{array} { l l l } k + 2 & 6 & - 4 \end{array} \right]$

Accepted Answer

A)  $\mathrm { k } = 3 ; \mathrm { p } = 3 ; \mathrm { q } = 3$ 
B)  $\mathrm { k } = - 3 ; \mathrm { p } = - 3 ; \mathrm { q } = - 3$ 
C)  $\mathrm { k } = 6 ; \mathrm { p } = 3 ; \mathrm { q } = 0$ 
D)  $\mathrm { k } = 3 ; \mathrm { p } = 5 ; \mathrm { q } = - 4$ 
A)  $\mathrm { k } = 3 ; \mathrm { p } = 3 ; \mathrm { q } = 3$ 
B)  $\mathrm { k } = - 3 ; \mathrm { p } = - 3 ; \mathrm { q } = - 3$ 
C)  $\mathrm { k } = 6 ; \mathrm { p } = 3 ; \mathrm { q } = 0$ 
D)  $\mathrm { k } = 3 ; \mathrm { p } = 5 ; \mathrm { q } = - 4$

Question 15

Solve the system for x and y using Cramerʹs rule. Assume a and b are nonzero constants.
-$$x + a y = \frac { 1 } { a }$$
$$x + b y = \frac { 1 } { b }$$

Accepted Answer

A)  $\left\{ \left( b + a , - \frac { 1 } { a b } \right) \right\}$ 
B)  $\left\{ \left( \frac { \mathrm { b } - \mathrm { a } } { \mathrm { ab } } , - \frac { 1 } { \mathrm { ab } } \right) \right\}$ 
C)  $\left\{ \left( \frac { a + b } { a b } , - \frac { 1 } { a b } \right) \right\}$ 
D)  $\left\{ \left( \frac { 1 } { a b } , \frac { a - b } { a b } \right) \right\}$ 
A)  $\left\{ \left( b + a , - \frac { 1 } { a b } \right) \right\}$ 
B)  $\left\{ \left( \frac { \mathrm { b } - \mathrm { a } } { \mathrm { ab } } , - \frac { 1 } { \mathrm { ab } } \right) \right\}$ 
C)  $\left\{ \left( \frac { a + b } { a b } , - \frac { 1 } { a b } \right) \right\}$ 
D)  $\left\{ \left( \frac { 1 } { a b } , \frac { a - b } { a b } \right) \right\}$

Question 16

Solve the problem.
-Given $A = \left[ \begin{array} { r r } 3 & - 1 \ - 2 & - 5 \end{array} \right]$, find $A ^ { 3 }$.

Accepted Answer

A) 
$\left[ \begin{array} { r r } 5 & - 47 \ - 94 & - 99 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 27 & - 1 \ - 8 & - 125 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 29 & - 21 \ - 42 & - 139 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 31 & - 15 \ - 38 & 91 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } 5 & - 47 \ - 94 & - 99 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 27 & - 1 \ - 8 & - 125 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 29 & - 21 \ - 42 & - 139 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 31 & - 15 \ - 38 & 91 \end{array} \right]$

Question 17

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

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
\frac { 1 } { 5 } & 0 & \frac { 1 } { 5 } \
- \frac { 1 } { 5 } & 0 & 0 \
- \frac { 1 } { 8 } & \frac { 1 } { 8 } & - \frac { 1 } { 8 }
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r } 
\frac { 1 } { 5 } & - \frac { 1 } { 5 } & - \frac { 1 } { 8 } \
0 & 0 & \frac { 1 } { 8 } \
\frac { 1 } { 5 } & 0 & - \frac { 1 } { 8 }
\end{array} \right]$$ 
C)  The inverse does not exist. 
D) 
$$\left[ \begin{array} { c c c } 
- \frac { 1 } { 5 } & - \frac { 1 } { 5 } & - \frac { 1 } { 8 } \
- \frac { 1 } { 8 } & 0 & \frac { 1 } { 8 } \
\frac { 1 } { 5 } & 0 & 0
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
\frac { 1 } { 5 } & 0 & \frac { 1 } { 5 } \
- \frac { 1 } { 5 } & 0 & 0 \
- \frac { 1 } { 8 } & \frac { 1 } { 8 } & - \frac { 1 } { 8 }
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r } 
\frac { 1 } { 5 } & - \frac { 1 } { 5 } & - \frac { 1 } { 8 } \
0 & 0 & \frac { 1 } { 8 } \
\frac { 1 } { 5 } & 0 & - \frac { 1 } { 8 }
\end{array} \right]$$ 
C)  The inverse does not exist. 
D) 
$$\left[ \begin{array} { c c c } 
- \frac { 1 } { 5 } & - \frac { 1 } { 5 } & - \frac { 1 } { 8 } \
- \frac { 1 } { 8 } & 0 & \frac { 1 } { 8 } \
\frac { 1 } { 5 } & 0 & 0
\end{array} \right]$$

Question 18

Find the indicated matrix.
-Let $B = \left[ \begin{array} { l l l l } - 1 & 6 & 6 & - 3 \end{array} \right]$. Find -4B.

Accepted Answer

A)  $\left[ \begin{array} { l l l l } - 3 & 4 & 4 & - 5 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l l } 4 & 6 & 6 & - 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l l } 4 & - 24 & - 24 & 12 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } - 4 & 24 & 24 & - 12 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l l l } - 3 & 4 & 4 & - 5 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l l } 4 & 6 & 6 & - 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l l } 4 & - 24 & - 24 & 12 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } - 4 & 24 & 24 & - 12 \end{array} \right]$

Question 19

Solve the system to find WS1U1B11S1U1B0 and WS1U1B12S1U1B0.
-Linear systems occur in the design of roof trusses for new homes and buildings. The simplest type of roof truss is a triangle. The truss shown in the figure is used to frame roofs of small buildings. If
A 192-pound force is applied at the peak of the truss, then the forces or weights W1 and W2
Exerted parallel to each rafter of the truss are determined by the following linear system of
Equations.

Accepted Answer

A)  $W _ { 1 } = 192 \mathrm { lb } ; W _ { 2 } = 235.15 \mathrm { lb }$ 
B)  $W _ { 1 } = 172.14 \mathrm { lb } ; W _ { 2 } = 140.55 \mathrm { lb }$ 
C)  $W _ { 1 } = 140.55 \mathrm { lb } ; W _ { 2 } = 172.14 \mathrm { lb }$ 
D)  $\mathrm { W } _ { 1 } = 70.28 \mathrm { lb } ; \mathrm { W } _ { 2 } = 86.07 \mathrm { lb }$ 
A)  $W _ { 1 } = 192 \mathrm { lb } ; W _ { 2 } = 235.15 \mathrm { lb }$ 
B)  $W _ { 1 } = 172.14 \mathrm { lb } ; W _ { 2 } = 140.55 \mathrm { lb }$ 
C)  $W _ { 1 } = 140.55 \mathrm { lb } ; W _ { 2 } = 172.14 \mathrm { lb }$ 
D)  $\mathrm { W } _ { 1 } = 70.28 \mathrm { lb } ; \mathrm { W } _ { 2 } = 86.07 \mathrm { lb }$

Question 20

Find the values of the variables for which the statement is true, if possible.
-$$\left[ \begin{array} { c c } 
x + 3 & y + 4 \
7 & 2
\end{array} \right] = \left[ \begin{array} { c c } 
7 & - 4 \
7 & k
\end{array} \right]$$

Accepted Answer

A)  $x = 7 ; y = - 4 ; k = 2$ 
B)  $x = - 4 ; y = 8 ; k = - 2$ 
C)  $x = 4 ; y = - 8 ; k = 2$ 
D)  $x = 4 ; y = 2 ; k = 7$ 
A)  $x = 7 ; y = - 4 ; k = 2$ 
B)  $x = - 4 ; y = 8 ; k = - 2$ 
C)  $x = 4 ; y = - 8 ; k = 2$ 
D)  $x = 4 ; y = 2 ; k = 7$

Solve the problem. -Find the equation of the line $y = a x + b$ that passes through the points $( - 3,5 )$ and $( - 4 , - 7 )$ .

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. - x+3y=-8 21x+6y=3

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

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - x+y=1 x-y=-4

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

Graph the solution set of the system of inequalities. -

Solve the problem. -Find the equation of the line $y = a x + b$ that passes through the points $( - 4,3 )$ and $( 2 , - 8 )$ .

Write the augmented matrix for the system. Do not solve the system. - 9x+7y=93 2y=6

Solve the system. - += -=-

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } - 5 & - 1 \\6 & 0\end{array} \right] \text { and } \left[ \begin{array} { r } 0 \frac { 1 } { 6 } \\- 1 \frac { 5 } { 6 }\end{array} \right]$

Solve the system. - -3x-y+9z=57 9x+7y-7z=13 -5x-5y+z=-37

Find the partial fraction decomposition for the rational expression. - $\frac { 4 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 7 } { 2 x ^ { 2 } - x - 1 }$

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { l l l } 5 & p + 3 & q - 7 \end{array} \right] = \left[ \begin{array} { l l l } k + 2 & 6 & - 4 \end{array} \right]$

Solve the system for x and y using Cramerʹs rule. Assume a and b are nonzero constants. - $x + a y = \frac { 1 } { a }$ $x + b y = \frac { 1 } { b }$

Solve the problem. -Given $A = \left[ \begin{array} { r r } 3 & - 1 \\ - 2 & - 5 \end{array} \right]$ , find $A ^ { 3 }$ .

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

Find the indicated matrix. -Let $B = \left[ \begin{array} { l l l l } - 1 & 6 & 6 & - 3 \end{array} \right]$ . Find -4B.

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { c c } x + 3 & y + 4 \\7 & 2\end{array} \right] = \left[ \begin{array} { c c } 7 & - 4 \\7 & k\end{array} \right]$

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 problem. -Find the equation of the line y=ax+by = a x + by=ax+b that passes through the points (−3,5)( - 3,5 )(−3,5) and (−4,−7)( - 4 , - 7 )(−4,−7) .

Decide whether or not the matrices are inverses of each other. - [101−10] and [01−110]\left[ \begin{array} { r r } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right][10−1​10​] and [0−1​110​]