Question 1

Write the system of equations associated with the augmented matrix. Do not solve.
-$\left[ \begin{array} { r r | r } 14 & 17 & 13 \ 15 & 6 & 12 \end{array} \right]$

Accepted Answer

A)  $17 x + 14 y = 13$
$15 x + 6 y = 12$ 
B)  $14 x + 17 y = 13$
$6 x + 15 y = 12$ 
C)  $14 x + 17 y = 13$
$15 x + 6 y = 12$ 
D)  $14 x + 17 y = - 13$
$15 x + 6 y = - 12$ 
A)  $17 x + 14 y = 13$
$15 x + 6 y = 12$ 
B)  $14 x + 17 y = 13$
$6 x + 15 y = 12$ 
C)  $14 x + 17 y = 13$
$15 x + 6 y = 12$ 
D)  $14 x + 17 y = - 13$
$15 x + 6 y = - 12$

Question 2

Evaluate the determinant.
$\left| \begin{array} { r r r } 
- 2 & - 6 & 0 \
3 & - 6 & 0 \
1 & - 5 & 0
\end{array} \right|$

Accepted Answer

A)  18 
B)  0 
C)   -28 
D)  8 
A)  18 
B)  0 
C)   -28 
D)  8

Question 3

For the equation, determine the constants A and B that make the equation an identity.
-$$\frac { x } { ( x + 2 ) ( x - 2 ) } = \frac { A } { ( x + 2 ) } + \frac { B } { ( x - 2 ) }$$

Accepted Answer

A)  $\mathrm { A } = \frac { 1 } { 2 } , \mathrm {~B} = 1$ 
B)  $A = 1 , B = \frac { 1 } { 2 }$ 
C)  $\mathrm { A } = - \frac { 1 } { 2 } , \mathrm {~B} = \frac { 1 } { 2 }$ 
D)  $\mathrm { A } = \frac { 1 } { 2 } , \mathrm {~B} = \frac { 1 } { 2 }$ 
A)  $\mathrm { A } = \frac { 1 } { 2 } , \mathrm {~B} = 1$ 
B)  $A = 1 , B = \frac { 1 } { 2 }$ 
C)  $\mathrm { A } = - \frac { 1 } { 2 } , \mathrm {~B} = \frac { 1 } { 2 }$ 
D)  $\mathrm { A } = \frac { 1 } { 2 } , \mathrm {~B} = \frac { 1 } { 2 }$

Question 4

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary.
-$$\begin{array} { l } 
6 x + 4 y = - 4 \
5 x + 7 y = 4
\end{array}$$

Accepted Answer

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

Question 5

Find the partial fraction decomposition for the rational expression.
$\frac { 31 x - 80 } { 6 x ^ { 2 } - 31 x + 40 }$

Accepted Answer

A)  $\frac { 5 } { 2 x + 5 } + \frac { 8 } { 3 x + 8 }$ 
B)  $\frac { 5 } { 2 x - 5 } + \frac { 8 } { 3 x - 8 }$ 
C)  $\frac { 8 } { 3 x - 8 } + \frac { - 5 } { 2 x - 5 }$ 
D)  $\frac { 5 } { 2 x - 5 } + \frac { - 8 } { 3 x - 8 }$ 
A)  $\frac { 5 } { 2 x + 5 } + \frac { 8 } { 3 x + 8 }$ 
B)  $\frac { 5 } { 2 x - 5 } + \frac { 8 } { 3 x - 8 }$ 
C)  $\frac { 8 } { 3 x - 8 } + \frac { - 5 } { 2 x - 5 }$ 
D)  $\frac { 5 } { 2 x - 5 } + \frac { - 8 } { 3 x - 8 }$

Question 6

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form.
-

Accepted Answer

A) 
$\begin{aligned}
y & \leq 0 \
x^{2}+y^{2} & \geq 16
\end{aligned}$ 
B) 
$\begin{aligned}
y & \leq 0 \
x^{2}+y^{2} & \leq 16
\end{aligned}$ 
C) 
$\begin{array}{c}
y \geq 0 \
x^{2}+y^{2} \leq 16
\end{array}$ 
D)  $y \geq 0$ 
$x ^ { 2 } + y ^ { 2 } \geq 16$ 
A) 
$\begin{aligned}
y & \leq 0 \
x^{2}+y^{2} & \geq 16
\end{aligned}$ 
B) 
$\begin{aligned}
y & \leq 0 \
x^{2}+y^{2} & \leq 16
\end{aligned}$ 
C) 
$\begin{array}{c}
y \geq 0 \
x^{2}+y^{2} \leq 16
\end{array}$ 
D)  $y \geq 0$ 
$x ^ { 2 } + y ^ { 2 } \geq 16$

Question 7

If the graphs of a system of two equations are both circles, what are the possible numbers of solutions (with real
coordinates) of this system?

Accepted Answer

0, 1, 2, o

Question 8

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

Accepted Answer

A)  $- 49$ 
B)  $- 37$ 
C)  $- 13$ 
D)  $- 61$ 
A)  $- 49$ 
B)  $- 37$ 
C)  $- 13$ 
D)  $- 61$

Question 9

Solve the system.
$\begin{array} { l } 
\frac { 9 } { x } + \frac { 1 } { y } = - \frac { 8 } { 7 } \
\frac { 4 } { x } - \frac { 1 } { y } = - \frac { 19 } { 63 }
\end{array}$

Accepted Answer

A)  $\{ ( - 7 , - 9 ) \}$ 
B)  $\{ ( 4,9 ) \}$ 
C)  $\{ ( 9,7 ) \}$ 
D)  $\{ ( - 9 , - 7 ) \}$ 
A)  $\{ ( - 7 , - 9 ) \}$ 
B)  $\{ ( 4,9 ) \}$ 
C)  $\{ ( 9,7 ) \}$ 
D)  $\{ ( - 9 , - 7 ) \}$

Question 10

Find the partial fraction decomposition for the rational expression.
$\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }$

Accepted Answer

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

Question 11

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.
-$\begin{array}{l}
y=\ln x \
x^{2}-y^{2}=7
\end{array}$

Accepted Answer

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

Question 12

Graph the solution set of the system of inequalities.
-$\begin{array}{r}
2 x+y \leq 4 \
x-1 \geq 0
\end{array}$

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Provide an appropriate response.
-Fill in the blank to complete the statement. In order for a matrix to have a determinant, it must be _______ matrix.

Accepted Answer

A)  a minor 
B)  an augmented 
C)  a rectangular 
D)  a square 
A)  a minor 
B)  an augmented 
C)  a rectangular 
D)  a square

Question 15

Graph the solution set of the system of inequalities.
-$\begin{array}{l}
y \geq 3 x \
y \leq 8
\end{array}$

Accepted Answer

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

Question 16

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
-$$\begin{array} { l } 
x + b y = a ^ { 2 } \
a x + y = b ^ { 2 }
\end{array}$$

Accepted Answer

A)  $\left\{ \left( \frac { a ^ { 2 } - b ^ { 3 } } { a + b } , \frac { a ^ { 2 } - b ^ { 3 } } { a + b } \right) \right\}$ 
B)  $\left\{ \left( - \frac { a ^ { 2 } - b ^ { 3 } } { a b } , - \frac { b ^ { 2 } - a ^ { 3 } } { a b } \right) \right\}$ 
C)  $\left\{ \left( \frac { b ^ { 3 } } { 1 - a b } , \frac { a ^ { 3 } } { 1 - a b } \right) \right\}$ 
D)  $\left\{ \left( \frac { a ^ { 2 } - b ^ { 3 } } { 1 - a b } , \frac { b ^ { 2 } - a ^ { 3 } } { 1 - a b } \right) \right\}$ 
A)  $\left\{ \left( \frac { a ^ { 2 } - b ^ { 3 } } { a + b } , \frac { a ^ { 2 } - b ^ { 3 } } { a + b } \right) \right\}$ 
B)  $\left\{ \left( - \frac { a ^ { 2 } - b ^ { 3 } } { a b } , - \frac { b ^ { 2 } - a ^ { 3 } } { a b } \right) \right\}$ 
C)  $\left\{ \left( \frac { b ^ { 3 } } { 1 - a b } , \frac { a ^ { 3 } } { 1 - a b } \right) \right\}$ 
D)  $\left\{ \left( \frac { a ^ { 2 } - b ^ { 3 } } { 1 - a b } , \frac { b ^ { 2 } - a ^ { 3 } } { 1 - a b } \right) \right\}$

Question 17

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
5 x - y + z = 8 \
7 x + y + z = 6
\end{array}$$
$$12 x + 2 z = 14$$

Accepted Answer

A)  $\{ ( x , - x - 1 , - 6 x + 7 ) \}$ 
B)  $\{ ( x , - x + 1,6 x - 7 ) \}$ 
C)  $\{ ( x , - x + 1 , - 6 x + 7 ) \}$ 
D)  $\{ ( x , - x + 1,6 x + 7 ) \}$ 
A)  $\{ ( x , - x - 1 , - 6 x + 7 ) \}$ 
B)  $\{ ( x , - x + 1,6 x - 7 ) \}$ 
C)  $\{ ( x , - x + 1 , - 6 x + 7 ) \}$ 
D)  $\{ ( x , - x + 1,6 x + 7 ) \}$

Question 18

A grain dealer sold to one customer 2 bushels of wheat, 9 of corn, and 6 of rye for $45.20; to another, 9 of wheat, 6 of corn, and 2 of rye for $35.60; and to a third, 6 of wheat, 2 of corn, and 9 of rye for $45.00. What was the price
Per bushel for corn?

Accepted Answer

A)  $2.10 
B)  $3.40 
C)  $2.40 
D)  $1.60 
A)  $2.10 
B)  $3.40 
C)  $2.40 
D)  $1.60

Question 19

Solve the system.
$\begin{array} { l } 
4 x - 7 y - z = - 17 \
x + 3 y - 5 z = - 9 \
3 x + y + z = 21
\end{array}$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 4,5,4 ) \}$ 
C)  $\{ ( - 4,4,8 ) \}$ 
D)  $\{ ( 4,4,5 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 4,5,4 ) \}$ 
C)  $\{ ( - 4,4,8 ) \}$ 
D)  $\{ ( 4,4,5 ) \}$

Question 20

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

Accepted Answer

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

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[ \begin{array} { r r | r } 14 & 17 & 13 \\ 15 & 6 & 12 \end{array} \right]$

Evaluate the determinant. $\left| \begin{array} { r r r } - 2 & - 6 & 0 \\3 & - 6 & 0 \\1 & - 5 & 0\end{array} \right|$

For the equation, determine the constants A and B that make the equation an identity. - $\frac { x } { ( x + 2 ) ( x - 2 ) } = \frac { A } { ( x + 2 ) } + \frac { B } { ( x - 2 ) }$

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - 6x+4y=-4 5x+7y=4

Find the partial fraction decomposition for the rational expression. $\frac { 31 x - 80 } { 6 x ^ { 2 } - 31 x + 40 }$

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form. -

If the graphs of a system of two equations are both circles, what are the possible numbers of solutions (with real coordinates) of this system?

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

Solve the system. +=- -=-

Find the partial fraction decomposition for the rational expression. $\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }$

Graph the solution set of the system of inequalities. - 2x+y\leq4 x-1\geq0 $Graph the solution set of the system of inequalities. - \begin{array}{r} 2 x+y \leq 4 \\ x-1 \geq 0 \end{array}$

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

Provide an appropriate response. -Fill in the blank to complete the statement. In order for a matrix to have a determinant, it must be _______ matrix.

Graph the solution set of the system of inequalities. - y\geq3x y\leq8 $Graph the solution set of the system of inequalities. - \begin{array}{l} y \geq 3 x \\ y \leq 8 \end{array}$

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - x+by= ax+y=

If the system has infinitely many solutions, write the solution set with x arbitrary. - 5x-y+z=8 7x+y+z=6 $12 x + 2 z = 14$

A grain dealer sold to one customer 2 bushels of wheat, 9 of corn, and 6 of rye for $45.20; to another, 9 of wheat, 6 of corn, and 2 of rye for $35.60; and to a third, 6 of wheat, 2 of corn, and 9 of rye for $45.00. What was the price Per bushel for corn?

Solve the system. 4x-7y-z=-17 x+3y-5z=-9 3x+y+z=21

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

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

Write the system of equations associated with the augmented matrix. Do not solve. - [14171315612]\left[ \begin{array} { r r | r } 14 & 17 & 13 \\ 15 & 6 & 12 \end{array} \right][1415​176​1312​]

Evaluate the determinant. ∣−2−603−601−50∣\left| \begin{array} { r r r } - 2 & - 6 & 0 \\3 & - 6 & 0 \\1 & - 5 & 0\end{array} \right|​−231​−6−6−5​000​​

For the equation, determine the constants A and B that make the equation an identity. - x(x+2)(x−2)=A(x+2)+B(x−2)\frac { x } { ( x + 2 ) ( x - 2 ) } = \frac { A } { ( x + 2 ) } + \frac { B } { ( x - 2 ) }(x+2)(x−2)x​=(x+2)A​+(x−2)B​

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - 6x+4y=-4 5x+7y=4

Find the partial fraction decomposition for the rational expression. 31x−806x2−31x+40\frac { 31 x - 80 } { 6 x ^ { 2 } - 31 x + 40 }6x2−31x+4031x−80​

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form. -

If the graphs of a system of two equations are both circles, what are the possible numbers of solutions (with real coordinates) of this system?

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. -objective function =x−5y= x - 5 y=x−5y ; minimum

Solve the system. +=- -=-

Find the partial fraction decomposition for the rational expression. −31x−1132x2+20x+42\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }2x2+20x+42−31x−113​

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. - y=x -=7

Graph the solution set of the system of inequalities. - 2x+y\leq4 x-1\geq0

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

Provide an appropriate response. -Fill in the blank to complete the statement. In order for a matrix to have a determinant, it must be _______ matrix.

Graph the solution set of the system of inequalities. - y\geq3x y\leq8

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - x+by= ax+y=

If the system has infinitely many solutions, write the solution set with x arbitrary. - 5x-y+z=8 7x+y+z=6 12x+2z=1412 x + 2 z = 1412x+2z=14

A grain dealer sold to one customer 2 bushels of wheat, 9 of corn, and 6 of rye for $45.20; to another, 9 of wheat, 6 of corn, and 2 of rye for $35.60; and to a third, 6 of wheat, 2 of corn, and 9 of rye for $45.00. What was the price Per bushel for corn?

Solve the system. 4x-7y-z=-17 x+3y-5z=-9 3x+y+z=21

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

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[ \begin{array} { r r | r } 14 & 17 & 13 \\ 15 & 6 & 12 \end{array} \right]$

Evaluate the determinant. $\left| \begin{array} { r r r } - 2 & - 6 & 0 \\3 & - 6 & 0 \\1 & - 5 & 0\end{array} \right|$

For the equation, determine the constants A and B that make the equation an identity. - $\frac { x } { ( x + 2 ) ( x - 2 ) } = \frac { A } { ( x + 2 ) } + \frac { B } { ( x - 2 ) }$

Find the partial fraction decomposition for the rational expression. $\frac { 31 x - 80 } { 6 x ^ { 2 } - 31 x + 40 }$

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

Find the partial fraction decomposition for the rational expression. $\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }$

Graph the solution set of the system of inequalities. - 2x+y\leq4 x-1\geq0 $Graph the solution set of the system of inequalities. - \begin{array}{r} 2 x+y \leq 4 \\ x-1 \geq 0 \end{array}$

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

Graph the solution set of the system of inequalities. - y\geq3x y\leq8 $Graph the solution set of the system of inequalities. - \begin{array}{l} y \geq 3 x \\ y \leq 8 \end{array}$

If the system has infinitely many solutions, write the solution set with x arbitrary. - 5x-y+z=8 7x+y+z=6 $12 x + 2 z = 14$