Question 1

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

Accepted Answer

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

Question 2

Solve the problem using matrices.
-John has a jarful of quarters and nickels. There are 109 coins in the jar. The value of the coins is $15.25. How many of each type of coin are there?

Accepted Answer

A)  49 quarters; 60 nickels 
B)  104 quarters; 5 nickels 
C)  54 quarters ; 55 nickels 
D)  60 quarters; 49 nickels 
A)  49 quarters; 60 nickels 
B)  104 quarters; 5 nickels 
C)  54 quarters ; 55 nickels 
D)  60 quarters; 49 nickels

Question 3

Graph the inequality.
-$-3 x-5 y \leq 15$

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Find the values of the variables for which the statement is true, if possible.
-$$\left[ \begin{array} { r r } 
6 & - 2 \
x & 0
\end{array} \right] = \left[ \begin{array} { c c } 
\mathrm { m } - 4 & \mathrm { n } + 4 \
8 & 0
\end{array} \right]$$

Accepted Answer

A)  $\mathrm { m } = - 6 ; \mathrm { n } = 10 ; \mathrm { x } = 0$ 
B)  $\mathrm { m } = 10 ; \mathrm { n } = - 6 ; \mathrm { x } = 8$ 
C)  $\mathrm { m } = 6 ; \mathrm { n } = - 2 ; \mathrm { x } = 8$ 
D)  $\mathrm { m } = - 10 ; \mathrm { n } = + 6 ; \mathrm { x } = - 8$ 
A)  $\mathrm { m } = - 6 ; \mathrm { n } = 10 ; \mathrm { x } = 0$ 
B)  $\mathrm { m } = 10 ; \mathrm { n } = - 6 ; \mathrm { x } = 8$ 
C)  $\mathrm { m } = 6 ; \mathrm { n } = - 2 ; \mathrm { x } = 8$ 
D)  $\mathrm { m } = - 10 ; \mathrm { n } = + 6 ; \mathrm { x } = - 8$

Question 6

Write the augmented matrix for the system. Do not solve the system.
$\begin{aligned}
- 5 x + 2 y & & = - 3 \
- 5 y - & 6 z & = 9 \
& 5 z & = - 5
\end{aligned}$

Accepted Answer

A)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & - 3 & 0 \
- 5 & - 6 & 0 & 9 \
5 & 0 & 0 & - 5
\end{array} \right]$ 
B)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & 0 & - 3 \
- 5 & - 6 & 0 & 9 \
- 5 & 0 & 0 & - 5
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & 0 & - 3 \
0 & - 5 & - 6 & 9 \
0 & 0 & 5 & - 5
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & - 3 & 0 \
- 5 & - 6 & 9 & 0 \
5 & - 5 & 0 & 0
\end{array} \right]$ 
A)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & - 3 & 0 \
- 5 & - 6 & 0 & 9 \
5 & 0 & 0 & - 5
\end{array} \right]$ 
B)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & 0 & - 3 \
- 5 & - 6 & 0 & 9 \
- 5 & 0 & 0 & - 5
\end{array} \right]$ 
C)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & 0 & - 3 \
0 & - 5 & - 6 & 9 \
0 & 0 & 5 & - 5
\end{array} \right]$ 
D)  $\left[ \begin{array} { r r r | r } 
- 5 & 2 & - 3 & 0 \
- 5 & - 6 & 9 & 0 \
5 & - 5 & 0 & 0
\end{array} \right]$

Question 7

Find the value of the determinant.
-$\left| \begin{array} { l l l } 5 & 5 & 3 \ 3 & 2 & 3 \ 1 & 5 & 2 \end{array} \right|$

Accepted Answer

A)  191 
B)  $- 31$ 
C)  31 
D)  $- 61$ 
A)  191 
B)  $- 31$ 
C)  31 
D)  $- 61$

Question 8

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points  (-2,4),(0,-4) , and  (3,2) .

Accepted Answer

A)  $y = - \frac { 8 } { 5 } x ^ { 2 } + \frac { 6 } { 5 } x - 4$ 
B)  $y = \frac { 6 } { 5 } x ^ { 2 } - \frac { 8 } { 5 } x - 4$ 
C)  $y = \frac { 8 } { 15 } x ^ { 2 } - \frac { 16 } { 5 } x - \frac { 4 } { 3 }$ 
D)  $y = - \frac { 2 } { 15 } x ^ { 2 } + \frac { 4 } { 5 } x - 7$ 
A)  $y = - \frac { 8 } { 5 } x ^ { 2 } + \frac { 6 } { 5 } x - 4$ 
B)  $y = \frac { 6 } { 5 } x ^ { 2 } - \frac { 8 } { 5 } x - 4$ 
C)  $y = \frac { 8 } { 15 } x ^ { 2 } - \frac { 16 } { 5 } x - \frac { 4 } { 3 }$ 
D)  $y = - \frac { 2 } { 15 } x ^ { 2 } + \frac { 4 } { 5 } x - 7$

Question 9

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
$\begin{array} { r r } 
- 5 x + 3 z & = 16 \
7 x + 5 y - 9 z & = - 26 \
- 3 x - 7 y & = - 45
\end{array}$

Accepted Answer

A)   ({2,4,7)} 
B)   {(6,7,6)} 
C)   {(1,-6,-7)} 
D)   {(1,6,7)} 
A)   ({2,4,7)} 
B)   {(6,7,6)} 
C)   {(1,-6,-7)} 
D)   {(1,6,7)}

Question 10

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-$$\begin{array} { l } 
3.2 x + \pi y = \sqrt { 5 } \
\sqrt { 3 } x - 7.84 y = 4
\end{array}$$

Accepted Answer

A)  $\{ ( 0.25269,0.45438 ) \}$ 
B)  $\{ ( 1.25432 , - 0.73113 ) \}$ 
C)  $\{ ( 0.98584 , - 0.29241 ) \}$ 
D)  $\{ ( 0,1 ) \}$ 
A)  $\{ ( 0.25269,0.45438 ) \}$ 
B)  $\{ ( 1.25432 , - 0.73113 ) \}$ 
C)  $\{ ( 0.98584 , - 0.29241 ) \}$ 
D)  $\{ ( 0,1 ) \}$

Question 11

Find the matrix product when possible.
-$\left[ \begin{array} { r r } - 1 & 3 \ 5 & 2 \end{array} \right] \left[ \begin{array} { l l l } 0 & - 2 & 7 \ 1 & - 3 & 2 \end{array} \right]$

Accepted Answer

A)  They cannot be multiplied. 
B) 
$$\left[ \begin{array} { r r } 
3 & 2 \
- 7 & - 16 \
- 1 & 39
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l l } 
0 & - 6 & 4 \
5 & - 6 & 4
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c c } 
3 & - 7 & - 1 \
2 & - 16 & 39
\end{array} \right]$$ 
A)  They cannot be multiplied. 
B) 
$$\left[ \begin{array} { r r } 
3 & 2 \
- 7 & - 16 \
- 1 & 39
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l l } 
0 & - 6 & 4 \
5 & - 6 & 4
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c c } 
3 & - 7 & - 1 \
2 & - 16 & 39
\end{array} \right]$$

Question 12

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

Accepted Answer

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

Question 13

Provide an appropriate response.
-For a certain system of two linear equations, the solution set is $\{ - 3 , - 6 \}$, and $D = 18$. Find the values of $D _ { X }$ and $\mathrm { D } _ { \mathrm { y } }$

Accepted Answer

A)  $\mathrm { D } _ { \mathrm { x } } = - \frac { 1 } { 6 } ; \mathrm { D } _ { \mathrm { y } } = - \frac { 1 } { 3 }$ 
B)  $\mathrm { D } _ { \mathrm { X } } = - 3 ; \mathrm { D } _ { \mathrm { y } } = - 6$ 
C)  $\mathrm { D } _ { \mathrm { x } } = - 54 ; \mathrm { D } _ { \mathrm { y } } = - 108$ 
D)  $\mathrm { D } _ { \mathrm { X } } = - 108 ; \mathrm { D } _ { \mathrm { y } } = - 54$ 
A)  $\mathrm { D } _ { \mathrm { x } } = - \frac { 1 } { 6 } ; \mathrm { D } _ { \mathrm { y } } = - \frac { 1 } { 3 }$ 
B)  $\mathrm { D } _ { \mathrm { X } } = - 3 ; \mathrm { D } _ { \mathrm { y } } = - 6$ 
C)  $\mathrm { D } _ { \mathrm { x } } = - 54 ; \mathrm { D } _ { \mathrm { y } } = - 108$ 
D)  $\mathrm { D } _ { \mathrm { X } } = - 108 ; \mathrm { D } _ { \mathrm { y } } = - 54$

Question 14

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$x - 3 y = 21$ 
$- 3 x - 4 y = 15$

Accepted Answer

A)  $\{ ( 2 , - 5 ) \}$ 
B)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
C)  $\{ ( 3 , - 6 ) \}$ 
D)  $\{ ( - 3 , - 5 ) \}$ 
A)  $\{ ( 2 , - 5 ) \}$ 
B)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
C)  $\{ ( 3 , - 6 ) \}$ 
D)  $\{ ( - 3 , - 5 ) \}$

Question 15

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { c } 
9 x - 5 y - z = 39 \
x + 7 y + 2 z = 56 \
3 x + y + z = 33
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 8,6,16 ) \}$ 
C)  $\{ ( 8,6,3 ) \}$ 
D)  $\{ ( 8,3,6 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 8,6,16 ) \}$ 
C)  $\{ ( 8,6,3 ) \}$ 
D)  $\{ ( 8,3,6 ) \}$

Question 16

Solve the system.
$\begin{array} { l } 
\frac { 3 } { x } + \frac { 1 } { y } = \frac { 16 } { 35 } \
\frac { - 8 } { x } - \frac { 6 } { y } = - \frac { 26 } { 35 }
\end{array}$

Accepted Answer

A)  $\{ ( - 7,5 ) \}$ 
B)  $\{ ( 5 , - 7 ) \}$ 
C)  $\{ ( 35 , - 7 ) \rangle$ 
D)  $\{ ( - 5,7 ) \}$ 
A)  $\{ ( - 7,5 ) \}$ 
B)  $\{ ( 5 , - 7 ) \}$ 
C)  $\{ ( 35 , - 7 ) \rangle$ 
D)  $\{ ( - 5,7 ) \}$

Question 17

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

Accepted Answer

A)  The inverse does not exist. 
B)  $$\left[ \begin{array} { r r r } 
- 1 & 1 & 0 \
4 & - 1 & - 1 \
- 2 & 0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l l } 
- 1 & - 1 & - 1 \
- 2 & - 1 & - 1 \
- 2 & - 2 & - 3
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { c c c } 
1 & 1 & 1 \
\frac { 1 } { 2 } & 1 & 1 \
\frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }
\end{array} \right]$$ 
A)  The inverse does not exist. 
B)  $$\left[ \begin{array} { r r r } 
- 1 & 1 & 0 \
4 & - 1 & - 1 \
- 2 & 0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { l l l } 
- 1 & - 1 & - 1 \
- 2 & - 1 & - 1 \
- 2 & - 2 & - 3
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { c c c } 
1 & 1 & 1 \
\frac { 1 } { 2 } & 1 & 1 \
\frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }
\end{array} \right]$$

Question 18

Evaluate the determinant.
$$\left| \begin{array} { r r r r } 
3 & - 2 & 2 & - 1 \
4 & 1 & 1 & 6 \
- 3 & 2 & - 2 & 1 \
5 & 0 & 3 & - 2
\end{array} \right|$$

Accepted Answer

A)  9 
B)  0 
C)  6 
D)  37 
A)  9 
B)  0 
C)  6 
D)  37

Question 19

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of c?
$\begin{array} { c | c } 
\text { Year } & \begin{array} { c } 
\text { Number of dog collars } \
\text { (in thousands) }
\end{array} \
\hline 0 & 20 \
\hline 6 & 14 \
\hline 18 & 23.6
\end{array}$

Accepted Answer

A)   23.6 
B)  14 
C)  20 
D)   12.8 
A)   23.6 
B)  14 
C)  20 
D)   12.8

Question 20

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
$\begin{array} { l } 
3 x ^ { 2 } + 3 y ^ { 2 } = 20 \
6 x ^ { 2 } + 6 y ^ { 2 } = 48
\end{array}$

Accepted Answer

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

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

Solve the problem using matrices. -John has a jarful of quarters and nickels. There are 109 coins in the jar. The value of the coins is $15.25. How many of each type of coin are there?

Graph the inequality. - $-3 x-5 y \leq 15$ $Graph the inequality. - -3 x-5 y \leq 15$

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { r r } 6 & - 2 \\x & 0\end{array} \right] = \left[ \begin{array} { c c } \mathrm { m } - 4 & \mathrm { n } + 4 \\8 & 0\end{array} \right]$

Write the augmented matrix for the system. Do not solve the system. -5x+2y =-3 -5y- 6z =9 5z =-5

Find the value of the determinant. - $\left| \begin{array} { l l l } 5 & 5 & 3 \\ 3 & 2 & 3 \\ 1 & 5 & 2 \end{array} \right|$

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points (-2,4),(0,-4) , and (3,2) .

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. -5x+3z =16 7x+5y-9z =-26 -3x-7y =-45

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - 3.2x+\piy= x-7.84y=4

Find the matrix product when possible. - $\left[ \begin{array} { r r } - 1 & 3 \\ 5 & 2 \end{array} \right] \left[ \begin{array} { l l l } 0 & - 2 & 7 \\ 1 & - 3 & 2 \end{array} \right]$

Graph the solution set of the system of inequalities. - y\geqx y\leq|x-5| $Graph the solution set of the system of inequalities. - \begin{array}{l} y \geq \log x \\ y \leq|x-5| \end{array}$

Provide an appropriate response. -For a certain system of two linear equations, the solution set is $\{ - 3 , - 6 \}$ , and $D = 18$ . Find the values of $D _ { X }$ and $\mathrm { D } _ { \mathrm { y } }$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - $x - 3 y = 21$ $- 3 x - 4 y = 15$

Solve the system by using the inverse of the coefficient matrix. - 9x-5y-z=39 x+7y+2z=56 3x+y+z=33

Solve the system. += -=-

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

Evaluate the determinant. $\left| \begin{array} { r r r r } 3 & - 2 & 2 & - 1 \\4 & 1 & 1 & 6 \\- 3 & 2 & - 2 & 1 \\5 & 0 & 3 & - 2\end{array} \right|$

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of c? Year Number of dog collars (in thousands) 0 20 6 14 18 23.6

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. 3+3=20 6+6=48

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

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

Solve the problem using matrices. -John has a jarful of quarters and nickels. There are 109 coins in the jar. The value of the coins is $15.25. How many of each type of coin are there?

Graph the inequality. - −3x−5y≤15-3 x-5 y \leq 15−3x−5y≤15

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

Find the values of the variables for which the statement is true, if possible. - [6−2x0]=[m−4n+480]\left[ \begin{array} { r r } 6 & - 2 \\x & 0\end{array} \right] = \left[ \begin{array} { c c } \mathrm { m } - 4 & \mathrm { n } + 4 \\8 & 0\end{array} \right][6x​−20​]=[m−48​n+40​]