Question 1

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { c } 
5 x + 4 y - z = 39 \
x + 5 y - 6 z = - 5 \
5 x + y + z = 31
\end{array}$$

Accepted Answer

A)  $\{ ( 3 , - 8 , - 8 ) \}$ 
B)  $\{ ( 3,8,8 ) \}$ 
C)  $\{ ( 4,6,8 ) \}$ 
D)  $\{ ( 8,8,8 ) \}$ 
A)  $\{ ( 3 , - 8 , - 8 ) \}$ 
B)  $\{ ( 3,8,8 ) \}$ 
C)  $\{ ( 4,6,8 ) \}$ 
D)  $\{ ( 8,8,8 ) \}$

Question 2

Find the matrix product when possible.
-Given $A = \left[ \begin{array} { r r } 5 & - 4 \ - 5 & 3 \end{array} \right]$, find $A ^ { 2 }$

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
45 & - 32 \
- 40 & 29
\end{array} \right]$$ 
B) 
$\left[ \begin{array} { r r } 41 & - 37 \ 34 & - 5 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 5 & - 8 \ - 10 & 11 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 25 & 16 \ 25 & 9 \end{array} \right]$ 
A) 
$$\left[ \begin{array} { r r } 
45 & - 32 \
- 40 & 29
\end{array} \right]$$ 
B) 
$\left[ \begin{array} { r r } 41 & - 37 \ 34 & - 5 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 5 & - 8 \ - 10 & 11 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 25 & 16 \ 25 & 9 \end{array} \right]$

Question 3

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-Find the dimensions of a rectangular enclosure with perimeter $34 \mathrm { ft }$ and area $60 \mathrm { ft } ^ { 2 }$.

Accepted Answer

A)  $10 \mathrm { ft }$ by $24 \mathrm { ft }$ 
B)  $12 \mathrm { ft }$ by $22 \mathrm { ft }$ 
C)  $12 \mathrm { ft }$ by $5 \mathrm { ft }$ 
D)  $10 \mathrm { ft }$ by $6 \mathrm { ft }$ 
A)  $10 \mathrm { ft }$ by $24 \mathrm { ft }$ 
B)  $12 \mathrm { ft }$ by $22 \mathrm { ft }$ 
C)  $12 \mathrm { ft }$ by $5 \mathrm { ft }$ 
D)  $10 \mathrm { ft }$ by $6 \mathrm { ft }$

Question 4

Use a graphing calculator to Express solutions with approximations to the nearest thousandth.
-$$\begin{array} { c } 
x + y + z = 2 \
x - y + 4 z = - 5 \
3 x + y + z = - 6
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 4,5,1 ) \}$ 
C)  $\{ ( 1,5 , - 4 ) \}$ 
D)  $\{ ( 1 , - 4,5 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 4,5,1 ) \}$ 
C)  $\{ ( 1,5 , - 4 ) \}$ 
D)  $\{ ( 1 , - 4,5 ) \}$

Question 5

Use a graphing calculator to find the value of the determinant.
-$$\left| \begin{array} { r r r } 
- 9.1 & - 6.4 & 6.5 \
6.4 & - 1.2 & 10 \
6.1 & - 3.4 & - 6.9
\end{array} \right|$$

Accepted Answer

A)  $194.416$ 
B)  $- 27.768$ 
C)  $- 1151.632$ 
D)  $152.808$ 
A)  $194.416$ 
B)  $- 27.768$ 
C)  $- 1151.632$ 
D)  $152.808$

Question 6

Solve the problem.
-Find the equation of the line $y = a x + b$ that passes through the points $( - 5,5 )$ and $( 3 , - 9 )$.

Accepted Answer

A)  $y = - \frac { 7 } { 4 } x - \frac { 15 } { 4 }$ 
B)  $y = - \frac { 15 } { 4 } x - \frac { 7 } { 4 }$ 
C)  $y = - \frac { 1 } { 2 } x + 1$ 
D)  $y = - \frac { 7 } { 4 } x + \frac { 15 } { 2 }$ 
A)  $y = - \frac { 7 } { 4 } x - \frac { 15 } { 4 }$ 
B)  $y = - \frac { 15 } { 4 } x - \frac { 7 } { 4 }$ 
C)  $y = - \frac { 1 } { 2 } x + 1$ 
D)  $y = - \frac { 7 } { 4 } x + \frac { 15 } { 2 }$

Question 7

Find the dimension of the matrix.
-$$\left[ \begin{array} { l l } 
6 & 0 \
0 & 6
\end{array} \right]$$

Accepted Answer

A)  4 
B)  6 
C)  $6 \times 6$ 
D)  $2 \times 2$ 
A)  4 
B)  6 
C)  $6 \times 6$ 
D)  $2 \times 2$

Question 8

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$y \geq - 4$$

Accepted Answer

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

Question 9

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

Accepted Answer

A)  $\{ ( - 0.75,4.50 ) \}$ 
B)  $\{ ( 0.75 , - 4.50 ) \}$ 
C)  $\{ ( 0.75,4.50 ) \}$ 
D)  $\{ ( - 0.75 , - 4.50 ) \}$ 
A)  $\{ ( - 0.75,4.50 ) \}$ 
B)  $\{ ( 0.75 , - 4.50 ) \}$ 
C)  $\{ ( 0.75,4.50 ) \}$ 
D)  $\{ ( - 0.75 , - 4.50 ) \}$

Question 10

Find the cofactor of the indicated element.
-$a _ { 13 }$
$$\left| \begin{array} { r r r } 
- 2 & - 1 & 0 \
1 & 2 & 3 \
4 & 4 & 2
\end{array} \right|$$

Accepted Answer

A)  $- 4$ 
B)  $- 10$ 
C)  4 
D)  $- 8$ 
A)  $- 4$ 
B)  $- 10$ 
C)  4 
D)  $- 8$

Question 11

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of
the concepts involved in graphing inequalities.
-

Accepted Answer

A)  $y \leq - 3 x - 2$ 
B)  $y \geq - 3 x - 2$ 
C)  $y \leq 3 x - 2$ 
D)  $y \geq 3 x - 2$ 
A)  $y \leq - 3 x - 2$ 
B)  $y \geq - 3 x - 2$ 
C)  $y \leq 3 x - 2$ 
D)  $y \geq 3 x - 2$

Question 12

Graph the solution set of the system of inequalities.
-$\begin{array}{l}
\frac{x^{2}}{9}+\frac{y^{2}}{25} \leq 1 \
\frac{x^{2}}{25}+\frac{y^{2}}{9} \geq 1
\end{array}$

Accepted Answer

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

Question 13

Find the matrix product when possible.
-$$\left[ \begin{array} { l l l } 
1 & - 7 & 3
\end{array} \right] \left[ \begin{array} { r r r } 
- 3 & - 5 & 5 \
4 & - 1 & 3 \
- 7 & - 2 & - 4
\end{array} \right]$$

Accepted Answer

A) 
$[-52-4-28]$ 
B) 
$\left[\begin{array}{r}
-52 \
-4 \
-28
\end{array}\right]$ 
C) 
$\left[\begin{array}{rrr}
-3 & 35 & 15 \
4 & 7 & 9 \
-7 & 14 & -12
\end{array}\right]$ 
D) 
 $\left[\begin{array}{rrr}
1 & -7 & 3 \
-3 & -5 & 5 \
4 & -1 & 3 \
-7 & -2 & -4
\end{array}\right]$ 
A) 
$[-52-4-28]$ 
B) 
$\left[\begin{array}{r}
-52 \
-4 \
-28
\end{array}\right]$ 
C) 
$\left[\begin{array}{rrr}
-3 & 35 & 15 \
4 & 7 & 9 \
-7 & 14 & -12
\end{array}\right]$ 
D) 
 $\left[\begin{array}{rrr}
1 & -7 & 3 \
-3 & -5 & 5 \
4 & -1 & 3 \
-7 & -2 & -4
\end{array}\right]$

Question 14

Find the values of the variables for which the statement is true, if possible.
-$$\left[ \begin{array} { c c } 
1 & - 3 \
x & 0
\end{array} \right] = \left[ \begin{array} { c c } 
\mathrm { m } - 4 & \mathrm { n } + 4 \
7 & 0
\end{array} \right]$$

Accepted Answer

A)  $\mathrm { m } = - 7 ; \mathrm { n } = 5 ; \mathrm { x } = 0$ 
B)  $\mathrm { m } = 1 ; \mathrm { n } = - 3 ; \mathrm { x } = 7$ 
C)  $m = 5 ; n = - 7 ; x = 7$ 
D)  $\mathrm { m } = - 5 ; \mathrm { n } = + 7 ; \mathrm { x } = - 7$ 
A)  $\mathrm { m } = - 7 ; \mathrm { n } = 5 ; \mathrm { x } = 0$ 
B)  $\mathrm { m } = 1 ; \mathrm { n } = - 3 ; \mathrm { x } = 7$ 
C)  $m = 5 ; n = - 7 ; x = 7$ 
D)  $\mathrm { m } = - 5 ; \mathrm { n } = + 7 ; \mathrm { x } = - 7$

Question 15

Solve the problem.
-The perimeter of a triangle is $69 \mathrm {~cm}$. The triangle is isosceles now, but if its base were lengthened by $2 \mathrm {~cm}$ and each leg were shortened by $4 \mathrm {~cm}$, it would be equilateral. Find the length of the base of the original triangle.

Accepted Answer

A)  $21 \mathrm {~cm}$ 
B)  $25 \mathrm {~cm}$ 
C)  $18 \mathrm {~cm}$ 
D)  $19 \mathrm {~cm}$ 
A)  $21 \mathrm {~cm}$ 
B)  $25 \mathrm {~cm}$ 
C)  $18 \mathrm {~cm}$ 
D)  $19 \mathrm {~cm}$

Question 16

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the
solution with y arbitrary.
-$$\begin{array} { l } 
x + y = 1 \
x - y = - 5
\end{array}$$

Accepted Answer

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

Question 17

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-A bakery sells three types of cakes, each requiring the amount of ingredients shown.

To fill its orders for these cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 53 eggs. How many cakes of each type were made?

Accepted Answer

A)  39 Cake I; 8 Cake II; 12 Cake III 
B)  15 Cake I; 8 Cake II; 5 Cake III 
C)  17 Cake I; 8 Cake II; 5 Cake III 
D)  5 Cake I; 8 Cake II; 15 Cake III 
A)  39 Cake I; 8 Cake II; 12 Cake III 
B)  15 Cake I; 8 Cake II; 5 Cake III 
C)  17 Cake I; 8 Cake II; 5 Cake III 
D)  5 Cake I; 8 Cake II; 15 Cake III

Question 18

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
x + 3 y = 26 \
- 3 x + 2 y = - 12
\end{array}$$

Accepted Answer

A)  Cramer's rule does not apply since $\mathrm { D } = 0$; 
B)  $\{ ( - 8,7 ) \}$ 
C)  $\{ ( 8,6 ) \}$ 
D)  $\{ ( 7,7 ) \}$ 
A)  Cramer's rule does not apply since $\mathrm { D } = 0$; 
B)  $\{ ( - 8,7 ) \}$ 
C)  $\{ ( 8,6 ) \}$ 
D)  $\{ ( 7,7 ) \}$

Question 19

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

Accepted Answer

A)  -5 
B)  -21 
C)  -39 
D)  -27 
A)  -5 
B)  -21 
C)  -39 
D)  -27

Question 20

Solve the equation for x.
-$$P ( 1 , - 2 ) , Q ( 1,3 ) , R ( 5 , - 1 )$$

Accepted Answer

A)  0 
B)  20 
C)  7 
D)  10 
A)  0 
B)  20 
C)  7 
D)  10

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 5x+4y-z=39 x+5y-6z=-5 5x+y+z=31

Find the matrix product when possible. -Given $A = \left[ \begin{array} { r r } 5 & - 4 \\ - 5 & 3 \end{array} \right]$ , find $A ^ { 2 }$

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. -Find the dimensions of a rectangular enclosure with perimeter $34 \mathrm { ft }$ and area $60 \mathrm { ft } ^ { 2 }$ .

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - x+y+z=2 x-y+4z=-5 3x+y+z=-6

Use a graphing calculator to find the value of the determinant. - $\left| \begin{array} { r r r } - 9.1 & - 6.4 & 6.5 \\6.4 & - 1.2 & 10 \\6.1 & - 3.4 & - 6.9\end{array} \right|$

Solve the problem. -Find the equation of the line $y = a x + b$ that passes through the points $( - 5,5 )$ and $( 3 , - 9 )$ .

Find the dimension of the matrix. - $\left[ \begin{array} { l l } 6 & 0 \\0 & 6\end{array} \right]$

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $y \geq - 4$ $Which method should be used to solve the system? Explain your answer, including a description of the first step. - y \geq - 4$

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. - y= 2x+y=3

Find the cofactor of the indicated element. - $a _ { 13 }$ $\left| \begin{array} { r r r } - 2 & - 1 & 0 \\1 & 2 & 3 \\4 & 4 & 2\end{array} \right|$

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Graph the solution set of the system of inequalities. - +\leq1 +\geq1

Find the matrix product when possible. - $\left[ \begin{array} { l l l } 1 & - 7 & 3\end{array} \right] \left[ \begin{array} { r r r } - 3 & - 5 & 5 \\4 & - 1 & 3 \\- 7 & - 2 & - 4\end{array} \right]$

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { c c } 1 & - 3 \\x & 0\end{array} \right] = \left[ \begin{array} { c c } \mathrm { m } - 4 & \mathrm { n } + 4 \\7 & 0\end{array} \right]$

Solve the problem. -The perimeter of a triangle is $69 \mathrm {~cm}$ . The triangle is isosceles now, but if its base were lengthened by $2 \mathrm {~cm}$ and each leg were shortened by $4 \mathrm {~cm}$ , it would be equilateral. Find the length of the base of the original triangle.

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=-5

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+3y=26 -3x+2y=-12

Solve the equation for x. - $P ( 1 , - 2 ) , Q ( 1,3 ) , R ( 5 , - 1 )$

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