Question 1

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
3 x + 5 y = - 10 \
- 3 x - 6 y = 9
\end{array}$$

Accepted Answer

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

Question 2

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { c } 
- 5 x - y - 5 z = - 43 \
4 x \quad + 4 z = 32 \
6 y + z = 24
\end{array}$$

Accepted Answer

A)  $\{ ( 2,6,3 ) \}$ 
B)  $\{ ( - 2,3,4 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( 2,3,6 ) \}$ 
A)  $\{ ( 2,6,3 ) \}$ 
B)  $\{ ( - 2,3,4 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( 2,3,6 ) \}$

Question 3

Solve the problem using matrices.
-Anne and Nancy use a metal alloy that is $21 \%$ copper to make jewelry. How many ounces of an alloy that is $14 \%$ copper must be mixed with an alloy that is $24 \%$ copper to form 100 ounces of the desired alloy?

Accepted Answer

A)  $30 \mathrm { oz }$ 
B)  $70 \mathrm { oz }$ 
C)  $75 \mathrm { oz }$ 
D)  $32 \mathrm { oz }$ 
A)  $30 \mathrm { oz }$ 
B)  $70 \mathrm { oz }$ 
C)  $75 \mathrm { oz }$ 
D)  $32 \mathrm { oz }$

Question 4

Provide an appropriate response.
-Suppose that $A$ is an $m \times n$ matrix and $B$ is a $\mathrm { p } \times \mathrm { q }$ matrix. Under what conditions will the sum $\mathrm { A } +$ B exist?

Accepted Answer

A)  If $n = p$ 
B)  If $\mathrm { m } = \mathrm { p }$ and $\mathrm { n } = \mathrm { q }$ 
C)  It is not possible for the sum to exist. 
D)  The sum will always exist. 
A)  If $n = p$ 
B)  If $\mathrm { m } = \mathrm { p }$ and $\mathrm { n } = \mathrm { q }$ 
C)  It is not possible for the sum to exist. 
D)  The sum will always exist.

Question 5

Find the partial fraction decomposition for the rational expression.
-$$\frac { 532 x ^ { 2 } + 342 x } { \left( x ^ { 2 } + 2 \right) ( x + 6 ) }$$

Accepted Answer

A)  $\frac { 82 x - 150 } { x ^ { 2 } + 2 } + \frac { - 450 } { x + 6 }$ 
B)  $\frac { 82 x - 150 } { x ^ { 2 } + 2 } + \frac { 450 } { x + 6 }$ 
C)  $\frac { 82 x } { x + 6 } + \frac { 300 } { x ^ { 2 } + 2 }$ 
D)  $\frac { 82 x + 450 } { x ^ { 2 } + 2 } + \frac { - 150 } { x + 6 }$ 
A)  $\frac { 82 x - 150 } { x ^ { 2 } + 2 } + \frac { - 450 } { x + 6 }$ 
B)  $\frac { 82 x - 150 } { x ^ { 2 } + 2 } + \frac { 450 } { x + 6 }$ 
C)  $\frac { 82 x } { x + 6 } + \frac { 300 } { x ^ { 2 } + 2 }$ 
D)  $\frac { 82 x + 450 } { x ^ { 2 } + 2 } + \frac { - 150 } { x + 6 }$

Question 6

Find the cofactor of the indicated element.
-
a 11 
$$\begin{array} { l } 
\left| \begin{array} { r r r } 
- 1 & 0 & 2 \
1 & 1 & 4 \
4 & 5 & 4
\end{array} \right|
\end{array}$$

Accepted Answer

A)  $- 16$ 
B)  $- 1$ 
C)  $- 11$ 
D)  16 
A)  $- 16$ 
B)  $- 1$ 
C)  $- 11$ 
D)  16

Question 7

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
5 x ^ { 2 } - 5 y ^ { 2 } = - 60 \
3 x ^ { 2 } + 4 y ^ { 2 } = 76
\end{array}$$

Accepted Answer

A)  $\{ ( 2,4 ) , ( 4,2 ) , ( - 2 , - 4 ) , ( - 4 , - 2 ) \}$ 
B)  $\{ ( - 2 , - 4 ) , ( - 4 , - 2 ) \}$ 
C)  $\{ ( 2 , - 4 ) , ( 2,4 ) \}$ 
D)  $\{ ( 2,4 ) , ( - 2,4 ) , ( 2 , - 4 ) , ( - 2 , - 4 ) \}$ 
A)  $\{ ( 2,4 ) , ( 4,2 ) , ( - 2 , - 4 ) , ( - 4 , - 2 ) \}$ 
B)  $\{ ( - 2 , - 4 ) , ( - 4 , - 2 ) \}$ 
C)  $\{ ( 2 , - 4 ) , ( 2,4 ) \}$ 
D)  $\{ ( 2,4 ) , ( - 2,4 ) , ( 2 , - 4 ) , ( - 2 , - 4 ) \}$

Question 8

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-Matt bought 3 pounds of oranges and 2 pounds of apples and paid $\$ 4.16$, before tax. Andy bought 4 pounds of oranges and 3 pounds of apples and paid $\$ 5.86$, before tax. Use this information to set up a matrix equation of the form $\mathrm { AX } = \mathrm { B }$, which can be solved to determine the price per pound for orai and apples. Solve this matrix equation to find the price per pound of oranges.
Use the fact that for $\mathrm { A } = \left[ \begin{array} { l l } 3 & 2 \ 4 & 3 \end{array} \right] , \mathrm { A } ^ { - 1 } = \left[ \begin{array} { r r } 3 & - 2 \ - 4 & 3 \end{array} \right]$.

Accepted Answer

A)  $\$ 0.74$ per pound 
B)  $\$ 0.76$ per pound 
C)  $\$ 0.72$ per pound 
D)  $\$ 0.79$ per pound 
A)  $\$ 0.74$ per pound 
B)  $\$ 0.76$ per pound 
C)  $\$ 0.72$ per pound 
D)  $\$ 0.79$ per pound

Question 9

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

Question 10

Solve the linear programming problem.
-An automotive glass plant must be able to fill orders for 230 truck windshields, 350 auto windshields, and 310 RV windshields. Because the first-shift crews are more experienced, they are
Each able to make 7 truck windshields, 12 auto windshields, and 9 RV windshields. The
Second-shift crews are able to produce 4 truck windshields, 12 auto windshields, and 7 RV
Windshields. The first-shift crews cost the company $360.00 per crew per shift, and the second-shift
Crews cost $310.00 per crew per shift. The company's facilities will handle at most 50 crews on each
Shift, and they must have at least 10 crews come in to keep the plant running. How many first-shift
And how many second-shift crews should be scheduled to minimize costs?

Accepted Answer

A)  28 first-shift crews and 10 second-shift crews 
B)  43 first-shift crews and 24 second-shift crews 
C)  51 first-shift crews and 17 second-shift crews 
D)  37 first-shift crews and 36 second-shift crews 
A)  28 first-shift crews and 10 second-shift crews 
B)  43 first-shift crews and 24 second-shift crews 
C)  51 first-shift crews and 17 second-shift crews 
D)  37 first-shift crews and 36 second-shift crews

Question 11

Provide an appropriate response
-When using the substitution or elimination method to solve a system of two equations, you 
end up with an equation stating 0 = 0. What does this indicate to you about the system of
equations?

Accepted Answer

Answers may vary. One possibility: The s

Question 12

Find the dimension of the matrix.
-$\left[ \begin{array} { l l l } - 8 & - 3 & - 3 \end{array} \right]$

Accepted Answer

A)  3 
B)  $3 \times 1$ 
C)  $- 8 , - 3 , - 3$ 
D)  $1 \times 3$ 
A)  3 
B)  $3 \times 1$ 
C)  $- 8 , - 3 , - 3$ 
D)  $1 \times 3$

Question 13

Find the dimension of the matrix.
-$$\left[ \begin{array} { r r r r r } 
2 & - 9 & - 9 & 6 & 4 \
5 & - 9 & - 2 & - 9 & - 6 \
9 & - 1 & - 6 & - 2 & - 9
\end{array} \right]$$

Accepted Answer

A)  $2 , - 9 , - 9,6,4,5,9$ 
B)  $5 \times 3$ 
C)  $3 \times 5$ 
D)  15 
A)  $2 , - 9 , - 9,6,4,5,9$ 
B)  $5 \times 3$ 
C)  $3 \times 5$ 
D)  15

Question 14

Write the augmented matrix for the system. Do not
-$$\begin{array} { r r } 
6 x + 8 y & = 5 \
3 y - 4 z = - 2 \
5 z = 2
\end{array}$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 5 & 0 \
3 & - 4 & 0 & - 2 \
5 & 0 & 0 & 2
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 0 & 5 \
0 & 3 & - 4 & - 2 \
0 & 0 & 5 & 2
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 0 & 5 \
3 & - 4 & 0 & - 2 \
2 & 0 & 0 & 2
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 5 & 0 \
3 & - 4 & - 2 & 0 \
5 & 2 & 0 & 0
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 5 & 0 \
3 & - 4 & 0 & - 2 \
5 & 0 & 0 & 2
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 0 & 5 \
0 & 3 & - 4 & - 2 \
0 & 0 & 5 & 2
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 0 & 5 \
3 & - 4 & 0 & - 2 \
2 & 0 & 0 & 2
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r r | r } 
6 & 8 & 5 & 0 \
3 & - 4 & - 2 & 0 \
5 & 2 & 0 & 0
\end{array} \right]$$

Question 15

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-$$\begin{array} { l } 
y = \ln ( 3 x - 2 ) \
x ^ { 2 } + 3 y ^ { 2 } = 7
\end{array}$$

Accepted Answer

A)  $\{ ( 0.74 , - 1.47 ) , ( 1.73,1.16 ) \}$ 
B)  $\{ ( 5.25,2.62 ) \}$ 
C)  $\{ ( 1.73 , - 1.16 ) , ( 0.74,1.47 ) \}$ 
D)  $\{ ( 0.77,1.46 ) , ( - 0.59 , - 1.49 ) \}$ 
A)  $\{ ( 0.74 , - 1.47 ) , ( 1.73,1.16 ) \}$ 
B)  $\{ ( 5.25,2.62 ) \}$ 
C)  $\{ ( 1.73 , - 1.16 ) , ( 0.74,1.47 ) \}$ 
D)  $\{ ( 0.77,1.46 ) , ( - 0.59 , - 1.49 ) \}$

Question 16

Find the partial fraction decomposition for the rational expression.
-$$\frac { 3 x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } + 12 x + 13 } { ( x + 2 ) ( x - 1 ) ^ { 4 } }$$

Accepted Answer

A)  $\frac { - 1 } { x + 2 } + \frac { 5 } { x - 1 } + \frac { 2 } { ( x - 1 ) ^ { 2 } } + \frac { 17 } { ( x - 1 ) ^ { 3 } } + \frac { 12 } { ( x - 1 ) ^ { 4 } }$ 
B)  $\frac { 3 } { x + 2 } + \frac { 1 } { x - 1 } + \frac { 6 } { ( x - 1 ) ^ { 2 } } + \frac { 2 } { ( x - 1 ) ^ { 3 } } + \frac { 1 } { ( x - 1 ) ^ { 4 } }$ 
C)  $\frac { 1 } { x + 2 } + \frac { 2 } { x - 1 } + \frac { 3 } { ( x - 1 ) ^ { 2 } } + \frac { 5 } { ( x - 1 ) ^ { 3 } } + \frac { 10 } { ( x - 1 ) ^ { 4 } }$ 
D)  $\frac { 5 } { x + 2 } + \frac { 7 } { x - 1 } + \frac { - 2 } { ( x - 1 ) ^ { 2 } } + \frac { 4 } { ( x - 1 ) ^ { 3 } } + \frac { 3 } { ( x - 1 ) ^ { 4 } }$ 
A)  $\frac { - 1 } { x + 2 } + \frac { 5 } { x - 1 } + \frac { 2 } { ( x - 1 ) ^ { 2 } } + \frac { 17 } { ( x - 1 ) ^ { 3 } } + \frac { 12 } { ( x - 1 ) ^ { 4 } }$ 
B)  $\frac { 3 } { x + 2 } + \frac { 1 } { x - 1 } + \frac { 6 } { ( x - 1 ) ^ { 2 } } + \frac { 2 } { ( x - 1 ) ^ { 3 } } + \frac { 1 } { ( x - 1 ) ^ { 4 } }$ 
C)  $\frac { 1 } { x + 2 } + \frac { 2 } { x - 1 } + \frac { 3 } { ( x - 1 ) ^ { 2 } } + \frac { 5 } { ( x - 1 ) ^ { 3 } } + \frac { 10 } { ( x - 1 ) ^ { 4 } }$ 
D)  $\frac { 5 } { x + 2 } + \frac { 7 } { x - 1 } + \frac { - 2 } { ( x - 1 ) ^ { 2 } } + \frac { 4 } { ( x - 1 ) ^ { 3 } } + \frac { 3 } { ( x - 1 ) ^ { 4 } }$

Question 17

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
x - 4 y = 10 \
- 3 x + 12 y = - 30
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{ \left\{ x , \frac { x } { 12 } - \frac { 5 } { 2 } \right) \right\}$ 
C)  $\left\{ \left\{ x , \frac { 5 } { 2 } - \frac { x } { 4 } \right) \right\}$ 
D)  $\left\{ \left\{ x , \frac { x } { 4 } - \frac { 5 } { 2 } \right) \right\}$ 
A)  $\varnothing$ 
B)  $\left\{ \left\{ x , \frac { x } { 12 } - \frac { 5 } { 2 } \right) \right\}$ 
C)  $\left\{ \left\{ x , \frac { 5 } { 2 } - \frac { x } { 4 } \right) \right\}$ 
D)  $\left\{ \left\{ x , \frac { x } { 4 } - \frac { 5 } { 2 } \right) \right\}$

Question 18

Find the partial fraction decomposition for the rational expression.
-$$\frac { 4 x - 27 } { ( x + 2 ) ( x - 5 ) }$$

Accepted Answer

A)  $\frac { 4 } { x + 2 } + \frac { - 27 } { x - 5 }$ 
B)  $\frac { 1 } { x - 5 } + \frac { - 5 } { x + 2 }$ 
C)  $\frac { 5 } { x + 2 } + \frac { 1 } { x - 5 }$ 
D)  $\frac { 5 } { x + 2 } + \frac { - 1 } { x - 5 }$ 
A)  $\frac { 4 } { x + 2 } + \frac { - 27 } { x - 5 }$ 
B)  $\frac { 1 } { x - 5 } + \frac { - 5 } { x + 2 }$ 
C)  $\frac { 5 } { x + 2 } + \frac { 1 } { x - 5 }$ 
D)  $\frac { 5 } { x + 2 } + \frac { - 1 } { x - 5 }$

Question 19

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { r } 
x - y + 2 z = 3 \
5 x + z = 3 \
x + 5 y + z = 18
\end{array}$$

Accepted Answer

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

Question 20

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 $40 \mathrm { yd }$ and area $91 \mathrm { yd } ^ { 2 }$.

Accepted Answer

A)  $14 \mathrm { yd }$ by $6 \mathrm { yd }$ 
B)  16 yd by 7 yd 
C)  $15 \mathrm { yd }$ by $6 \mathrm { yd }$ 
D)  $13 \mathrm { yd }$ by $7 \mathrm { yd }$ 
A)  $14 \mathrm { yd }$ by $6 \mathrm { yd }$ 
B)  16 yd by 7 yd 
C)  $15 \mathrm { yd }$ by $6 \mathrm { yd }$ 
D)  $13 \mathrm { yd }$ by $7 \mathrm { yd }$

Solve the system by using the inverse of the coefficient matrix. - 3x+5y=-10 -3x-6y=9

Solve the system by using the inverse of the coefficient matrix. - -5x-y-5z=-43 4x+4z=32 6y+z=24

Solve the problem using matrices. -Anne and Nancy use a metal alloy that is $21 \%$ copper to make jewelry. How many ounces of an alloy that is $14 \%$ copper must be mixed with an alloy that is $24 \%$ copper to form 100 ounces of the desired alloy?

Provide an appropriate response. -Suppose that $A$ is an $m \times n$ matrix and $B$ is a $\mathrm { p } \times \mathrm { q }$ matrix. Under what conditions will the sum $\mathrm { A } +$ B exist?

Find the partial fraction decomposition for the rational expression. - $\frac { 532 x ^ { 2 } + 342 x } { \left( x ^ { 2 } + 2 \right) ( x + 6 ) }$

Find the cofactor of the indicated element. - a 11 -1 0 2 1 1 4 4 5 4

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - 5-5=-60 3+4=76

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

Provide an appropriate response -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating 0 = 0. What does this indicate to you about the system of equations?

Find the dimension of the matrix. - $\left[ \begin{array} { l l l } - 8 & - 3 & - 3 \end{array} \right]$

Find the dimension of the matrix. - $\left[ \begin{array} { r r r r r } 2 & - 9 & - 9 & 6 & 4 \\5 & - 9 & - 2 & - 9 & - 6 \\9 & - 1 & - 6 & - 2 & - 9\end{array} \right]$

Write the augmented matrix for the system. Do not - 6x+8y =5 3y-4z=-2 5z=2

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. - y=(3x-2) +3=7

Find the partial fraction decomposition for the rational expression. - $\frac { 3 x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } + 12 x + 13 } { ( x + 2 ) ( x - 1 ) ^ { 4 } }$

If the system has infinitely many solutions, write the solution set with x arbitrary. - x-4y=10 -3x+12y=-30

Find the partial fraction decomposition for the rational expression. - $\frac { 4 x - 27 } { ( x + 2 ) ( x - 5 ) }$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-y+2z=3 5x+z=3 x+5y+z=18

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 $40 \mathrm { yd }$ and area $91 \mathrm { yd } ^ { 2 }$ .

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