Question 1

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

Accepted Answer

A)  $\{ ( 0,2 ) , ( 0 , - 2 ) , ( 2 , - 6 ) , ( - 2,6 ) \}$ 
B)  $\{ ( 2,0 ) , ( - 2,0 ) , ( 2 , - 6 ) , ( - 2,6 ) \}$ 
C)  $\{ ( 2 , - 6 ) , ( - 2,6 ) \}$ 
D)  $\{ ( 2,0 ) , ( - 2,0 ) \}$ 
A)  $\{ ( 0,2 ) , ( 0 , - 2 ) , ( 2 , - 6 ) , ( - 2,6 ) \}$ 
B)  $\{ ( 2,0 ) , ( - 2,0 ) , ( 2 , - 6 ) , ( - 2,6 ) \}$ 
C)  $\{ ( 2 , - 6 ) , ( - 2,6 ) \}$ 
D)  $\{ ( 2,0 ) , ( - 2,0 ) \}$

Question 2

Find the values of the variables for which the statement is true, if possible.
-

Accepted Answer

A)  $\mathrm { m } = - 3 ; \mathrm { x } = - 4 ; \mathrm { n } = - 8 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
B)  $\mathrm { m } = - 6 ; \mathrm { x } = - 8 ; \mathrm { n } = - 3 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
C)  $\mathrm { m } = - 3 ; \mathrm { x } = - 8 ; \mathrm { n } = - 6 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
D)  $\mathrm { m } = - 3 ; \mathrm { x } = - 4 ; \mathrm { n } = - 6 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
A)  $\mathrm { m } = - 3 ; \mathrm { x } = - 4 ; \mathrm { n } = - 8 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
B)  $\mathrm { m } = - 6 ; \mathrm { x } = - 8 ; \mathrm { n } = - 3 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
C)  $\mathrm { m } = - 3 ; \mathrm { x } = - 8 ; \mathrm { n } = - 6 ; \mathrm { y } = 1 ; \mathrm { p } = 7$ 
D)  $\mathrm { m } = - 3 ; \mathrm { x } = - 4 ; \mathrm { n } = - 6 ; \mathrm { y } = 1 ; \mathrm { p } = 7$

Question 3

Find the inverse, if it exists, for the matrix.
-$\left[ \begin{array} { r r } 2 & 0 \ - 6 & 5 \end{array} \right]$

Accepted Answer

A)  The inverse does not exist. 
B) 
$\left[ \begin{array} { l l } \frac { 1 } { 2 } & 0 \ \frac { 3 } { 5 } & \frac { 1 } { 5 } \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } \frac { 1 } { 2 } & 0 \ - \frac { 3 } { 5 } & \frac { 1 } { 5 } \end{array} \right]$ 
D) 
$\left[ \begin{array} { l l } \frac { 1 } { 5 } & 0 \ \frac { 3 } { 5 } & \frac { 1 } { 2 } \end{array} \right]$ 
A)  The inverse does not exist. 
B) 
$\left[ \begin{array} { l l } \frac { 1 } { 2 } & 0 \ \frac { 3 } { 5 } & \frac { 1 } { 5 } \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } \frac { 1 } { 2 } & 0 \ - \frac { 3 } { 5 } & \frac { 1 } { 5 } \end{array} \right]$ 
D) 
$\left[ \begin{array} { l l } \frac { 1 } { 5 } & 0 \ \frac { 3 } { 5 } & \frac { 1 } { 2 } \end{array} \right]$

Question 4

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last
variable be the arbitrary variable.
-$$
\begin{array}{l}
7 x-2 y+6 z=4 \
-8 x+6 y+4 z=8 \
35 x-10 y+30 z=-2
\end{array}
$$

Accepted Answer

A)  $ \{(4,8,-2)\} $ 
B)  $ \{(-5 z+6,7 z-10, z)\} $ 
C)  $ \varnothing $ 
D)  $ \{(5,5,-1)\} $ 
A)  $ \{(4,8,-2)\} $ 
B)  $ \{(-5 z+6,7 z-10, z)\} $ 
C)  $ \varnothing $ 
D)  $ \{(5,5,-1)\} $

Question 5

Solve the problem.
-A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 95 black, 100
White and 80 red wires. How many of each type of cable were made?

Accepted Answer

A)  15 A; 16 B; 15 C 
B) 15 A; 20 B; 15 C 
C) 48A; 20 B; 12 C 
D) 20 A; 15 B; 15 C 
A)  15 A; 16 B; 15 C 
B) 15 A; 20 B; 15 C 
C) 48A; 20 B; 12 C 
D) 20 A; 15 B; 15 C

Question 6

Solve the system by substitution.
-$$\begin{array} { r } 
x + 8 y = 23 \
2 x + 8 y = 30
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 7,1 ) \}$ 
C)  $\{ ( 7,2 ) \}$ 
D)  $\{ ( 8,7 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 7,1 ) \}$ 
C)  $\{ ( 7,2 ) \}$ 
D)  $\{ ( 8,7 ) \}$

Question 7

Find the cofactor of the indicated element.
-$a_{22}$
$\left|\begin{array}{rrr}
1 & 5 & 1 \
-2 & -1 & 1 \
4 & 2 & 3
\end{array}\right|$

Accepted Answer

A)  1 
B)  -1 
C)  $- 2$ 
D)  2 
A)  1 
B)  -1 
C)  $- 2$ 
D)  2

Question 8

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { c } 
x + 3 y = - 21 \
- 2 x + 4 y = - 28
\end{array}$$

Accepted Answer

A)  $\{ ( 0 , - 7 ) \}$ 
B)  $\{ ( 1 , - 8 ) \}$ 
C)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
D)  $\{ ( 7,0 ) \}$ 
A)  $\{ ( 0 , - 7 ) \}$ 
B)  $\{ ( 1 , - 8 ) \}$ 
C)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
D)  $\{ ( 7,0 ) \}$

Question 9

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { r } 
x - y + 5 z = 17 \
4 x + z = 4 \
x + 5 y + z = 19
\end{array}$$

Accepted Answer

A)  $\{ ( 4,0,3 ) \}$ 
B)  $\{ ( 0,3,4 ) \}$ 
C)  Cramer's rule does not apply since $\mathrm { D } = 0$; $\varnothing$ 
D)  $\{ ( 4,3,0 ) \}$ 
A)  $\{ ( 4,0,3 ) \}$ 
B)  $\{ ( 0,3,4 ) \}$ 
C)  Cramer's rule does not apply since $\mathrm { D } = 0$; $\varnothing$ 
D)  $\{ ( 4,3,0 ) \}$

Question 10

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

Accepted Answer

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

Question 11

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$x y = 11$$
$2 y ^ { 2 } - x ^ { 2 } = 11$

Accepted Answer

A)  $\left\{ \left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) \right\}$ 
B)  $\left\{ \left( - 22 , - \frac { 1 } { 2 } \right) , ( 11,1 ) \right\}$ 
C)  $\{ ( 2 \mathrm { i } \sqrt { 11 } , - \mathrm { i } \sqrt { 11 } ) , ( - 2 \mathrm { i } \sqrt { 11 } , \mathrm { i } \sqrt { 11 } ) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \}$ 
D)  $\left\{ \left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , \left( - \mathrm { i } \sqrt { 22 } , \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \right\}$ 
A)  $\left\{ \left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) \right\}$ 
B)  $\left\{ \left( - 22 , - \frac { 1 } { 2 } \right) , ( 11,1 ) \right\}$ 
C)  $\{ ( 2 \mathrm { i } \sqrt { 11 } , - \mathrm { i } \sqrt { 11 } ) , ( - 2 \mathrm { i } \sqrt { 11 } , \mathrm { i } \sqrt { 11 } ) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \}$ 
D)  $\left\{ \left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , \left( - \mathrm { i } \sqrt { 22 } , \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \right\}$

Question 12

Solve the problem using matrices.
-Jerry has three pieces of string. The total length is $80 \mathrm {~cm}$. The sum of the two longer pieces is $69 \mathrm {~cm}$. The middle piece is twice as long as the shortest piece. What are the lengths of the three pieces?

Accepted Answer

A)  $5.5 \mathrm {~cm} ; 11 \mathrm {~cm} ; 63.5 \mathrm {~cm}$ 
B)  $11 \mathrm {~cm} ; 47 \mathrm {~cm} ; 57 \mathrm {~cm}$ 
C)  No solution 
D)  $11 \mathrm {~cm} ; 22 \mathrm {~cm} ; 47 \mathrm {~cm}$ 
A)  $5.5 \mathrm {~cm} ; 11 \mathrm {~cm} ; 63.5 \mathrm {~cm}$ 
B)  $11 \mathrm {~cm} ; 47 \mathrm {~cm} ; 57 \mathrm {~cm}$ 
C)  No solution 
D)  $11 \mathrm {~cm} ; 22 \mathrm {~cm} ; 47 \mathrm {~cm}$

Question 13

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

Accepted Answer

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

Question 14

Solve the system.
-$$\begin{array} { c } 
x - y + 3 z = 1 \
2 x + z = 0 \
x + 4 y + z = - 4
\end{array}$$

Accepted Answer

A)  $\{ ( 0,0 , - 1 ) \}$ 
B)  $\{ ( 0 , - 1,0 ) \}$ 
C)  $\{ ( 0 , - 1,1 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 0,0 , - 1 ) \}$ 
B)  $\{ ( 0 , - 1,0 ) \}$ 
C)  $\{ ( 0 , - 1,1 ) \}$ 
D)  $\varnothing$

Question 15

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 5 x + 3 y = 8 \
- 2 x + 4 y = 20
\end{array}$$

Accepted Answer

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

Question 16

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
5 x - 7 y = 30 \
2 x - 5 y = 12
\end{array}$$

Accepted Answer

A)  $\{ ( 6,1 ) \}$ 
B)  $\{ ( 6,0 ) \}$ 
C)  $\{ ( 5,1 ) \}$ 
D)  Cramer's rule does not apply since $\mathrm { D } = 0 ; \varnothing$ 
A)  $\{ ( 6,1 ) \}$ 
B)  $\{ ( 6,0 ) \}$ 
C)  $\{ ( 5,1 ) \}$ 
D)  Cramer's rule does not apply since $\mathrm { D } = 0 ; \varnothing$

Question 17

Solve the linear programming problem.
-An automotive glass plant must be able to fill orders for 230 truck windshields, 350 auto windshields, and $310 \mathrm { RV }$ windshields. Because the first-shift crews are more experienced, they are each able to make 7 truck windshields, 12 auto windshields, and $9 \mathrm { RV }$ 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)  51 first-shift crews and 17 second-shift crews 
B)  43 first-shift crews and 24 second-shift crews 
C)  28 first-shift crews and 10 second-shift crews 
D)  37 first-shift crews and 36 second-shift crews 
A)  51 first-shift crews and 17 second-shift crews 
B)  43 first-shift crews and 24 second-shift crews 
C)  28 first-shift crews and 10 second-shift crews 
D)  37 first-shift crews and 36 second-shift crews

Question 18

Solve the system by elimination.
-$$\begin{array} { l } 
- 7 x - 34 = - 5 y \
- 2 x - 3 y = - 8
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 2,5 ) \}$ 
C)  $\{ ( - 3,5 ) \}$ 
D)  $\{ ( - 2,4 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 2,5 ) \}$ 
C)  $\{ ( - 3,5 ) \}$ 
D)  $\{ ( - 2,4 ) \}$

Question 19

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)  $y \geq 3 x + 5$ 
B)  $y \geq - 3 x + 5$ 
C)  $y \leq 3 x + 5$ 
D)  $y \leq - 3 x + 5$ 
A)  $y \geq 3 x + 5$ 
B)  $y \geq - 3 x + 5$ 
C)  $y \leq 3 x + 5$ 
D)  $y \leq - 3 x + 5$

Question 20

Use a graphing calculator to solve the system. Express solutions with approximations to the nearest thousandth.
-$$\frac { 9 } { 7 } x + y = 5$$
$0.4 x - y = 3$

Accepted Answer

A)  $\{ ( 4.746 , - 1.102 ) \}$ 
B)  $\{ ( 2.258,2.097 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( - 2.947,8.789 ) \}$ 
A)  $\{ ( 4.746 , - 1.102 ) \}$ 
B)  $\{ ( 2.258,2.097 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( - 2.947,8.789 ) \}$

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

Find the values of the variables for which the statement is true, if possible. -

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r } 2 & 0 \\ - 6 & 5 \end{array} \right]$

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. - 7x-2y+6z=4 -8x+6y+4z=8 35x-10y+30z=-2

Solve the problem. -A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 95 black, 100 White and 80 red wires. How many of each type of cable were made?

Solve the system by substitution. - x+8y=23 2x+8y=30

Find the cofactor of the indicated element. - $a_{22}$ $\left|\begin{array}{rrr}1 & 5 & 1 \\-2 & -1 & 1 \\4 & 2 & 3\end{array}\right|$

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+3y=-21 -2x+4y=-28

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x-y+5z=17 4x+z=4 x+5y+z=19

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

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - $x y = 11$ $2 y ^ { 2 } - x ^ { 2 } = 11$

Solve the problem using matrices. -Jerry has three pieces of string. The total length is $80 \mathrm {~cm}$ . The sum of the two longer pieces is $69 \mathrm {~cm}$ . The middle piece is twice as long as the shortest piece. What are the lengths of the three pieces?

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

Solve the system. - x-y+3z=1 2x+z=0 x+4y+z=-4

Solve the system by using the inverse of the coefficient matrix. - -5x+3y=8 -2x+4y=20

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 5x-7y=30 2x-5y=12

Solve the system by elimination. - -7x-34=-5y -2x-3y=-8

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

Use a graphing calculator to solve the system. Express solutions with approximations to the nearest thousandth. - $\frac { 9 } { 7 } x + y = 5$ $0.4 x - y = 3$

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

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Exam 9: Systems and Matrices