Question 1

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$x + y < - 5$$

Accepted Answer

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

Question 2

20Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to
the nearest thousandth.
-$$\begin{array} { l } 
\sqrt { 3 } x - 0.9 y + z = - 8 \
\frac { 2 } { 3 } x - y + 2 z = \frac { 5 } { 6 }
\end{array}$$

Accepted Answer

A)  $\{ ( 3.628,5.029,3.719 ) \}$ 
B)  $\{ ( - 4.259,3.107,2.517 ) \}$ 
C)  $\{ ( - 5.072,3.306,3.760 ) \}$ 
D)  $\{ ( 4.729,7.426,3.652 ) \}$ 
A)  $\{ ( 3.628,5.029,3.719 ) \}$ 
B)  $\{ ( - 4.259,3.107,2.517 ) \}$ 
C)  $\{ ( - 5.072,3.306,3.760 ) \}$ 
D)  $\{ ( 4.729,7.426,3.652 ) \}$

Question 3

Use a graphing calcula tor to find the inverse of the matrix. Give five decimal places, if necessary.
-$$A = \left[ \begin{array} { c c c } 
1 / 3 & 2 / 7 & 4 / 5 \
17 / 11 & 5 & 0 \
18 / 13 & - 5 / 6 & 8 / 7
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
2.65952 & 0.15830 & - 1.86167 \
- 9.04238 & - 0.33824 & 6.32966 \
3.37129 & 0.05484 & - 1.48490
\end{array} \right]$$ 
B)

$\left[ \begin{array} { r r r } - 1.10557 & 0.19216 & 0.77390 \ 0.34172 & 0.14061 & - 0.23920 \ 1.58861 & - 0.13028 & - 0.23703 \end{array} \right]$
$$\left[ \begin{array} { l l l } 
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{array} \right]$$ 
C) 
$\left[\begin{array}{lll}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{array}\right]$ 
D)  
$$\left[ \begin{array} { c c c } 
3 & 3.5 & 1.25 \
0.64706 & 5 & 0 \
0.72222 & - 1.2 & 1.14286
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
2.65952 & 0.15830 & - 1.86167 \
- 9.04238 & - 0.33824 & 6.32966 \
3.37129 & 0.05484 & - 1.48490
\end{array} \right]$$ 
B)

$\left[ \begin{array} { r r r } - 1.10557 & 0.19216 & 0.77390 \ 0.34172 & 0.14061 & - 0.23920 \ 1.58861 & - 0.13028 & - 0.23703 \end{array} \right]$
$$\left[ \begin{array} { l l l } 
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{array} \right]$$ 
C) 
$\left[\begin{array}{lll}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{array}\right]$ 
D)  
$$\left[ \begin{array} { c c c } 
3 & 3.5 & 1.25 \
0.64706 & 5 & 0 \
0.72222 & - 1.2 & 1.14286
\end{array} \right]$$

Question 4

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective
function.
-$$\text { objective function } = 4 \mathrm { x } + 8 \mathrm { y } ; \text { maximum }$$

Accepted Answer

A)  112 
B)  102 
C)  120 
D)  76 
A)  112 
B)  102 
C)  120 
D)  76

Question 5

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
-$$\begin{array} { l } 
x + \frac { 1 } { a } y = a \
\frac { 1 } { b } x + y = b
\end{array}$$

Accepted Answer

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

Question 6

Use a graphing calculator to Express solutions with approximations to the nearest thousandth.
-$$\begin{array} { l } 
x - y + 2 z = 4 \
5 x + z = 0 \
x + 3 y + z = - 12
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 0 , - 4,4 ) \}$ 
C)  $\{ ( 0 , - 4,0 ) \}$ 
D)  $\{ ( 0,0 , - 4 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 0 , - 4,4 ) \}$ 
C)  $\{ ( 0 , - 4,0 ) \}$ 
D)  $\{ ( 0,0 , - 4 ) \}$

Question 7

Graph the solution set of the system of inequalities.
-$$\begin{array} { l } 
y \geq - x ^ { 2 } - 6 x - 4 \
y \geq x ^ { 2 } + 6 x + 4
\end{array}$$

Accepted Answer

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

Question 8

Find the cofactor of the indicated element.
-
a 12 
$$\begin{array} { l } 
\left| \begin{array} { r r r } 
- 2 & - 1 & 0 \
5 & 5 & 1 \
3 & 4 & 5
\end{array} \right|
\end{array}$$

Accepted Answer

A)  15 
B)  5 
C)  $- 22$ 
D)  22 
A)  15 
B)  5 
C)  $- 22$ 
D)  22

Question 9

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$x ^ { 2 } + ( y + 3 ) ^ { 2 } \leq 25$$

Accepted Answer

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

Question 10

Provide an appropriate response
-$$\begin{array} { l } 
2 x - 7 y = - 17 \
5 x + 3 y = 19
\end{array}$$ If your friend were going to solve this system of equations by first eliminating y, what
general suggestions would you make so your friend could start on this in a systematic
way?

Accepted Answer

Answers may vary. One possibility: Multi

Question 11

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
\frac { 8 } { x } + \frac { 1 } { y } = \frac { 17 } { 9 } \
\frac { 4 } { x } - \frac { 1 } { y } = \frac { 10 } { 9 }
\end{array}$$

Accepted Answer

A)  $\{ ( 4 , - 9 ) \}$ 
B)  $\{ ( - 4,9 ) \}$ 
C)  $\{ ( - 9,4 ) \}$ 
D)  $\{ ( 4,8 ) \}$ 
A)  $\{ ( 4 , - 9 ) \}$ 
B)  $\{ ( - 4,9 ) \}$ 
C)  $\{ ( - 9,4 ) \}$ 
D)  $\{ ( 4,8 ) \}$

Question 12

Solve the system by elimination.
-$$\begin{array} { l } 
6 x + 4 y = 30 \
4 x + 2 y = 20
\end{array}$$

Accepted Answer

A)  $\{ ( 5,0 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 4,1 ) \}$ 
D)  $\{ ( 5,1 ) \}$ 
A)  $\{ ( 5,0 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 4,1 ) \}$ 
D)  $\{ ( 5,1 ) \}$

Question 13

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.
-$$\begin{array} { l } 
x - z = - 2 \
y + z = 10 \
x + z = 3
\end{array}$$

Accepted Answer

A)  $\left\{ \left\{ \frac { 5 } { 2 } , \frac { 11 } { 2 } , \frac { 9 } { 2 } \right) \right\}$ 
B)  $\left\{ \left( \frac { 1 } { 2 } , \frac { 15 } { 2 } , \frac { 5 } { 2 } \right) \right\}$ 
C)  $\{ ( - 2,10,0 ) \}$ 
D)  $\{ ( 1,15,5 ) \}$ 
A)  $\left\{ \left\{ \frac { 5 } { 2 } , \frac { 11 } { 2 } , \frac { 9 } { 2 } \right) \right\}$ 
B)  $\left\{ \left( \frac { 1 } { 2 } , \frac { 15 } { 2 } , \frac { 5 } { 2 } \right) \right\}$ 
C)  $\{ ( - 2,10,0 ) \}$ 
D)  $\{ ( 1,15,5 ) \}$

Question 14

Provide an appropriate response
-The solution set of the system $y _ { 1 } = 2 x + 3$ and $y _ { 2 } = - x + 9$ is $\{ ( 2,7 ) \}$. Using slopes and $y$-intercepts, determine which of the two calculator-generated screens, left or right, is the appropriate one for this system.

Accepted Answer

A)  Right 
B)  Left 
C)  Neither is correct. 
D)  Both could be correct. 
A)  Right 
B)  Left 
C)  Neither is correct. 
D)  Both could be correct.

Question 15

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { r l } 
- 4 & 2 \
4 & 0
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 2 } & \frac { 1 } { 2 } \
0 & \frac { 1 } { 4 }
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { l l } 
\frac { 1 } { 2 } & \frac { 1 } { 4 } \
\frac { 1 } { 2 } & 0
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c } 
0 \frac { 1 } { 4 } \
\frac { 1 } { 2 } \frac { 1 } { 2 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
0 & - \frac { 1 } { 4 } \
- \frac { 1 } { 2 } & \frac { 1 } { 2 }
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 2 } & \frac { 1 } { 2 } \
0 & \frac { 1 } { 4 }
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { l l } 
\frac { 1 } { 2 } & \frac { 1 } { 4 } \
\frac { 1 } { 2 } & 0
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c } 
0 \frac { 1 } { 4 } \
\frac { 1 } { 2 } \frac { 1 } { 2 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
0 & - \frac { 1 } { 4 } \
- \frac { 1 } { 2 } & \frac { 1 } { 2 }
\end{array} \right]$$

Question 16

Solve the system by elimination.
-$$\begin{array} { l } 
4 x - 7 y = - 17 \
2 x - 2 y = - 10
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( - 7,0 ) \}$ 
C)  $\{ ( - 6,0 ) \}$ 
D)  $\{ ( - 6 , - 1 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( - 7,0 ) \}$ 
C)  $\{ ( - 6,0 ) \}$ 
D)  $\{ ( - 6 , - 1 ) \}$

Question 17

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

Accepted Answer

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

Question 18

Graph the solution set of the system of inequalities.
-$$\begin{array} { l } 
| x | \leq 5 \
y \leq 3 \
y \leq x ^ { 2 } - 3
\end{array}$$

Accepted Answer

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

Question 19

Provide an appropriate response.
-Suppose that you are solving a system of three linear equations by the Gauss-Jordan method and obtain the following augmented matrix.
$$\left[ \begin{array} { r r r | r } 
1 & - 8 & - 10 & - 17 \
0 & 1 & - 6 & - 3 \
0 & 0 & 0 & - 20
\end{array} \right]$$
What conclusion can you draw about the solutions of the system?

Accepted Answer

A)  The system has infinitely many solutions. 
B)  The system has exactly one solution. 
C)  The system has no solutions. 
D)  There is not enough information to draw any conclusion about the solutions of the system. 
A)  The system has infinitely many solutions. 
B)  The system has exactly one solution. 
C)  The system has no solutions. 
D)  There is not enough information to draw any conclusion about the solutions of the system.

Question 20

Find the values of the variables for which the statement is true, if possible.
-$$\left[ \begin{array} { l l l } 
\mathrm { a } & - 1 & 6 \
1 & - 3 & \mathrm { f }
\end{array} \right] + \left[ \begin{array} { r r r } 
- 2 & \mathrm {~b} & 7 \
\mathrm {~d} & - 7 & - 8
\end{array} \right] = \left[ \begin{array} { l l l } 
- 6 & 4 & \mathrm { c } \
- 9 & \mathrm { e } & - 7
\end{array} \right]$$

Accepted Answer

A)  $a = - 4 ; b = 5 ; c = 13 ; d = - 10 ; e = - 10 ; f = 1$ 
B)  $a = - 8 ; b = 3 ; c = 1 ; d = - 8 ; e = 4 ; f = - 15$ 
C)  $a = 3 ; b = - 4 ; c = 24 ; d = - 9 ; e = 21 ; f = \frac { 7 } { 8 }$ 
D)  $a = 4 ; b = - 5 ; c = - 1 ; d = 10 ; e = - 4 ; f = - 1$ 
A)  $a = - 4 ; b = 5 ; c = 13 ; d = - 10 ; e = - 10 ; f = 1$ 
B)  $a = - 8 ; b = 3 ; c = 1 ; d = - 8 ; e = 4 ; f = - 15$ 
C)  $a = 3 ; b = - 4 ; c = 24 ; d = - 9 ; e = 21 ; f = \frac { 7 } { 8 }$ 
D)  $a = 4 ; b = - 5 ; c = - 1 ; d = 10 ; e = - 4 ; f = - 1$

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $x + y < - 5$

20Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to the nearest thousandth. - x-0.9y+z=-8 x-y+2z=

Use a graphing calcula tor to find the inverse of the matrix. Give five decimal places, if necessary. - $A = \left[ \begin{array} { c c c } 1 / 3 & 2 / 7 & 4 / 5 \\17 / 11 & 5 & 0 \\18 / 13 & - 5 / 6 & 8 / 7\end{array} \right]$

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

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - x-y+2z=4 5x+z=0 x+3y+z=-12

Graph the solution set of the system of inequalities. - y\geq--6x-4 y\geq+6x+4 $Graph the solution set of the system of inequalities. - \begin{array} { l } y \geq - x ^ { 2 } - 6 x - 4 \\ y \geq x ^ { 2 } + 6 x + 4 \end{array}$

Find the cofactor of the indicated element. - a 12 -2 -1 0 5 5 1 3 4 5

Provide an appropriate response - 2x-7y=-17 5x+3y=19 If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

If the system has infinitely many solutions, write the solution set with x arbitrary. - += -=

Solve the system by elimination. - 6x+4y=30 4x+2y=20

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. - x-z=-2 y+z=10 x+z=3

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r l } - 4 & 2 \\4 & 0\end{array} \right]$

Solve the system by elimination. - 4x-7y=-17 2x-2y=-10

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - 2+xy+2=2 3-5xy+3=3

Graph the solution set of the system of inequalities. - |x|\leq5 y\leq3 y\leq-3 $Graph the solution set of the system of inequalities. - \begin{array} { l } | x | \leq 5 \\ y \leq 3 \\ y \leq x ^ { 2 } - 3 \end{array}$

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

Which method should be used to solve the system? Explain your answer, including a description of the first step. - x+y<−5x + y < - 5x+y<−5

20Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to the nearest thousandth. - x-0.9y+z=-8 x-y+2z=

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. - objective function =4x+8y; maximum \text { objective function } = 4 \mathrm { x } + 8 \mathrm { y } ; \text { maximum } objective function =4x+8y; maximum

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

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - x-y+2z=4 5x+z=0 x+3y+z=-12

Graph the solution set of the system of inequalities. - y\geq--6x-4 y\geq+6x+4

Find the cofactor of the indicated element. - a 12 -2 -1 0 5 5 1 3 4 5

Which method should be used to solve the system? Explain your answer, including a description of the first step. - x2+(y+3)2≤25x ^ { 2 } + ( y + 3 ) ^ { 2 } \leq 25x2+(y+3)2≤25

Provide an appropriate response - 2x-7y=-17 5x+3y=19 If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

If the system has infinitely many solutions, write the solution set with x arbitrary. - += -=

Solve the system by elimination. - 6x+4y=30 4x+2y=20

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. - x-z=-2 y+z=10 x+z=3

Find the inverse, if it exists, for the matrix. - [−4240]\left[ \begin{array} { r l } - 4 & 2 \\4 & 0\end{array} \right][−44​20​]

Solve the system by elimination. - 4x-7y=-17 2x-2y=-10

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - 2+xy+2=2 3-5xy+3=3

Graph the solution set of the system of inequalities. - |x|\leq5 y\leq3 y\leq-3

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

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $x + y < - 5$

Graph the solution set of the system of inequalities. - y\geq--6x-4 y\geq+6x+4 $Graph the solution set of the system of inequalities. - \begin{array} { l } y \geq - x ^ { 2 } - 6 x - 4 \\ y \geq x ^ { 2 } + 6 x + 4 \end{array}$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r l } - 4 & 2 \\4 & 0\end{array} \right]$

Graph the solution set of the system of inequalities. - |x|\leq5 y\leq3 y\leq-3 $Graph the solution set of the system of inequalities. - \begin{array} { l } | x | \leq 5 \\ y \leq 3 \\ y \leq x ^ { 2 } - 3 \end{array}$