Deck 6: Systems of Equations and Matrices

Question 1

Match the inequality with a graph.
$$2 y \leq - 7 x$$

A)

B)

C)

D)

Answer 1

A

Question 2

Find the graph that matches the given system of equations.
$$\begin{array} { c } 3 x + y = - 1 \\x + y = - 7\end{array}$$

A)

B)

C)

D)

Answer 2

B

Question 3

Match the system of inequalities with a graph.
A)

B)

C)

D)

Answer 3

C

Question 4

Find the graph that matches the given system of equations.
$$\begin{array} { l } x + y = 9 \\y = x - 1\end{array}$$

A)

B)

C)

D)

Question 5

Provide the proper response.
$$\begin{array} { l } 2 x - y = 0 \\- 6 x + 2 y = 0 \\\text { A) }\end{array}$$

A)

B)

C)

D)

Question 6

Match the system of inequalities with a graph.
$$\begin{array} { l } y \leq x - 2 \\y \geq x\end{array}$$

A)

B)

C)

D)

Question 7

Match the inequality with a graph.
$$- x > y$$

A)

B)

C)

D)

Question 8

Find the graph that matches the given system of equations.
$2 x + y = - 13$ $y = 2 x + 3$ A)

A)

B)

C)

D)

Question 9

Match the system of inequalities with a graph.
A)

B)

C)

D)

Question 10

Match the inequality with a graph.
$$- 4 y \leq x$$

A)

B)

C)

D)

Question 11

Find the graph that matches the given system of equations.
$$\begin{array} { l } x = - 3 y \\x - y = - 4\end{array}$$

A)

B)

C)

D)

Question 12

Match the inequality with a graph.
A)

B)

C)

D)

Question 13

Match the system of inequalities with a graph.
$$\begin{array} { l } 2 y - x \leq 2 \\y + 2 x \leq 4 \\y \geq - x\end{array}$$

A)

B)

C)

D)

Question 14

Provide an appropriate response.
Equating the numerators in the process of partial fraction decomposition, your friend has obtained the algebraic expression $6 x - 7 = A ( 3 x - 4 ) + B ( x + 7 ) + C ( x + 7 )$ . From this equation, what can you say about the original expression?

A) The original expression has a repeating linear factor in the denominator.
B) The original expression has a non-reducible quadratic factor in the denominator.
C) The original expression has a numerator of $( 6 x - 7 ) ( 3 x - 4 ) ( x + 7 ) ( x + 7 )$ .
D) The original expression has the factor $\left( x ^ { 2 } + 7 \right)$ in the denominator.

Question 15

Find the graph that matches the given system of equations.
$$\begin{array} { l } 3 x + 4 y = 7 \\5 x - 2 y = - 23\end{array}$$

A)

B)

C)

D)

Question 16

Match the system of inequalities with a graph.
$$\begin{array} { l } y \leq 4 x - 5 \\y \geq - x\end{array}$$

A)

B)

C)

D)

Question 17

Match the inequality with a graph.
$$2 x - 5 y \geq - 1$$

A)

B)

C)

D)

Question 18

Match the system of inequalities with a graph.
A)

B)

C)

D)

Question 19

Match the inequality with a graph.
A)

B)

C)

D)

Question 20

Find the graph that matches the given system of equations.
$$\begin{array} { l } 3 x + y = 9 \\2 x - 3 y = - 5\end{array}$$

A)

B)


C)

D)

Question 21

Match the inequality with a graph.
$$7 - 3 x > y$$

A)

B)

C)

D)

Question 22

Provide an appropriate response.
Equating the numerators in the process of partial fraction decomposition, your friend has obtained the algebraic expression $5 x - 8 = A ( 3 x - 3 ) + B ( x + 4 )$ . From this equation, what can you say about the original expression?

A) The denominator of the original expression is $3 x ^ { 2 } + 9 x - 12$ .
B) The original expression has repeated linear factors in the denominator.
C) The original expression has a non-reducible quadratic factor in the denominator.
D) The numerator of the original expression is $3 x ^ { 2 } + 9 x - 12$ .

Question 23

Rewrite the expression using a determinant. Show all your work.
$$\frac { a + b } { c }$$

Question 24

Find the graph that matches the given system of equations.
$$\begin{array} { l } 3 x + 4 y = - 7 \\2 x - 3 y = - 16\end{array}$$

A)

B)

C)

D)

Question 25

Provide an appropriate answer.
$$\text { Let } A = \left[ \begin{array} { c c } - 1 & 3 \\- 3 & 1\end{array} \right] \text { and } B = \left[ \begin{array} { c c } - 5 & - 1 \\- 5 & - 4\end{array} \right] \text {. Show that } A B \neq B A \text {. Show all your work. }$$

Question 26

Match the system of inequalities with a graph.
A)

B)

C)

D)

Question 27

Match the system of inequalities with a graph.
A)

B)

C)

D)

Question 28

Provide an appropriate response.
You are finding the partial fraction decomposition of the expression $\frac { x ^ { 3 } - 7 x + 12 } { x ^ { 2 } - 13 x + 40 }$ .
You should ...

A) Solve the equation $x ^ { 3 } - 7 x + 12 = A ( x - 5 ) + B ( x - 8 )$
B) Solve the equation $x ^ { 3 } - 7 x + 12 = ( A x + B ) \left( x ^ { 2 } - 13 x + 40 \right)$
C) Solve the equation $x ^ { 2 } - 13 x + 40 = ( A x + B ) \left( x ^ { 3 } - 7 x + 12 \right)$
D) Use long division to divide the numerator by the denominator

Question 29

Solve the problem.
Evaluate $| \mathrm { A } |$ by expanding across the third row. Show all your work.
$$A = \left[ \begin{array} { r r r r } - 1 & 1 & 0 & 2 \\- 5 & 2 & 4 & - 5 \\0 & 5 & 0 & - 1 \\1 & 2 & - 1 & 0\end{array} \right]$$

Question 30

Find the graph that matches the given system of equations.
$$\begin{array} { l } 3 x - y = 13 \\- 6 x + 2 y = - 26\end{array}$$

A)

B)

C)

D)

Question 31

Match the inequality with a graph.
$2 x+5 y>-4$

A)

B)


C)

D)

Question 32

Provide an appropriate response.
The partial fraction decomposition of an expression is $2 x + \frac { 4 } { 2 x - 7 } - \frac { 6 } { 2 x ^ { 2 } - 7 }$ . From this decomposition, what can you say about the original expression?

A) The numerator of the original expression was $2 x ^ { 2 } + 4 x - 6$ .
B) The numerator of the original expression was of a degree greater than the denominator of the original expression.
C) The denominator of the original expression contained a nonreducible quadratic factor.
D) The denominator of the original expression contained repeated linear factors.

Question 33

Provide an appropriate response.
You are graphing $2 x ^ { 2 } + 3 y ^ { 2 } > 5$ . You have drawn the graph of $2 x ^ { 2 } + 3 y ^ { 2 } = 5$ . What single point will allow you to determine the correct region?

A) $( 3,5 )$
B) $( 2,5 )$
C) The origin
D) $( 2,3 )$

Question 34

Find the graph that matches the given system of equations.
$$\begin{array} { l } 2 x + y = 8 \\y = 2 x + 0\end{array}$$

A)

B)

C)

D)

Question 35

Match the system of inequalities with a graph.
$$\begin{array} { l } 3 y - x \leq - 4 , \\y + 4 x \leq 6 \\- y \geq x - 3\end{array}$$

A)

B)
C)

D)

Question 36

Provide an appropriate response.
Which one of the following is a description of the graph of the inequality? $( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } < 4$

A) the region inside the circle with center $( 4 , - 7 )$ and radius 2
B) the region outside a circle with center $( 4 , - 7 )$ and radius 2
C) the region outside a circle with center $( - 4,7 )$ and radius 2
D) the region inside a circle with center $( - 4,7 )$ and radius 2

Question 37

Provide an appropriate response.
Which one of the following is a description of the graph of the inequality?
$$( x + 9 ) ^ { 2 } + ( y - 2 ) ^ { 2 } > 16$$

A) the region inside a circle with center $( - 9,2 )$ and radius 4
B) the region outside a circle with center $( 9 , - 2 )$ and radius 4
C) the region inside the circle with center $( 9 , - 2 )$ and radius 4
D) the region outside a circle with center $( - 9,2 )$ and radius 4

Question 38

Provide an appropriate response.
You are finding the partial fraction decomposition of the expression $\frac { x ^ { 2 } - 11 x + 24 } { \left( x ^ { 2 } - 10 x + 21 \right) ( x + 7 ) }$ . You should ...

A) Solve the equation $x ^ { 2 } - 11 x + 24 = A ( x + 7 ) + B ( x - 3 ) + C ( x + 7 )$
B) Use long division to divide the numerator by the denominator
C) Solve the equation $x ^ { 2 } - 11 x + 24 = A ( x - 7 ) + B ( x - 3 ) + C ( x + 7 )$
D) Solve the equation $x ^ { 2 } - 11 x + 24 = ( A x + B ) \left( x ^ { 2 } - 10 x + 21 \right) + C ( x + 7 )$

Question 39

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?

Question 40

Provide an appropriate answer.
$$\text { Let } A = \left[ \begin{array} { r r } - 1 & 3 \\- 3 & - 3\end{array} \right] \text { and } B = \left[ \begin{array} { r r } - 3 & 3 \\4 & - 4\end{array} \right] \text {. Show that } A ( A + B ) = A ^ { 2 } + A B \text {. Show all your work. }$$

Question 41

Provide an appropriate response.
Consider the linear equation in three variables $x - y + z = 3$ . Find a pair of linear equations that, when considered together with the given equation, will form a system having no solution.

Question 42

Given matrix A, find the indicated cofactor.
Find $\mathrm { A } _ { 34 }$ .
$$A = \left[ \begin{array} { r r r r } 2 & 0 & 4 & - 1 \\9 & 2 & 0 & 1 \\0 & - 7 & 4 & - 2 \\- 3 & 2 & 0 & 0\end{array} \right]$$

A) 0
B) 192
C) 96
D) $- 96$

Question 43

Decompose into partial fractions.
$$\frac { 7 x ^ { 2 } - 69 x + 18 } { \left( x ^ { 2 } + 3 x - 9 \right) ( x - 7 ) }$$

A) $\frac { 2 x } { x ^ { 2 } + 3 x - 9 } - \frac { 9 } { x - 7 }$
B) $\frac { 9 x } { x ^ { 2 } + 3 x - 9 } + \frac { 2 } { x - 7 }$
C) $\frac { 9 x } { x - 7 } - \frac { 2 } { x ^ { 2 } + 3 x - 9 }$
D) $\frac { 9 x } { x ^ { 2 } + 3 x - 9 } - \frac { 2 } { x - 7 }$

Question 44

Decompose into partial fractions.
$$\frac { 4 x ^ { 2 } - 2 x + 6 } { \left( x ^ { 2 } - 9 \right) ( x - 1 ) }$$

A) $\frac { 5 } { x - 3 } + \frac { 3 } { x + 3 } - \frac { 2 } { x - 1 }$
B) $\frac { 5 } { x ^ { 2 } - 9 } - \frac { 1 } { x - 1 }$
C) $\frac { 3 } { x - 3 } + \frac { 2 } { x + 3 } - \frac { 1 } { x - 1 }$
D) $\frac { 2 } { x - 3 } + \frac { 3 } { x + 3 } + \frac { 1 } { x - 1 }$

Question 45

Provide an appropriate response.
Consider the linear equation in three variables $x - y + z = 2$ . Find a pair of linear equations that, when considered together with the given equation, will form a system having infinitely many solutions.

Question 46

Solve the system.
$$\begin{array} { l } 2 x + 3 y + z = 5 \\4 x - 3 y - z = 13 \\3 x + y + 2 z = 2\end{array}$$

A) $( 3 , - 4,1 )$
B) $( 3,1 , - 4 )$
C) $( - 4,1,3 )$
D) No solution

Question 47

Solve the system of equations using Gaussian elimination or Gauss-Jordan elimination.
$$\begin{array} { l } 2 x + 6 y = 4 \\3 x + 9 y = 6\end{array}$$

A) $\left( \frac { 2 } { 3 } y , y \right)$
B) $( 2 - 3 y , y )$
C) $\left( \frac { 3 } { 2 } y , y \right)$
D) No solution

Question 48

Solve the problem.
An airline with two types of airplanes, $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ , has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip, airplane $\mathrm { P } _ { 1 }$ costs $\$ 10,000$ to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane $P _ { 2 }$ costs $\$ 8500$ to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

A) $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes
B) $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes
C) $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes
D) $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes

Question 49

Solve.
Jim wants to plan a meal with 82 grams of carbohydrates and 1260 calories. If green beans have 7 grams of carbohydrates and 30 calories per half cup serving and if french fried shrimp have 9
Grams of carbohydrates and 190 calories per three-ounce serving, how many servings of green
Beans and shrimp should he use?

A)7 half cups of beans and 9 three-ounce helpings of shrimp
B)9 half cups of beans and 7 three-ounce helpings of shrimp
C)4 half cups of beans and 6 three-ounce helpings of shrimp
D)6 half cups of beans and 4 three-ounce helpings of shrimp

Question 50

Solve the problem.
Evaluate $| \mathrm { A } |$ by expanding down the second column. Show all your work.
$$A = \left[ \begin{array} { r r r } 2 & - 5 & - 4 \\2 & 0 & - 2 \\5 & 1 & - 3\end{array} \right]$$

Question 51

Perform the matrix operation.
Given $A = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } - 2 & 0 \\ - 1 & 2 \end{array} \right]$ , find $A B$ .

A)
$\left[ \begin{array} { l l } 6 & - 1 \\ 4 & - 8 \end{array} \right]$
B)
$\left[ \begin{array} { l l } - 1 & 6 \\ - 8 & 4 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 2 & - 6 \\ - 2 & 1 \end{array} \right]$
D)
$\left[ \begin{array} { r r } 2 & 0 \\ - 3 & 4 \end{array} \right]$

Question 52

Find a system of inequalities with the given graph.

A) $y \leq - 2 x + 8$
B) $y \leq - 2 x + 8$
C) $y \leq - 2 x - 4$
D) $y \geq - 2 x + 8$ $y \geq 2 x + 8$ $y \geq 2 x - 4$ $y \geq 2 x - 4$ $y \leq 2 x - 4$

Question 53

Solve using the elimination method. Use a graphing calculator to check your answer.
$$\begin{array} { l } - 3 x + 4 y = 24 \\- 6 x = 22 - 8 y\end{array}$$

A) $( 3,5 )$
B) $( 4,9 )$
C) $( 4,5 )$
D) No solution

Question 54

Solve the problem.
Evaluate $| \mathrm { A } |$ by expanding across the second row. Show all your work.
$$A = \left[ \begin{array} { r r r } 1 & - 3 & - 4 \\0 & 2 & - 1 \\5 & 1 & - 3\end{array} \right]$$

Question 55

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 $0 = 7$ . What does this indicate to you about the system of equations?

Question 56

Solve the problem.
Evaluate $| \mathrm { A } |$ by expanding down the third column. Show all your work.
$$A = \left[ \begin{array} { r r r r } - 2 & 0 & 0 & 2 \\3 & 1 & 0 & 0 \\4 & 5 & 6 & 4 \\- 2 & 2 & - 1 & 0\end{array} \right]$$

Question 57

Solve the problem.
Evaluate $| \mathrm { A } |$ by expanding down the third column. Show all your work.
$$A = \left[ \begin{array} { r r r } - 5 & - 4 & - 2 \\1 & 4 & 3 \\- 5 & 1 & - 3\end{array} \right]$$

Question 58

Perform the matrix operation.
Given $A = \left[ \begin{array} { r r } 3 & - 7 \\ 9 & 4 \end{array} \right]$ and $I = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$ , find IA.

A)
$\left[ \begin{array} { r r } 9 & 4 \\ 3 & - 7 \end{array} \right]$
B)
$\left[ \begin{array} { r r } 0 & - 4 \\ 0 & 13 \end{array} \right]$
C)
$\left[ \begin{array} { r r } 3 & - 7 \\ 9 & 4 \end{array} \right]$
D)
$\left[ \begin{array} { r r } - 7 & 9 \\ 4 & 3 \end{array} \right]$

Question 59

Rewrite the expression using a determinant. Show all your work.
$$a x ^ { 2 } + b x$$

Question 60

Solve the problem.
The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the 3 years considered.

Fit a quadratic function $f ( x ) = a x ^ { 2 } + b x + c$ to the data. What is the value of $c$ ?

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

Question 61

$$\text { Let } u \text { represent } \frac { 1 } { x } \text {, } v \text { represent } \frac { 1 } { y } \text {, and } w \text { represent } \frac { 1 } { z } \text {. Solve first for } u , v \text {, and } w \text {. Then solve the system of equations. }$$
$$\begin{array} { l } \frac { 5 } { x } - \frac { 1 } { y } + \frac { 3 } { z } = - \frac { 25 } { 8 } \\- \frac { 1 } { x } - \frac { 8 } { y } + \frac { 7 } { z } = - \frac { 19 } { 8 } \\\frac { 4 } { x } - \frac { 1 } { y } - \frac { 1 } { z } = - \frac { 17 } { 8 }\end{array}$$

A) $( - 2,4,8 )$
B) $\left( - \frac { 1 } { 2 } , \frac { 1 } { 4 } , - \frac { 1 } { 8 } \right)$
C) $( - 2,4 , - 8 )$
D) $\left( - \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } \right)$

Question 62

Write the augmented matrix for the system.
$$\begin{array} { r } 5 x + 8 z = 37 \\8 y + 9 z = 7 \\4 x + 9 y + 3 z = 51\end{array}$$

A) $\left[ \begin{array} { l l | l } 5 & 0 & 8 \\ 0 & 8 & 9 \\ 4 & 9 & 3 \end{array} \right]$
B) $\left[ \begin{array} { l l r | r } 5 & 0 & 4 & 37 \\ 0 & 8 & 9 & 7 \\ 8 & 9 & 3 & 51 \end{array} \right]$
C) $\left[ \begin{array} { r r r | r } 5 & 0 & 8 & 37 \\ 0 & 8 & 9 & 7 \\ 4 & 9 & 3 & 51 \end{array} \right]$
D) $\left[ \begin{array} { r r r | r } 5 & 8 & 0 & 37 \\ 8 & 9 & 0 & 7 \\ 4 & 9 & 3 & 51 \end{array} \right]$

Question 63

Solve using the substitution method. Use a graphing calculator to check your answer.
$$\begin{array} { l } x + 4 y = 36 \\y = - 18 + 2 x\end{array}$$

A) $( 12,6 )$
B) $( - 10,12 )$
C) No solution
D) Infinitely many solutions

Question 64

Solve using Cramer's rule.
$$\begin{array} { c } x - 3 y = - 9 \\- 3 x - 4 y = 14\end{array}$$

A) $( - 7,2 )$
B) $( 6,2 )$
C) $( - 6,1 )$
D) No solution

Question 65

Graph the system of inequalities.
$$\begin{array} { l } y \leq x ^ { 2 } - 3 \\y \geq 1 - x ^ { 2 }\end{array}$$

A)

B)

C)

D)

Question 66

Solve the problem.
The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the 3 years considered.

Fit a quadratic function $f ( x ) = a x ^ { 2 } + b x + c$ to the data. What is the value of $b$ ?

A) $- 1.2$
B) $1.9$
C) $- 1.6$
D) $2.3$

Question 67

Find the product, if possible.
$$\left[ \begin{array} { r r r } 3 & 5 & 9 \\8 & - 7 & 6\end{array} \right] \left[ \begin{array} { r } - 9 \\6 \\- 5\end{array} \right]$$

A)
$$[ - 42 - 144 ]$$
B)
$$\left[ \begin{array} { r } - 42 \\- 144\end{array} \right]$$
C)
$$\left[ \begin{array} { r } - 144 \\- 42\end{array} \right]$$
D)
$$\left[ \begin{array} { r r r } 3 & 5 & 9 \\8 & - 7 & 6 \\- 9 & 6 & - 5\end{array} \right]$$

Question 68

Determine whether the system is consistent or inconsistent and whether the equations are dependent or independent.
$$\begin{array} { l } 6 = 3 x + 3 y \\6 y + 6 x = 11\end{array}$$

A) Consistent; independent
B) Inconsistent; dependent
C) Consistent; dependent
D) Inconsistent; independent

Question 69

Graph the inequality.
$|x+y| \geq 2$

A)

B)

C)

D)

Question 70

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

A)
$$\left[ \begin{array} { r } - 4 \\0 \\- 15\end{array} \right]$$
B)
[14 ]
C)
$[ - 19 ]$
D)
$\left[ \begin{array} { l l l } - 4 & 0 & - 15 \end{array} \right]$