Deck 4: Systems of Equations and Inequalities

Question 1

$\text { Sixteen }$ pounds of mixed birdseed sells for $7.2 per pound. The mixture is obtained from two kinds of birdseed, with one variety priced at $6.2 per pound and the other at $7.8 per pound. How many pounds of each variety of birdseed are used in the mixture?

A) $\text { Seven }$ of the $6.2 per pound birdseeds and ten of the $7.8 per pound birdseed.
B) $7.8 of the eleven per pound birdseeds and $30$ of the $\$ 7.8$ per pound birdseed.
C) $\text { Seven }$ of the $\$ 6.2$ per pound birdseeds and $\text { eleven }$ of the $\$ 7.8$ per pound birdseed.
D) $\text { Ten }$ of the $\$ 6.2$ per pound birdseeds and $\text { six}$ of the $\$ 7.8$ per pound birdseed.
E) $\text { Six }$ of the $\$ 6.2$ per pound birdseeds and $\text { ten }$ of the $\$ 7.8$ per pound birdseed.

Answer 1

$\text { Six }$ of the $\$ 6.2$ per pound birdseeds and $\text { ten }$ of the $\$ 7.8$ per pound birdseed.

Question 2

Graph the equations in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the number of solutions. $\left\{ \begin{array} { l } 5 x + y = - 8 \\- 5 x - y = 8\end{array} \right.$

A)The system is consistent and there is only one solution.
B)The system is inconsistent.

C)The system is consistent and there are infinite number of solutions.

D)The system is consistent and there is only one solution.
E)The system is consistent and there is only one solution.

Answer 2

The system is consistent and there are infinite number of solutions.

Question 3

Determine which of the following ordered pairs, $( 3 , - 7 )$ , $( 8,4 )$ , $( 8 , - 3 )$ , $( - 6 , - 7 )$ , or $( 9 , - 8 )$ is a solution to the system of equations below. $\left\{ \begin{array} { l } 6 x - 9 y = 81 \\- 2 x - y = 1\end{array} \right.$

A) $( 9 , - 8 )$
B) $( 8 , - 3 )$
C) $( 8,4 )$
D) $( 3 , - 7 )$
E) $( - 6 , - 7 )$

Answer 3

$( 3 , - 7 )$

Question 4

A total of $\$ 15,000$ is invested in two bonds that pay $5.5 \%$ and $6 \%$ simple interest. The annual interest is $\$ 865$ . How much is invested in the $6 \%$ bond?

A) $\$ 7,000$
B) $\$ 4,000$
C) $\$ 3,000$
D) $\$ 9,000$
E) $\$ 8,000$

Question 5

Find two positive integers that satisfy the requirements that the sum of the larger number and $\text { four }$ times the smaller number is $73$ and their difference is $3$ .

A) $34$ and $31$
B) $39$ and $36$
C) $37$ and $34$
D) $38$ and $35$
E) $17$ and $14$

Question 6

Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } 9 x + 5 y = - 66 \\5 x + 9 y = - 74\end{array} \right.$

A) (2,-5)
B) (-2,-8)
C) (-4,-6)
D) (0,-5)
E) (5,-5)

Question 7

Graph the equations below and determine the solution if it exists. $\left\{ \begin{array} { l } - 5 x - 5 y = - 10 \\10 x + 10 y = 20\end{array} \right.$

A) $\text { The system is consistent and has an finite number of solutions. }$
B) $\text { The system is consistent and has an infinite number of solutions. }$
C) $\text { The system is inconsistent. }$

D) $\text { The system is consistent and has one solution whose value for } x \text { is }-3 \text { and value for } y \text { is } 2 \text {. }$

E) $\text { The system is consistent and has one solution whose value for } x \text { is } 3 \text { and value for } y \text { is }-2$

Question 8

Determine whether the system is consistent or inconsistent.

A)The system is consistent.
B)The system is inconsistent.

Question 9

State the number of solutions of the system of linear equations $\left\{ \begin{array} { l } y = 3 x - 7 \\y = 7 x - 3\end{array} \right.$ without solving the system.

A)infinitely many solutions
B)one solution
C)no solution

Question 10

Graph the equations below and determine the solution if it exists. $\left\{ \begin{array} { l } 4.5 x + 4 y = 20.5 \\13.5 x + 12 y = 61.5\end{array} \right.$

A)The system is consistent and has an infinite number of solutions.
B)The system is consistent and has one solution whose value for x is 1 and the value for y is 4.5 .

C)The system is inconsistent.

D)The system is consistent and has one solution whose value for x is 4.5 and the value for y is 4 .

E)The system is consistent and has one solution whose value for x is 1 and the value for y is 2.25 .

Question 11

You are setting up a small business and have made an initial investment of $\$ 18,000$ . The unit cost of the guitar you are producing is $37.5 , and the selling price is $77.5 . How many guitars must you sell to break even?

A) 450 guitars
B) 600 guitars
C) 900 guitars
D) 1050 guitars
E) 1100guitars

Question 12

Determine which of the following ordered pairs, $( - 1.5 , - 1.5 )$ , $( - 0.5,9 )$ , $( 5.5 , - 3 )$ , $( 4.5,1 )$ , or $( 1.5,3.5 )$ is a solution to the system of equations below. $\left\{ \begin{array} { l } 6 x - 9 y = 4.5 \\x + 3 y = - 6\end{array} \right.$

A) $( 4.5,1 )$
B) $( - 1.5 , - 1.5 )$
C) $( 5.5 , - 3 )$
D) $( 1.5,3.5 )$
E) $( - 0.5,9 )$

Question 13

Use the graph of the equation $\left\{ \begin{array} { l } 4 x - y = 8 \\- 8 x + 2 y = - 12\end{array} \right.$ to determine whether the system has any solutions. Find any solutions that exist.

A) (2,0)
B) (4,8)
C) (0,2)
D)infinitely many solutions
E)no solution

Question 14

Use the graph of the equation $\left\{ \begin{array} { l } 8 x - 5 y = 14 \\3 x + 5 y = 8\end{array} \right.$ to determine whether the system has any solutions. Find any solutions that exist.

A) $\left( 2 , \frac { 2 } { 5 } \right)$
B) $\left( \frac { 2 } { 5 } , 2 \right)$
C) $( 2,2 )$
D)infinitely many solutions
E)no solution

Question 15

Graph the equations below and determine the solution if it exists. $\left\{ \begin{array} { l } - 2 x - y = 2 \\10 x + 5 y = 30\end{array} \right.$

A)The system is consistent and has one solution whose value for x is 3 and the value for y is 2 .

B)The system is inconsistent.

C)The system is consistent and has one solution whose value for x is 2 and the value for y is $25$ . $40$
D)The system is consistent and has one solution whose value for x is $47$ and the value for y is 5 .

E)The system is consistent and has an infinite number of solutions

Question 16

Use the graphs of the equations to determine whether the system has any solutions. Find any solutions that exist.
$\left\{ \begin{array} { l } - 5 x - 5 y = - 10 \\10 x + 10 y = 20\end{array} \right.$

A) $\text { The system is consistent and has one solution whose value for } x \text { is }-3 \text { and value for } y \text { is } 2$
B) $\text { The system is consistent and has an finite number of solutions. }$
C) $\text { The system is consistent and has an infinite number of solutions. }$
D) $\text { The system is consistent and has one solution whose value for } x \text { is } 3 \text { and value for } y \text { is }-2 \text {. }$
E) $\text { The system is inconsistent. }$

Question 17

Use the graphical method to solve the system of equations below. $\left\{ \begin{array} { l } y = x + 2 \\y = - x + 4\end{array} \right.$

A) (1,3)

B) (2,2.5)

C) (2.3 ,2.5)

D) (2.5 ,2.5)

E) (1,3)

Question 18

Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } 11 x - y = 20 \\- 8 x - 15 y = - 46\end{array} \right.$

A) (0,-6)
B) (8,-2)
C) (2,0)
D) (9,-1)
E) (2,2)

Question 19

Which of the following systems of equations below has the solution (1,4) ?
$\left\{ \begin{array} { l } 4 x - 6 y = - 20 \\3 x - 5 y = - 17\end{array} , \left\{ \begin{array} { l } 4 x - 6 y = 24 \\3 x - 5 y = 19\end{array} , \left\{ \begin{array} { l } 4 x - 6 y = 16 \\3 x - 5 y = 12\end{array} , \left\{ \begin{array} { l } 4 x - 6 y = 36 \\3 x - 5 y = 29\end{array} , \left\{ \begin{array} { l } 4 x - 6 y = 26 \\3 x - 5 y = 21\end{array} \right. \right. \right. \right. \right.$

A) $\left\{ \begin{array} { l } 4 x - 6 y = 36 \\3 x - 5 y = 29\end{array} \right.$
B) $\left\{ \begin{array} { l } 4 x - 6 y = 26 \\3 x - 5 y = 21\end{array} \right.$
C) $\left\{ \begin{array} { l } 4 x - 6 y = 24 \\3 x - 5 y = 19\end{array} \right.$
D) $\left\{ \begin{array} { l } 4 x - 6 y = - 20 \\3 x - 5 y = - 17\end{array} \right.$
E) $\left\{ \begin{array} { l } 4 x - 6 y = 16 \\3 x - 5 y = 12\end{array} \right.$

Question 20

Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } \frac { 3 } { 5 } x - \frac { 3 } { 2 } y = - \frac { 21 } { 10 } \\\\\frac { 5 } { 2 } x + \frac { 2 } { 7 } y = \frac { 335 } { 14 }\end{array} \right.$

A) (9,5)
B) (7,3)
C) (5,5)
D) (3,7)
E) (1,1)

Question 21

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } 4 r - 5 s = 5 \\2 r - 5 s = 15\end{array} \right.$

A) $r = 9 \text { and } s = - 9$
B) $r = - 1 \text { and } s = - 3$
C) $r = - 8 \text { and } s = - 8$
D) $r = - 5 \text { and } s = - 5$
E) $r = 2 \text { and } s = 4$

Question 22

Decide whether the system $\left\{ \begin{array} { c } 3 x - 6 y = 6 \\- 12 x + 24 y = - 24\end{array} \right.$ is consistent or inconsistent.

A)inconsistent
B)consistent

Question 23

An airplane flying into a headwind travels 2016 miles in 4 hours and 12 minutes. On the return flight, the same distance is traveled in 4 hours. Find the speed of the plane in still air, assuming that both the speed of the plane and the speed of the air remains constant throughout the round trip. Round to the nearest miles.

A)468 miles per hour
B)492 miles per hour
C)516 miles per hour
D)480 miles per hour
E)504 miles per hour

Question 24

A fundraising dinner was held on two consecutive nights. On the first night, 160 adult tickets and 237 children's tickets were sold, for a total of $3,478.25. On the second night, 193 adult tickets and 314 children's tickets were sold, for a total of $4,399.5. Find the price of each type of ticket.

A)The adult tickets were $13 and the children's tickets were $7.25.
B)The adult tickets were $11 and the children's tickets were $6.5.
C)The adult tickets were $14.25 and the children's tickets were $7.25.
D)The adult tickets were $11 and the children's tickets were $7.25.
E)The adult tickets were $14.25 and the children's tickets were $6.5.

Question 25

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - 2 x + y = 7 \\2 x - 5 y = - 27\end{array} \right.$

A) $x = - 3 \text { and } y = - 2$
B) $x = - 1 \text { and } y = 5$
C) $x = - 1 \text { and } y = - 2$
D) $x = - 1 \text { and } y = 0$
E)infinitely many solutions

Question 26

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { l } - 4 x + y = - 4 \\12 x - 3 y = 3\end{array} \right.$

A) $x = 8 \text { and } y = - 8$
B) $x = 1 \text { and } y = 0$
C) $x = 9 \text { and } y = 4$
D) $x = - 4 \text { and } y = 0$
E)no solution

Question 27

Decide whether the system $\left\{ \begin{array} { c } 5 x - 6 y = 4 \\- 15 x + 18 y = 14\end{array} \right.$ is consistent or inconsistent.

A)inconsistent
B)consistent

Question 28

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } x + y = - 6 \\- x + 2 y = - 3\end{array} \right.$

A) (-2,-4)
B) (5,0)
C) (-3,5)
D) (-2,-3)
E) (-3,-3)

Question 29

Find two positive integers that satisfy the requirements that the difference of four times the smaller number and the larger number is 14 and the sum of the smaller number and four times the larger number is 335 .

A) 23 and 110
B) 23 and 78
C) 31 and $78$
D) $31$ and $110$
E) $31$ and $62$

Question 30

How many liters of a 34% alcohol solution must be mixed with 84% solution to obtain 16 liters of a 71.5% solution?

A)2 liters of a 34% alcohol solution and 14 liters of 84% alcohol solution is required.
B)10 liters of a 34% alcohol solution and 6 liters of 84% alcohol solution is required.
C)12 liters of a 34% alcohol solution and 4 liters of 84% alcohol solution is required.
D)4 liters of a 34% alcohol solution and 12 liters of 84% alcohol solution is required.
E)6 liters of a 34% alcohol solution and 10 liters of 84% alcohol solution is required.

Question 31

Solve the system of linear equations below by any convenient method. $\left\{ \begin{array} { c } 2 x - 2 y = 2 \\- 2 x + y = - 2\end{array} \right.$

A) (-2,-1)
B) (-1,0)
C) (1,0)
D) (3,-1)
E) (2,1)

Question 32

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { l } 9 x + 4 y = - 108 \\- 7 x + 2 y = 38\end{array} \right.$

A) (1,7)
B) (-8,-9)
C) (1,0)
D) (-2,2)
E) (1,-7)

Question 33

To open a small business, you need an initial investment of $150,000. Each week your costs will be about $8,850. Your projected weekly revenue is $9,700. How many weeks will it take to break even? Round to the nearest number of weeks.

A)17 weeks
B)182 weeks
C)177 weeks
D)16 weeks
E)179 weeks

Question 34

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - 9 u - 8 v = - 8 \\9 u + 2 v = 2\end{array} \right.$

A) $u = 8 \text { and } v = - 2$
B) $u = - 3 \text { and } v = 5$
C) $u = - 6 \text { and } v = 7$
D) $u = 0 \text { and } v = 1$
E)no solution

Question 35

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - x + 2 y = 3 \\- 2 x + 4 y = 8\end{array} \right.$

A) $x = 9 \text { and } y = - 8$
B) $x = - 5 \text { and } y = - 1$
C) $x = 2 \text { and } y = 6$
D) $x = - 4 \text { and } y = 8$
E)Infinitely many solutions

Question 36

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } \frac { 5 } { 6 } b - \frac { 11 } { 12 } m = - \frac { 11 } { 18 } \\\\- \frac { 1 } { 2 } b + \frac { 3 } { 4 } m = \frac { 2 } { 3 }\end{array} \right.$

A) $b = - \frac { 1 } { 4 } \text { and } m = \frac { 7 } { 4 }$
B) $b = \frac { 11 } { 12 } \text { and } m = \frac { 3 } { 2 }$
C) $b = - \frac { 1 } { 12 } \text { and } m = \frac { 1 } { 3 }$
D) $b = - \frac { 11 } { 12 } \text { and } m = \frac { 1 } { 2 }$
E) $b = \frac { 5 } { 4 } \text { and } m = - \frac { 3 } { 2 }$

Question 37

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { l } \frac { 1 } { 2 } x + \frac { 1 } { 3 } y = \frac { 1 } { 9 } \\\\3 x - 4 y = - \frac { 13 } { 3 }\end{array} \right.$

A) $\left( \frac { 1 } { 3 } , \frac { 5 } { 6 } \right)$
B) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 2 } \right)$
C) $\left( \frac { 1 } { 3 } , - \frac { 5 } { 6 } \right)$
D) $\left( - \frac { 1 } { 3 } , \frac { 5 } { 6 } \right)$
E) $\left( - \frac { 4 } { 3 } , - \frac { 1 } { 3 } \right)$

Question 38

Solve the system of linear equations by the method of elimination and identify the grey line in the graph with its linear equation. $\left\{ \begin{array} { c } 6 x - 6 y = - 48 \\x + 4 y = 12\end{array} \right.$

A)The solution is (-4,4) and the equation of the grey line is $6 x - 6 y = - 48$ .
B)The solution is (-4,4) and the equation of the grey line is $x + 4 y = - 20$ .
C)The solution is (-4,4) and the equation of the grey line is $6 x - 6 y = - 48$ ..
D)The solution is $( 4 , - 4 )$ and the equation of the grey line is $6 x - 6 y = - 48$ .
E)The solution is $( 4 , - 4 )$ and the equation of the grey line is $6 x - 6 y = - 48$ ..

Question 39

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } \frac { 5 } { 6 } x - \frac { 11 } { 12 } y = - \frac { 11 } { 18 } \\\\- \frac { 1 } { 2 } x + \frac { 3 } { 4 } y = \frac { 2 } { 3 }\end{array} \right.$

A) $\left( - \frac { 11 } { 12 } , \frac { 1 } { 2 } \right)$
B) $\left( \frac { 11 } { 12 } , \frac { 3 } { 2 } \right)$
C) $\left( - \frac { 1 } { 12 } , \frac { 1 } { 3 } \right)$
D) $\left( - \frac { 1 } { 4 } , \frac { 7 } { 4 } \right)$
E) $\left( \frac { 5 } { 4 } , - \frac { 3 } { 2 } \right)$

Question 40

Determine the value of k such that the system of linear equations below is inconsistent. $\left\{ \begin{array} { c } 7 x + 18 y = - 3 \\5 x + k y = 3\end{array} \right.$

A) $k = - \frac { 60 } { 7 }$
B) $k = - \frac { 55 } { 6 }$
C) $k = \frac { 90 } { 7 }$
D) $k = \frac { 99 } { 4 }$
E) $k = \frac { 40 } { 9 }$

Question 41

The slope and y -intercept of the line $y = m x + b$ that best fits the three noncollinear points $( 2,2 ) , ( 3,3 )$ and $( 5,4 )$ are given by the solution of the following system of linear equations. $\left\{ \begin{array} { c } 77 m + 12 b = 46 \\12 m + 5 b = 8\end{array} \right.$ Solve the system and find the equation of the best-fitting line. Round parameters to the nearest thousandth.

A) $y = 0.415 x + 1.188$
B) $y = 0.412 x - 1.188$
C) $y = 0.417 x + 1.188$
D) $y = 0.556 x + 0.266$
E) $y = 0.42 x + 1.188$

Question 42

Form the coefficient matrix for the system of linear equations below.
$\left\{ \begin{array} { r } 6 x + 3 y + 7 z = 0 \\x + 6 y + z = 7 \\8 x + 9 y + 7 z = - 9\end{array} \right.$

A) $\left[ \begin{array} { c c c c } 6 & 3 & 7 & 0 \\1 & 6 & 1 & 7 \\8 & 9 & 7 & - 9\end{array} \right]$
B) $\left[ \begin{array} { l l l } 6 & 3 & 7 \\1 & 6 & 1 \\8 & 9 & 7\end{array} \right]$
C) $\left[ \begin{array} { c c c } 6 & 1 & 8 \\3 & 6 & 9 \\7 & 1 & 7 \\0 & 7 & - 9\end{array} \right]$
D) $\left[ \begin{array} { l l l } 6 & 1 & 8 \\3 & 6 & 9 \\7 & 1 & 7\end{array} \right]$
E) $\left[ \begin{array} { c } 0 \\7 \\- 9\end{array} \right]$

Question 43

Write the system of linear equations represented by the augmented matrix below. Use variables x , y , z , v , and w, if necessary.
$\left[ \begin{array} { c c c : c } - 9 & - 5 & - 7 & 9 \\2 & - 6 & - 6 & - 8 \\5 & - 2 & - 9 & 0\end{array} \right]$

A) $\left\{ \begin{array} { c } - 9 x - 7 z = 2 \\2 x - 6 z = - 14 \\5 x - 9 z = - 9\end{array} \right.$
B) $\left\{ \begin{array} { c } - 9 x - 5 y - 7 z = 9 \\2 x - 6 y - 6 z = - 8 \\5 x - 2 y - 9 z = 0\end{array} \right.$
C) $\left\{ \begin{array} { r } - 9 x - 5 y - 7 z = 0 \\2 x - 6 y - 6 z = 0 \\5 x - 2 y - 9 z = 0\end{array} \right.$
D) $\left\{ \begin{array} { c } - 9 x - 5 y = 2 \\2 x - 6 y = - 14 \\5 x - 2 y = - 9\end{array} \right.$
E) $\left\{ \begin{array} { r } - 9 x - 5 y - 7 z + 9 v = 0 \\2 x - 6 y - 6 z - 8 v = 0 \\5 x - 2 y - 9 z + 0 v = 0\end{array} \right.$

Question 44

Determine which ordered triple below is a solution of the system of linear equations. $\left\{ \begin{array} { r } 3 x - 3 y - 3 z = 3 \\x - 3 y - 3 z = 7 \\2 x - 3 y - 3 z = 5\end{array} \right.$
$( - 4,1 , - 2 ) , ( 1 , - 4 , - 2 ) , ( - 4 , - 2,1 ) , ( - 2,1 , - 4 )$ or $( 1 , - 2 , - 4 )$

A) $( 1 , - 2 , - 4 )$
B) $( 1 , - 4 , - 2 )$
C) $( - 2,1 , - 4 )$
D) $( - 4 , - 2,1 )$
E) $( - 4,1 , - 2 )$

Question 45

The sum of the measures of two angles of a triangle is the measure of the third angle. The measure of the second angle is than the measure of the third angle. Find the measures of the three angles.

A)The first angle is , the second angle is , and the third angle is .
B)The first angle is , the second angle is , and the third angle is .
C)The first angle is , the second angle is , and the third angle is .
D)The first angle is , the second angle is , and the third angle is .
E)The first angle is , the second angle is , and the third angle is .

Question 46

Form the augmented matrix for the system of linear equations below. $\left\{ \begin{array} { r } 6 x + y + z = 2 \\4 x + 4 y + 4 z = 9 \\8 x + y + 4 z = - 7\end{array} \right.$

A) $\left[ \begin{array} { c } 2 \\9 \\- 7\end{array} \right]$
B) $\left[ \begin{array} { l l l } 6 & 4 & 8 \\1 & 4 & 1 \\1 & 4 & 4\end{array} \right]$
C) $\left[ \begin{array} { c c c c } 6 & 1 & 1 & 2 \\4 & 4 & 4 & 9 \\8 & 1 & 4 & - 7\end{array} \right]$
D) $\left[ \begin{array} { l l l } 6 & 1 & 1 \\4 & 4 & 4 \\8 & 1 & 4\end{array} \right]$
E) $\left[ \begin{array} { c c c } 6 & 4 & 8 \\1 & 4 & 1 \\1 & 4 & 4 \\2 & 9 & - 7\end{array} \right]$

Question 47

Determine the order of the given matrix below.
$\left[ \begin{array} { c c c } - 4 & 4 & 0 \\2 & - 5 & 5 \\- 5 & 0 & - 3\end{array} \right]$

A) 1
B) 3×1
C) 3×3
D) 3
E) 1×3

Question 48

Find a system of linear equations below that has the point $( - 1,0,4 )$ as a solution. $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 12 \\2 x - 6 y + 8 z = 30 \\7 x - 3 y + 6 z = 17\end{array} \right.$ , $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 30 \\2 x - 6 y + 8 z = 17 \\7 x - 3 y + 6 z = 12\end{array} \right.$ , $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 17 \\2 x - 6 y + 8 z = 12 \\7 x - 3 y + 6 z = 30\end{array} \right.$ , $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 12 \\2 x - 6 y + 8 z = 17 \\7 x - 3 y + 6 z = 30\end{array} \right.$ or $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 30 \\2 x - 6 y + 8 z = 12 \\7 x - 3 y + 6 z = 17\end{array} \right.$

A) $\left\{ \begin{array} { c } 4 x - 2 y + 4 = 17 \\2 x - 6 y + 8 z = 12 \\7 x - 3 y + 6 z = 30\end{array} \right.$
B) $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 12 \\2 x - 6 y + 8 z = 30 \\7 x - 3 y + 6 z = 17\end{array} \right.$
C) $\left\{ \begin{array} { l } 4 x - 2 y + 4 z = 30 \\2 x - 6 y + 8 z = 12 \\7 x - 3 y + 6 z = 17\end{array} \right.$
D) $\left\{ \begin{array} { c } 4 x - 2 y + 4 = 12 \\2 x - 6 y + 8 z = 17 \\7 x - 3 y + 6 z = 30\end{array} \right.$
E) $\left\{ \begin{array} { c } 4 x - 2 y + 4 = 30 \\2 x - 6 y + 8 z = 17 \\7 x - 3 y + 6 z = 12\end{array} \right.$

Question 49

Solve the system of linear equations below. $\left\{ \begin{array} { c } 2 x - 3 y - 6 z = - 34 \\4 x - y + 5 z = 17 \\3 x + 6 z = 24\end{array} \right.$

A) $( 5,0 , - 2 )$
B) $( - 2,0,5 )$
C) $( 5 , - 2,0 )$
D) $( 0,5 , - 2 )$
E) $( - 2,5,0 )$

Question 50

A coffee manufacturer sells a 18 -pound package of coffee that consists of three flavors of coffee. Vanilla flavored coffee costs $2.25 per pound, Hazelnut flavored coffee costs $2.5 per pound, and French Roast flavored coffee costs $3 per pound. The package contains the same amount of Hazelnut coffee as French Roast coffee. The cost of the 18 -pound package is $46.5 . How many pounds of $\text { Vanilla }$ flavored coffee are there in the package?

A)There are $6$ pounds of $\text { Vanilla }$ flavored coffee in the package.
B)There are $4$ $\text { pounds }$ of $\text { Vanilla }$ flavored coffee in the package.
C)There are $7$ $\text { pounds }$ of $\text { Vanilla }$ flavored coffee in the package.
D)There are 5 $\text { pounds }$ of $\text { Vanilla }$ flavored coffee in the package.
E)There are $3$ $\text { pounds }$ of $\text { Vanilla }$ flavored coffee in the package.

Question 51

Use back-substitution to solve the system of linear equations below. $\left\{ \begin{aligned}x - y - z & = 3 \\2 y - z & = - 12 \\z & = - 2\end{aligned} \right.$

A) $( 6,7,2 )$
B) $( 6 , - 7 , - 2 )$
C) $( - 7,6 , - 2 )$
D) $( 7,6,2 )$
E) $( - 6 , - 7 , - 2 )$

Question 52

Find the position equation $s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ for an object that has distance $s = 24 \text { feet }$ at t=1 second, $s = 47 \text { feet }$ at t=2 seconds, and $s = 78 \mathrm { feet }$ at $t = 3$ seconds.

A) $s = 9 t ^ { 2 } + 4 t + 11$
B) $s = 4 t ^ { 2 } + 11 t + 9$
C) $s = 3 t ^ { 2 } + 9 t + 4$
D) $s = 11 t ^ { 2 } + 4 t + 9$
E) $s = 4 t ^ { 2 } + 3 t + 11$

Question 53

Solve the system of linear equations below. $\left\{ \begin{aligned}2 x + 3 y & = - 18 \\z & = 4 \\x - 8 y - z & = 6\end{aligned} \right.$

A) $( 4 , - 6 , - 2 )$
B) $( - 6,4 , - 2 )$
C) $( - 2,4 , - 6 )$
D) $( 4,0,1 )$
E) $( - 6 , - 2,4 )$

Question 54

Write the system of linear equations represented by the augmented matrix below. Use variables x , y , z , v , and w.
$\left[ \begin{array} { c c c c : c } - 4 & - 6 & - 7 & 7 & - 9 \\- 4 & - 3 & - 3 & - 7 & 9 \\- 2 & - 1 & - 5 & 7 & - 2 \\- 7 & - 7 & - 7 & - 7 & 3\end{array} \right]$

A) $\left\{ \begin{array} { l } - 4 x - 6 y - 7 z + 7 w = - 9 \\- 4 x - 3 y - 3 z - 7 w = 9 \\- 2 x - y - 5 z + 7 w = - 2 \\- 7 x - 7 y - 7 z - 7 w = 3\end{array} \right.$
B) $\left\{ \begin{array} { l } - 4 x - 6 y - 7 z + 7 w = - 16 \\- 4 x - 3 y - 3 z - 7 w = 6 \\- 2 x - y - 5 z + 7 w = - 7 \\- 7 x - 7 y - 7 z - 7 w = 10\end{array} \right.$
C) $\left\{ \begin{array} { r } - 4 x - 6 y - 7 z + 7 w - 9 = 0 \\- 4 x - 3 y - 3 z - 7 w + 9 = 0 \\- 2 x - y - 5 z + 7 w - 2 = 0 \\- 7 x - 7 y - 7 z - 7 w + 3 = 0\end{array} \right.$
D) $\left\{ \begin{array} { l } - 4 x - 6 y - 7 z = - 9 \\- 4 x - 3 y - 3 z = 9 \\- 2 x - y - 5 z = - 2 \\- 7 x - 7 y - 7 z = 3\end{array} \right.$
E) $\left\{ \begin{array} { r } - 4 x - 6 y - 7 z + 7 w = 0 \\- 4 x - 3 y - 3 z - 7 w = 0 \\- 2 x - y - 5 z + 7 w = 0 \\- 7 x - 7 y - 7 z - 7 w = 0\end{array} \right.$

Question 55

Solve the system of linear equations below. $\left\{ \begin{array} { r } x - 3 y + 4 z = 13 \\3 x - 2 z = - 8 \\2 x - 5 y - z = - 9\end{array} \right.$

A) $( 4,0,1 )$
B) $( 4,1,0 )$
C) $( 0,4,1 )$
D) $( 1,0,4 )$
E) $( 0,1,4 )$

Question 56

You receive a total of 1,350 a year in interest from three investments. The interest rates for the three investments are $5 \% , 6.5 \%$ and 7.5% . The 6.5% investment is half of the 5% investment, and the 7.5% investment is 3000 less than the 5% investment. What is the amount of the $6.5 \%$ investment?

A) $\$ 5,000$
B) $\$ 6,000$
C) $\$ 10,000$
D) $\$ 7,000$
E) $\$ 3,000$

Question 57

Determine the order of the given matrix below.
$\left[ \begin{array} { c } - 4 \\- 1 \\0 \\4\end{array} \right]$

A) 1×4
B) 4
C) 4×1
D) 6
E)3

Question 58

Solve the system of linear equations below. $\left\{ \begin{array} { r } x + y - 2 z = - 8 \\x - y - 3 z = - 5 \\2 x + 4 z = - 4\end{array} \right.$

A) $\left\{ \begin{array} { l } 3 x - y > 4 \\x + 6 y \leq 9\end{array} \right.$
B) $( - 4 , - 2,1 )$
C) $( - 4,1 , - 2 )$
D) $( - 2 , - 4,1 )$
E) $( - 2,1 , - 4 )$

Question 59

Solve the system of linear equations below. $\left\{ \begin{array} { c } 4 x + 3 y + z = 6 \\5 x - 4 y - 2 z = - 4 \\2 x + 5 y - 3 z = 38\end{array} \right.$

A) $( 4 , - 6,0 )$
B) $( 0 , - 6,4 )$
C) $( 0,4 , - 6 )$
D) $( - 6,4,0 )$
E) $( - 6,0,4 )$

Question 60

14 pounds of mixed nuts sells for $6.21 per pound. The mixture is obtained from two kinds of nuts, peanuts priced at $5.5 per pound and cashews at $6.5 per pound. How many pounds of each variety of nut are used in the mixture?

A)10 pounds of peanuts and 4 pounds of cashews are used in the mixture.
B)6 pounds of peanuts and 4 pounds of cashews are used in the mixture
C)4 pounds of peanuts and 6 pounds of cashews are used in the mixture.
D)4 pounds of peanuts and 11 pounds of cashews are used in the mixture.
E)4 pounds of peanuts and 10 pounds of cashews are used in the mixture.

Question 61

Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest.
$\left[ \begin{array} { c c c } - 1 & 0 & - 1 \\0 & 1 & 0 \\1 & 2 & - 2\end{array} \right]$

A) 3
B) $- 1$
C) $- 3$
D) $2$
E) $1$

Question 62

Find the determinant of the matrix below. $\left[ \begin{array} { c c } 5 & - 5 \\1 & 1\end{array} \right]$

A) 25
B) 10
C) -20
D) -10
E) -25

Question 63

Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest. Round your answer to three decimals places.
$\left[ \begin{array} { c c c } 0.3 & 0.4 & - 0.3 \\- 0.4 & 0.2 & - 0.4 \\- 0.7 & 0.2 & - 0.1\end{array} \right]$

A) 0.117
B) 0.096
C) 0.25
D) 0.072
E) 0.075

Question 64

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. Use variables x , y , and z . $\left[ \begin{array} { l l l : l } 1 & 2 & 1 & - 5 \\0 & 1 & 2 & - 6 \\0 & 0 & 1 & - 1\end{array} \right]$

A) $\left\{ \begin{array} { r } x + 2 y + z = - 5 \\y + 2 z = - 6 \\z = - 1\end{array} \right.$
$( - 4,4 , - 1 )$
B) $\left\{ \begin{array} { r } x + 2 y + z = - 5 \\y + 2 z = - 6 \\z = - 1\end{array} \right.$
$( - 4,4 , - 1 )$ .
C) $\left\{ \begin{array} { r } x + 2 y + z = - 5 \\y + 2 z = - 6 \\z = - 1\end{array} \right.$
$( 4,4 , - 1 )$
D) $\left\{ \begin{array} { r } x + 2 y + 2 z = - 5 \\y + z = - 6 \\z = - 1\end{array} \right.$
$( 4,4 , - 1 )$
E) $\left\{ \begin{array} { r } x + 2 y + z = - 5 \\y + 2 z = - 6 \\z = - 1\end{array} \right.$
$( 4 , - 4 , - 1 )$ .

Question 65

Describe the elementary row operation used to transform the first matrix $\left[ \begin{array} { c c c } 3 & 5 & 6 \\0 & 2 & 2 \\0 & - 3 & - 4\end{array} \right]$ into the second matrix $\left[ \begin{array} { l l l } 3 & 5 & 6 \\0 & 2 & 2 \\0 & 3 & 2\end{array} \right]$ .

A)Add 4 times the third row to the second row.
B)Add 4 times the second row to the third row.
C)Add 3 times the third row to the second row.
D)Add 3 times the second row to the third row.
E)Add 2 times the second row to the third row.

Question 66

Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest.
$\left[ \begin{array} { c c c } 1 & 0 & 4 \\1 & - 4 & 2 \\- 1 & 2 & 3\end{array} \right]$

A) -5
B) 11
C) 8
D) -24
E) -4

Question 67

Solve for x in the matrix below by using elementary row operations to form a row-equivalent matrix.
$\left[ \begin{array} { c c c c } 6 & 3 & - 2 & 0 \\- 2 & - 3 & - 3 & - 9 \\4 & 0 & - 3 & 8\end{array} \right]$
$\left[ \begin{array} { c c c c } 6 & 3 & - 2 & 0 \\- 2 & - 3 & - 3 & - 9 \\- 2 & - 3 & x & y\end{array} \right]$

A) -6
B) -1
C) -15
D) 15
E) 10

Question 68

Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest.
$\left[ \begin{array} { c c c } 6 & 8 & - 5 \\- 7 & 3 & 8 \\- 8 & - 2 & 5\end{array} \right]$

A) -158
B) -356
C) 266
D) -236
E) -266

Question 69

Use matrices to solve the system of linear equations below.
$\left\{ \begin{array} { r } 4 x - 2 y - 3 z = 1 \\x - z = - 1 \\5 x - 3 y - 4 z = 2\end{array} \right.$

A) $( 0 , - 2,1 )$
B) $( 0,1 , - 2 )$
C) $( - 2,0,1 )$
D) $( 1,0 , - 2 )$
E) $( 1 , - 2,0 )$

Question 70

Evaluate the determinant of the matrix.
$\left[ \begin{array} { c c c } - 0.7 & - 0.6 & - 0.9 \\- 0.9 & - 0.5 & 0.5 \\1 & - 9 & 3\end{array} \right]$

A) -11.94
B) -11.76
C) -11.52
D) -12.42
E) -13.284

Question 71

Use Cramer s Rule to solve the system of linear equations below. $\left\{ \begin{array} { l } x - 4 y - 2 z = 13 \\3 x + 2 y - z = - 12 \\4 x - 3 y + 2 z = 10\end{array} \right.$

A) $( - 6,0,3 )$
B) $( - 1 , - 4,1 )$
C) $( - 5,4,3 )$
D) $( 5,1,0 )$
E) $( - 3,4 , - 3 )$

Question 72

A corporation borrowed $\$ 1,330,000$ to expand its line of clothing. Some of the money was borrowed at $8 \%$ some at $11 \%$ and the remainder at $14 \%$ . The annual interest payment to the lenders was $\$ 127,400$ . The amount borrowed at $8 \%$ was $4$ times the amount borrowed at $14 \%$ . How much was borrowed at the $14 \%$ rate?

A) $\$ 210,000$
B) $\$ 430,000$
C) $\$ 840,000$
D) $\$ 160,000$
E) $\$ 280,000$

Question 73

Use matrices to solve the system of linear equations below. $\left\{ \begin{array} { l } 2 x - y = - 3 \\x - 2 y = 6\end{array} \right.$

A) (-2,-5)
B) (-4,-5)
C) (-5,4)
D) (-2,-1)
E) (2,4)

Question 74

Use matrices to solve the system of linear equations below.
$\left\{ \begin{array} { l } 3 x + 3 y = 18 \\2 x + 3 y = 13 \\4 x + z = 24\end{array} \right.$

A) $( 4,1,5 )$
B) $( 5,1,4 )$
C) $( 5,4,1 )$
D) $( 4,5,1 )$
E) $( 1,5,4 )$

Question 75

Use Cramer s Rule to solve the system of linear equations below. $\left\{ \begin{array} { l } - 2 x + 4 y = 18 \\x + 2 y = 11\end{array} \right.$

A) (-2,-2)
B) (-2,3)
C) (0,4)
D) (3,-1)
E) (1,5)

Question 76

Convert the matrix to row-echelon form. $\left[ \begin{array} { c c c c } 1 & - 3 & 2 & 3 \\- 2 & 9 & - 6 & - 6 \\- 1 & 6 & - 5 & 0\end{array} \right]$

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

Question 77

The sum of three positive numbers is $65$ . The second number is 5 $\text { less }$ than the first, and the third is 5 times the first. What is the $\text { third }$ number?

A) $10$
B) $50$
C) $12$
D) $20$
E) $24$

Question 78

Use matrices to solve the system of linear equations below.
$\left\{ \begin{array} { l } - 2 x - y = - 13 \\x + 2 y = 11\end{array} \right.$

A) (-2,2)
B) (3,-1)
C) (5,3)
D) (5,-2)
E) (-1,4)

Question 79

A grocer wishes to mix three kinds of nuts to obtain $40$ pounds of a mixture priced at $\$ 4.45$ per pound. Peanuts cost $\$ 3$ per pound, almonds cost $\$ 5$ per pound, and pistachios cost $\$ 5.5$ per pound. Half of the mixture is composed of peanuts and almonds. How many pounds of $\text { peanuts }$ should the grocer use?

A) $16$ pounds
B) $20$ pounds
C) $28$ pounds
D) $15$ pounds
E) $30$ pounds

Question 80

Use matrices to solve the system of linear equations below.
$\left\{ \begin{array} { r } x - 2 z = 7 \\4 x - 4 y - z = 9 \\x - 2 y - 3 z = 20\end{array} \right.$

A) $( - 3 , - 4 , - 5 )$
B) $( - 5 , - 4 , - 3 )$
C) $( - 3 , - 5 , - 4 )$
D) $( - 4 , - 5 , - 3 )$
E) $( 1,5,4 )$