Deck 6: Systems of Equations and Inequalities

Question 1

Solve the system of equations below. $\left\{ \begin{array} { l } y = x ^ { 2 } - 3 x - 6 \\y = - 2 x - 6\end{array} \right.$

A) $( 0 , - 6 )$ and $( 4 , - 14 )$
B) $( 0 , - 6 )$ and $( 1 , - 8 )$
C) $( - 3,0 )$ and $( 1 , - 8 )$
D) $( - 3,0 )$ and $( 0 , - 6 )$
E) $( - 3,0 )$ and $( 4 , - 14 )$

Answer 1

$( 0 , - 6 )$ and $( 1 , - 8 )$

Question 2

You invest $2500 in a fishing lure business. A lure costs $1.47 to produce and will be sold for $4.97. How many lures must you sell to break even?

A)1198 lures
B)715 lures
C)1701 lures
D)389 lures
E)504 lures

Answer 2

715 lures

Question 3

Solve the system graphically. $\left\{ \begin{array} { l } 3 x + 4 y = 4 \\y ^ { 2 } - x ^ { 2 } = 9\end{array} \right.$

A) $\approx ( - 1.8,2.4 ) , \approx ( 2.8 , - 1.1 )$
B) $\approx ( - 2.8 , - 1.1 ) , \approx ( 1.8,2.4 )$
C) $\text { no solution }$
D) $\approx ( - 3.4,4.5 )$
E) $\text { no solution }$

Answer 3

$\text { no solution }$

Question 4

Determine which ordered pair is a solution of the system. $$\left\{ \begin{array} { l } 3 x - 4 y = 8 \\4 x - 7 y = 19\end{array} \right.$$

A)(-5, -4)
B)(4, -5)
C)(2, 7)
D)(-4, -5)
E)(-7, 2)

Question 5

Solve the system by the method of substitution. $\left\{ \begin{array} { c } x - y = - 7 \\x ^ { 2 } - y = - 5\end{array} \right.$

A)(2, -5), (-1, -8)
B)(2, 9), (-1, 6)
C)(2, 5), (-1, 2)
D)(2, 5)
E)no real solution

Question 6

A total of $50,000 is invested in two funds paying 8% and 13% simple interest. The total yearly interest is $5100. How much is invested at the 8% rate?

A)$22,000
B)$26,000
C)$19,000
D)$28,000
E)$30,000

Question 7

Solve the system by the method of substitution. $\left\{ \begin{array} { l } - \frac { 1 } { 2 } x + \frac { 1 } { 8 } y = 1 \\\\- \frac { 1 } { 3 } x + \frac { 1 } { 6 } y = 1\end{array} \right.$

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

Question 8

Solve the system below by method of substitution, if possible. $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 25 \\- x + y = 12\end{array} \right.$

A)no solution
B) $( 5,17 )$
C) $( 2,14 )$
D) $( 5,17 )$ , $( 2,14 )$
E) $( 1,13 )$

Question 9

Find the sales necessary to break even (R - C = 0) for the cost C of producing x units and the revenue R obtained by selling x units. (Round to the nearest whole unit.) $C = 8.8 \sqrt { x } + 2000 \quad R = 7.9 x$

A)236 units
B)236 units or 272 units
C)272 units
D)259 units
E)no real solution

Question 10

Solve each system of equations by the substitution method. $7 x - y = 6$ $- x - 7 y = 3$

A) $\left( - \frac { 39 } { 50 } , \frac { 27 } { 50 } \right)$
B) $\left( \frac { 9 } { 10 } , - \frac { 3 } { 10 } \right)$
C) $\left( \frac { 39 } { 50 } , - \frac { 27 } { 50 } \right)$
D) $\left( \frac { 9 } { 16 } , - \frac { 13 } { 16 } \right)$
E)no solution

Question 11

Determine whether the system of equations below has one solution, two solutions, or no solution. $$\left\{ \begin{array} { l } y = x ^ { 2 } - 16 x + 11 \\y = - x ^ { 2 } - 8 x - 11\end{array} \right.$$

A)two solutions
B)no solution
C)one solution

Question 12

Solve the system of equations below by the substitution method. $5 x + 8 y = 13$ $- 5 x - 8 y = - 11$

A) $\left( \frac { 1 } { 8 } , - \frac { 1 } { 5 } \right)$
B) $\left( - \frac { 1 } { 5 } , \frac { 1 } { 8 } \right)$
C) $\left( - \frac { 3 } { 2 } , - \frac { 12 } { 5 } \right)$
D) $\left( - \frac { 1 } { 8 } , \frac { 1 } { 5 } \right)$
E)no solution

Question 13

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { c } x + 4 y ^ { 2 } = 28 \\- x + y = 5\end{array} \right.$

A)(-8, -3)
B)(-8, 7)
C)(-8, -2)
D)(-3, 8)
E)(-8, 3)

Question 14

Solve the system by the method of substitution. $\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 625 \\7 x - 24 y & = 0\end{aligned} \right.$

A)(24, 7), (-24, -7)
B)(24, 7)
C)(24, 7), (-24, 7), (-24, -7), (24, -7)
D)(7, 24), (7, -24)
E)no real solution

Question 15

You are offered two different jobs. Company A offers an annual salary of $32,500 plus a year-end bonus of 2.5% of your total sales. Company B offers a salary of $28,000 plus a year-end bonus of 5.5% of your total sales. What is the amount you must sell in one year to earn the same salary working for either company?

A)$130,000
B)$170,000
C)$150,000
D)$110,000
E)$120,000

Question 16

Determine which ordered pair is a solution of the system. $\left\{ \begin{aligned}6 x - 6 y & = - 9 \\- 2 x - 7 y & = 2\end{aligned} \right.$

A) $\left( - \frac { 25 } { 18 } , - \frac { 5 } { 9 } \right)$
B) $\left( - \frac { 17 } { 18 } , - \frac { 5 } { 9 } \right)$
C)(-8, 39)
D) $\left( - \frac { 25 } { 18 } , \frac { 1 } { 9 } \right)$
E)(-8, -14)

Question 17

Solve the system by the method of substitution. $\left\{ \begin{array} { l } y = x ^ { 3 } + 2 x ^ { 2 } - 1 \\y = x ^ { 2 } + 12 x - 1\end{array} \right.$

A)(-4, -33), (-3, -28), (1, 12)
B)(-4, -33), (-1, -12)
C)(3, 44), (1, 12)
D)(-4, -33), (3, 44), (0, -1)
E)no real solution

Question 18

The sales of various types of lawn and garden items vary according to the season. At a certain home improvement store, the monthly sales H of garden hoses (hoses sold per month) declines from July to October whereas the monthly sales of lawn rakes R (rakes sold per month) increase during this same interval. The sales of these two items during the calendar months July-October are modeled by the equations: H(t) = 64 - 6t
R(t) = 17t - 97,
Where t is the month (t = 7 corresponds to July). In which month does the number of rakes sold equal the number of hoses sold?

A)August
B)September
C)October
D)November
E)July

Question 19

Solve the system by the method of elimination. $\left\{ \begin{aligned}5 x - y & = 43 \\- 8 x - y & = - 74\end{aligned} \right.$

A) $( 3,50 )$
B) $( 9,2 )$
C) $\left( - \frac { 31 } { 13 } , - \frac { 1106 } { 13 } \right)$
D) $\left( - 2 , \frac { 41 } { 5 } \right)$
E)inconsistent

Question 20

Use a graphing utility to find the point(s) of intersection of the graphs. $\left\{ \begin{aligned}y & = e ^ { x + 2 } \\x - y + 3 & = 0\end{aligned} \right.$

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

Question 21

Use the statements below to write a system of equations. Solve the system by elimination. The sum of twice a number $r$ and a number $s$ is 4. The difference of $r$ and $s$ is 11.

A) $r = 6 , s = - 5$
B) $r = 6 , s = 5$
C) $r = 5 , s = 6$
D) $r = 5 , s = - 6$
E) $r = - 5 , s = 6$

Question 22

Solve the system of linear equations. $\left\{ \begin{aligned}- 3 x + 2 y - z & = 9 \\- x - 3 y + z & = 12 \\x - 2 y + 4 z & = 13\end{aligned} \right.$

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

Question 23

Solve each system of equations by the elimination method. $- 4 x - 7 y = - 16$ $- 5 x + 2 y = 18$

A) $\left( \frac { 152 } { 43 } , - \frac { 94 } { 43 } \right)$
B) $\left( - \frac { 158 } { 43 } , \frac { 8 } { 43 } \right)$
C) $\left( - \frac { 94 } { 43 } , \frac { 152 } { 43 } \right)$
D) $\left( \frac { 152 } { 27 } , - \frac { 94 } { 27 } \right)$
E)no solution

Question 24

Solve the system by the method of elimination. $\left\{ \begin{array} { c } \frac { 1 } { 4 } x + \frac { 1 } { 8 } y = - \frac { 5 } { 8 } \\2 x + y = - 5\end{array} \right.$

A) $( - 4,3 )$
B) $( - 2 , - 1 )$
C) $( 8 , - 21 )$
D) $( a , - 5 - 2 a ) \text { (dependent) }$
E)inconsistent

Question 25

The supply and demand equations for a small LCD television are given by $\left\{ \begin{array} { l l } p + 0.6 x = 1436 & \text { Demand } \\p - 0.2 x = 300 & \text { Supply }\end{array} \right.$ where $p$ is the price (in dollars) and $x$ represents the number of televisions. For how many units will the quantity demanded equal the quantity supplied? What price corresponds to this value?

A)1795 units at $359.00
B)2170 units at $134.00
C)1420 units at $134.00
D)2170 units at $584.00
E)1420 units at $584.00

Question 26

Solve the system below by elimination if possible. Then state whether the system below is consistent or inconsistent. $$\left\{ \begin{array} { r } 4 b + m = 0 \\- 8 b - 2 m = 3\end{array} \right.$$

A)The system is consistent and its solution is $( - 5,3 )$
B)The system is consistent and its solution is $( 5,3 )$
C)The system is consistent and its solution is $( - 3,5 )$
D)The system is consistent and its solution is $( 3,5 )$
E)The system is inconsistent and no solution exists.

Question 27

Find the equilibrium point of the demand and supply equations. (The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations.) Demand $~~~~~~~~~~~~~~~~~~$ Supply
$p = 30 - 0.04 x \quad p = 0.8 x - 390$

A) $( 500,10 )$
B) $( 200,22 )$
C) $( 899,329.2 )$
D) $( 600,6 )$
E)inconsistent

Question 28

Solve the system of equations below, if possible. $\left\{ \begin{array} { c } 4 x + 2 y + 5 z = - 3 \\x + y + z = 0 \\3 x + y + 4 z = 1\end{array} \right.$

A) $( - 6,1,6 )$
B) $( 6 , - 6,1 )$
C) $( 1,6 , - 6 )$
D) $( 1 , - 6,6 )$
E)The system is inconsistent.

Question 29

Determine which of the following systems of equations is in row-echelon form. System I: $\left\{ \begin{aligned}x - 8 y + 6 z & = 2 \\y - 7 z & = - 19 \\z & = 17\end{aligned} \right.$ System II: $$\left\{ \begin{aligned}x - 6 y - 8 z & = 2 \\2 y + 5 z & = - 1 \\z & = - 3\end{aligned} \right.$$ System III: $$\left\{ \begin{aligned}x - 4 y - z & = 20 \\y + 3 z & = 14 \\z & = 9\end{aligned} \right.$$

A)None of the systems are in row-echelon form.
B)Only system I is in row-echelon form.
C)Only systems I and III are in row-echelon form.
D)Only systems I and II are in row-echelon form.
E)Only systems II and III are in row-echelon form.

Question 30

Solve the system by elimination. $\left\{ \begin{array} { c } 6 x + 8 y = 1 \\- 4 x - 3 y = - 14\end{array} \right.$

A) $\left( 1 , - \frac { 5 } { 8 } \right)$
B) $\left( 3 , \frac { 2 } { 3 } \right)$
C) $\left( \frac { 109 } { 14 } , - \frac { 40 } { 7 } \right)$
D) $\left( a , \frac { 1 - 6 a } { 8 } \right) \text { (dependent) }$
E)inconsistent

Question 31

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l } 6 x + 3 y + 3 z = - 81 \\6 x - y - z = - 21 \\5 x - 5 y + 5 z = - 25\end{array} \right.$

A)(9, 6, -51)
B)(-6, -8, -7)
C)(-8, 7, 6)
D)(9, 6, -7)
E)(-7, -8, -6)

Question 32

Find the least squares regression line y = ax + b for the points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \mathbf { K } , \left( x _ { n } , y _ { n } \right)$ by solving the system for a and b. $\begin{array} { r } n b + \left( \sum _ { i = 1 } ^ { n } x _ { i } \right) a = \left( \sum _ { i = 1 } ^ { n } y _ { i } \right) \\\\\left( \sum _ { i = 1 } ^ { n } x _ { i } \right) b + \left( \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } \right) a = \left( \sum _ { i = 1 } ^ { n } x _ { i } y _ { i } \right)\end{array}$ Points: $( 1,13 ) , ( 3,15 ) , ( 6,18 ) , ( 6,19 )$ space

A)y = 11.81x + 1.11
B)y = -1.09x + 11.81
C)y = -4.30x - 7.24
D)y = 1.11x + 11.81
E)y = 4.01x - 7.24

Question 33

Solve the system of linear equations. $\left\{ \begin{array} { c } x + y + z = - 2 \\x + y - 3 z = 6 \\- y + 7 z = - 7\end{array} \right.$

A) $( 6 , - 8,0 )$
B) $( - 8,6,0 )$
C) $( 7 , - 7 , - 2 )$
D) $( - 7,7 , - 2 )$
E) $( 10 , - 9 , - 3 )$

Question 34

One acetic acid solution is 80% water and another is 45% water. How many liters of each solution should be mixed to produce 10 liters of a solution that is 73% water?

A)5 liters of the 80% solution and 5 liters of the 45% solution
B)8 liters of the 80% solution and 2 liters of the 45% solution
C)5 liters of the 80% solution and 5 liters of the 45.0% solution
D)3 liters of the 80% solution and 7 liters of the 45% solution
E)7 liters of the 80% solution and 3 liters of the 45% solution

Question 35

Use back-substitution to solve the system of linear equations. $$\left\{ \begin{array} { r } - x + 7 y - 5 z = 51 \\5 y - 8 z = 82 \\z = - 9\end{array} \right.$$

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

Question 36

Solve the system of linear equations. $\left\{ \begin{aligned}4 x + 3 y - z & = - 20 \\x + 3 z & = - 2 \\2 x + 4 y + 3 z & = - 20\end{aligned} \right.$

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

Question 37

Solve the system by the method of elimination. $$\left\{ \begin{array} { l } 0.02 x - 0.02 y = 0.17 \\0.01 x + 0.05 y = 0.18\end{array} \right.$$

A) $\left( 7 , - \frac { 3 } { 2 } \right)$
B) $\left( 2 , \frac { 16 } { 5 } \right)$
C) $\left( \frac { 121 } { 12 } , \frac { 19 } { 12 } \right)$
D) $\left( a , \frac { - 17 + 2 a } { 2 } \right) \text { (dependent) }$
E)inconsistent

Question 38

An airplane flying into a headwind travels 328 miles in 2 hours and 44 minutes. On the return flight, the distance is traveled in 2 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant.

A)plane speed = 157 mph; wind speed = 29 mph
B)plane speed = 157 mph; wind speed = 22 mph
C)plane speed = 119 mph; wind speed = 11 mph
D)plane speed = 119 mph; wind speed = 22 mph
E)plane speed = 142 mph; wind speed = 22 mph

Question 39

A total of $29,000 is invested in two corporate bonds that pay 12% and 10.5% simple interest. The total annual interest is $3195. How much is invested in the 10.5% bond?

A)$17,000
B)$16,000
C)$19,000
D)$18,000
E)$10,000

Question 40

The concentration $C$ (in parts per million) of carbon dioxide in the atmosphere is measured at the Mauna Loa Observatory in Hawaii. The greatest monthly carbon dioxide concentration for each year from 2002 to 2006 is shown in the table. $$\begin{array} { | c | c | c | } \hline \text { Year } & t & \text { Concentration, } C \\\hline 2002 & 0 & 375.55 \\\hline 2003 & 1 & 378.35 \\\hline 2004 & 2 & 380.63 \\\hline 2005 & 3 & 382.26 \\\hline 2006 & 4 & 384.92 \\\hline\end{array}$$ Solve the following system for $a$ and $b$ to find the least squares regression line $y = a t + b$ for the data. Let $t$ represent the year, with $t = 0$ corresponding to 2002. $$\left\{ \begin{array} { r } 5 b + 10 a = 1901.71 \\10 b + 30 a = 3826.07\end{array} \right.$$ Use the least square regression line to predict the largest monthly carbon dioxide concentration in 2011. Round your answer to the nearest hundredths part per million.

A)402.99 parts per million
B)387.14 parts per million
C)400.73 parts per million
D)393.93 parts per million
E)396.20 parts per million

Question 41

Sketch the graph of the inequality below. $x < - 3$

A)
B)
C)
D)
E)

Question 42

Solve the system of equations below: $8 x - 8 y - 8 z = - 24$ $2 x + 2 y - 2 z = 2$ $x - 2 y - z = - 5$

A) $( 8 , - 9,5 )$
B) $( - 9 , - 5,8 )$
C) $( c - 1,2 , c )$
D) $( 2 , c - 1 , c )$
E) $( 2 , c + 1 , c )$

Question 43

Find the equation of the circle $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$ that passes through the points $( 3,4 ) , ( 0,1 ) , ( 6,1 )$ .

A) $x ^ { 2 } + y ^ { 2 } - 6 x - 2 y + 1 = 0$
B) $x ^ { 2 } + y ^ { 2 } - 3 x - y + 1 = 0$
C) $x ^ { 2 } + y ^ { 2 } - 6 x - 2 y + 19 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 6 x - 2 y - 9 = 0$
E) $x ^ { 2 } + y ^ { 2 } - 3 x - y - 9 = 0$

Question 44

A real estate company borrows $700,000. Some of the money is borrowed at 5%, some at 9%, and some at 11% simple annual interest. How much is borrowed at the 5% rate when the total annual interest is $55,000 and the amount borrowed at 9% is the same as the amount borrowed at 11%?

A)$700,000
B)$300,000
C)$500,000
D)$200,000
E)$900,000

Question 45

A mixture of 6 gallons of chemical A, 7 gallons of chemical B, and 10 gallons of chemical C is required to kill a crop destroying insect. Commercial spray X contains 1, 2, and 3 parts of these chemicals, respectively. Commercial spray Y contains only chemical C. Commercial spray Z contains chemicals A, B, and C in equal amounts. How much of commercial spray $Y$ is needed to obtain the desired mixture?

A)14 gallons
B)1 gallon
C)16 gallons
D)2 gallons
E)6 gallons

Question 46

Sketch the graph of the inequality below. $x + y > 3$

A)
B)
C)
D)
E)

Question 47

Find an equation of the form $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( 1,7 ),$ $( - 4 , - 18 ),$ and $( - 1,3 ).$

A) $y = - x ^ { 2 } + 2 x + 6$
B) $y = - x ^ { 2 } + 6 x + 2$
C) $y = 2 x ^ { 2 } - x + 6$
D) $y = 6 x ^ { 2 } - x + 2$
E) $y = 6 x ^ { 2 } + 2 x - 1$

Question 48

Sketch the graph of the inequality below. $y > - x - 3$

A)
B)
C)
D)
E)

Question 49

Write an inequality for the shaded region shown in the figure.

A) $y \leq 2 - x ^ { 2 }$
B) $y \geq ( x - 2 ) ^ { 2 }$
C) $y \geq 2 - x ^ { 2 }$
D) $y \geq x ^ { 2 } + 2$
E) $y \geq x ^ { 2 } - 2$

Question 50

Which of the following three ordered triples are of the given form below. $\left( a , a + 3 , \frac { 1 } { 2 } a - 5 \right)$ Triple I : $( 2,1 , - 6 )$
Triple II : $( - 1 , - 3 , - 8 )$
Triple III : $( 4,7 , - 1 )$

A)Only triple I and II are in the given form.
B)Only triple II and III are in the given form.
C)Only triple III is in the given form.
D)Only triple I and III are in the given form.
E)None of the triples are in the given form.

Question 51

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. $( 0,2 ) , ( 1,2 ) , ( 2,4 )$

A) $y = x ^ { 2 } + x - 2$
B) $y = x ^ { 2 } - x - 2$
C) $y = 2 x ^ { 2 } - x + 2$
D) $y = 2 x ^ { 2 } - x + 1$
E) $y = x ^ { 2 } - x + 2$

Question 52

A residential building contractor borrowed $24,000 to complete a new home. Some of the money was borrowed at 5%, some at 7%, and some at 8%. How much was borrowed at each rate if the annual interest owed was $1480 and the amount borrowed at 7% is two times more than the amount borrowed at 8%?

A)$10,000 at 5%; $10,000 at 7%; $4000 at 8%
B)$12,000 at 5%; $8000 at 7%; $4000 at 8%
C)$10,000 at 5%; $9000 at 7%; $5000 at 8%
D)$11,000 at 5%; $6000 at 7%; $6000 at 8%
E)$11,000 at 5%; $10,000 at 7%; $5000 at 8%

Question 53

Solve the system of equations below: $- 6 x + y + 7 z = 20$ $7 x - 2 y - 9 z = - 25$

A) $( - c - 3 , c + 2 , c )$
B) $( c - 3 , - c + 2 , c )$
C) $( c + 2 , - c - 3 , c )$
D) $( - c + 2 , c - 3 , c )$
E) $( - 3 c + 2 , c - 1 , c )$

Question 54

Solve the system of equations below: $- 7 x + 6 y + 5 z = 0$ $- 2 x - 2 y - 2 z = 0$ $2 x - 7 y - 3 z = 0$

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

Question 55

Which of the following systems of equations has as a solution the ordered triple $( 3,1,2 )$ System I: $$\left\{ \begin{array} { c } 3 x + 5 y - 3 z = 7 \\- 5 x + 6 y - 5 z = - 15 \\- 6 x - 2 y - 6 z = - 36\end{array} \right.$$ System II: $$\left\{ \begin{array} { l } 5 x - 5 y - 3 z = 4 \\- 5 x - 2 y + z = - 15 \\6 x + y - 5 z = 9\end{array} \right.$$ System III: $$\left\{ \begin{array} { r } 6 x - 3 y + 2 z = 19 \\3 x + y - 6 z = - 2 \\5 x + 4 y - 2 z = 15\end{array} \right.$$

A)Only system III has the ordered triple $( 3,1,2 )$ as a solution.
B)Only systems II and III have the ordered triple $( 3,1,2 )$ as a solution.
C)Only systems I and II have the ordered triple $( 3,1,2 )$ as a solution.
D)Each of the systems I, II, and III has the ordered triple $( 3,1,2 )$ as a solution.
E)Only system I has the ordered triple $( 3,1,2 )$ as a solution.

Question 56

Graph the solution set of the system of inequalities below. $\left\{ \begin{aligned}x - y & \leq 2 \\- x - y & \leq 2 \\- y & \geq 0\end{aligned} \right.$

A)
B)
C)
D)
E)

Question 57

Use a graphing utility to graph the inequality. Shade the region representing the solution. $y > 3 - \frac { 2 } { 5 } x$

A)
B)
C)
D)
E)

Question 58

A chemist needs 8 liters of a 45% acid solution. The solution is to be mixed from three solutions whose acid concentrations are 10%, 20%, and 50%. How many liters of the 50% solution should the chemist use if trying to use as little as possible of the 50% solution?

A) $1 \frac { 1 } { 3 }$ liters
B) $3 \frac { 2 } { 3 }$ liters
C) $6 \frac { 2 } { 3 }$ liters
D) $2 \frac { 1 } { 3 }$ liters
E) $2 \frac { 2 } { 3 }$ liters

Question 59

The federal debt of the United States as a percentage of the Gross Domestic Product (GDP) from 2001 to 2005 is shown in the table. In the table, $x$ represents the year, with $x = 0$ corresponding to 2002. $$\begin{array} { | c | c | } \hline \text { Year, } x & \% \text { of GDP } \\\hline - 1 & 57.4 \\\hline 0 & 59.7 \\\hline 1 & 62.6 \\\hline 2 & 63.7 \\\hline 3 & 64.3 \\\hline\end{array}$$ Find the least squares regression parabola $y = a x ^ { 2 } + b x + c$ for the data by solving the following system. $$\left\{ \begin{aligned}5 c + 5 b + 15 a & = 307.7 \\5 c + 15 b + 35 a & = 325.5 \\15 c + 35 b + 99 a & = 953.5\end{aligned} \right.$$ Use the model to predict the federal debt as percents of GDP in 2006. Round to the nearest tenth percents.

A)56.5%
B)61.9%
C)64.2%
D)52.7%
E)59.6%

Question 60

Solve the system of linear equations. $\left\{ \begin{array} { c } x + y + z + w = 0 \\- 3 x - 2 y + z - w = - 5 \\- 5 x + 4 z - w = - 20 \\- 5 x + y + z + 4 w = - 15\end{array} \right.$

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

Question 61

Find which of the following system of inequalities has a right triangle as a graphed solution set. System I: $\left\{ \begin{array} { l } y \leq \frac { 1 } { 5 } x + 4 \\y \geq 0 \\x \geq 0\end{array} \right.$ System II: $\left\{ \begin{array} { l } y \leq \frac { 4 } { 5 } x - 4 \\y \geq 0 \\x \geq 0\end{array} \right.$ System III: $\left\{ \begin{array} { l } y \leq \frac { 4 } { 5 } x + 5 \\y \geq 0 \\x \geq 0\end{array} \right.$

A)Only systems I and III have a right triangle as a graphed solution set.
B)Only systems II and III have a right triangle as a graphed solution set.
C)No systems have a right triangle as a graphed solution set.
D)Only system III has a right triangle as a graphed solution set.
E)Only systems I and II have a right triangle as a graphed solution set.

Question 62

Find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $z = 3 x + 4 y$ Constraints: $$\left\{ \begin{array} { l } y \geq 0 \\x - y \geq - 2 \\2 x + 3 y \geq 6 \\5 x + 2 y \leq 25\end{array} \right.$$

A) $\text { minimum } = 0 \text { at } ( 0,0 ) ; \text { maximum } = 29 \text { at } ( 3,5 )$
B) $\text { minimum } = 9 \text { at } ( 3,0 ) ; \text { maximum } = 15 \text { at } ( 5,0 )$
C) $\text { minimum } = 8 \text { at } ( 0,2 ) ; \text { maximum } = 27 \text { at } ( 3,5 )$
D) $\text { minimum } = - 11 \text { at } ( 3,5 ) ; \text { maximum } = 15 \text { at } ( 5,0 )$
E) $\text { minimum } = 8 \text { at } ( 0,2 ) ; \text { maximum } = 29 \text { at } ( 3,5 )$

Question 63

Derive a set of inequalities to describe the region. Triangle: vertices at (0, 0), (3, 0), (3, 5)

A) $\left\{ \begin{array} { l } y \leq \frac { 5 } { 3 } x \\y \geq 0 \\x \leq 3\end{array} \right.$
B) $\left\{ \begin{array} { l } y \leq \frac { 5 } { 3 } x \\x \leq 3\end{array} \right.$
C) $\left\{ \begin{array} { l } y \geq \frac { 5 } { 3 } x \\y \geq 0 \\x \leq 3\end{array} \right.$
D) $\left\{ \begin{array} { l } y \leq \frac { 5 } { 3 } x \\x \geq 0\end{array} \right.$
E) $\left\{ \begin{array} { l } y \leq \frac { 3 } { 5 } x \\y \geq 0 \\x \leq 3\end{array} \right.$

Question 64

Which of the following vertices of the constraint region shown is a maximum value of the objective function below. $z = - 2 x - y - 5$

A)vertices A and D
B)only vertex C
C)vertices A and B
D)vertices C and D
E)only vertex A

Question 65

For the given supply and demand equations, find the consumer surplus. Round to the nearest dollar. Demand Supply
P = 160 - 0.00006x p = 110 + 0.00003x

A)$9,259,259
B)$10,185,185
C)$11,111,111
D)$13,888,889
E)$4,629,630

Question 66

Find the consumer surplus for the pair of demand and supply equations below. $\begin{array} { l l } p = 75 - 0.00002 x & \text { Demand } \\p = 30 + 0.00003 x & \text { Suppply }\end{array}$

A)$12,150,000
B)$8,100,000
C)$24,300,000
D)$4,050,000
E)$6,075,000

Question 67

Write a system of inequalities whose solution set is graphed below.

A) $\left\{ \begin{array} { l } 0 \leq x \leq 3 \\0 \leq y \leq 2\end{array} \right.$
B) $\left\{ \begin{array} { l } 0 \leq x \leq 2 \\0 \leq y \leq 3\end{array} \right.$
C) $\left\{ \begin{array} { c } 0 \leq x \leq 3 \\- 2 \leq y \leq 0\end{array} \right.$
D) $\left\{ \begin{array} { r } - 3 \leq x \leq 0 \\0 \leq y \leq 2\end{array} \right.$
E) $\left\{ \begin{array} { r } 0 \leq x \leq 2 \\- 3 \leq y \leq 0\end{array} \right.$

Question 68

Which of the following vertices of the constraint region shown is a minimum value of the objective function below. $$z = - 6 x - 4 y + 2$$

A)only vertex B
B)vertices B and C
C)vertices A and B
D)only vertex D
E)only vertex C

Question 69

A furniture company produces tables and chairs. Each table requires 2.5 hours in the assembly center and 1.75 hours in the finishing center. Each chair requires 1 hour in the assembly center and 1.25 hours in the finishing center. The company's assembly center is available 16 hours per day, and its finishing center is available 12 hours per day. Let $x$ and $y$ be the number of tables and chairs produced per day, respectively. Find a system of inequalities describing all possible production levels.

A) $\left\{ \begin{aligned}x + 1.25 y & \leq 16 \\2.5 x + 1.75 y & \leq 12 \\x & \geq 0 \\y & \geq 0\end{aligned} \right.$
B) $\left\{ \begin{aligned}1.75 x + 1.25 y & \leq 16 \\2.5 x + y & \leq 12 \\x & \geq 0 \\y & \geq 0\end{aligned} \right.$
C) $\left\{ \begin{aligned}2.5 x + 1.25 y & \leq 16 \\x + 1.75 y & \leq 12 \\x & \geq 0 \\y & \geq 0\end{aligned} \right.$
D) $\left\{ \begin{aligned}2.5 x + 1.75 y & \leq 16 \\x + 1.25 y & \leq 12 \\x & \geq 0 \\y & \geq 0\end{aligned} \right.$
E) $\left\{ \begin{aligned}2.5 x + y & \leq 16 \\1.75 x + 1.25 y & \leq 12 \\x & \geq 0 \\y & \geq 0\end{aligned} \right.$

Question 70

You plan to invest up to $60,000 in two different interest-bearing accounts. Each account is to contain at least $8000. Moreover, one account should have at least three times the amount that is in the other account. Find a system of inequalities that describes the amount that you can invest in each account.

A) $\left\{ \begin{aligned}x + y & \leq 60,000 \\x & \geq 8000 \\y & \geq 8000 \\x - 3 y & \leq 0\end{aligned} \right.$
B) $\left\{ \begin{aligned}x + y & \geq 60,000 \\x & \geq 8000 \\y & \geq 8000 \\x - 3 y & \leq 0\end{aligned} \right.$
C) $\left\{ \begin{aligned}x + y & \geq 60,000 \\x & \leq 8000 \\y & \leq 8000 \\x - 3 y & \geq 0\end{aligned} \right.$
D) $\left\{ \begin{aligned}x + y & \leq 60,000 \\x & \geq 8000 \\y & \geq 8000 \\x - 3 y & \geq 0\end{aligned} \right.$
E) $\left\{ \begin{aligned}x + y & \geq 60,000 \\x & \geq 8000 \\y & \geq 8000 \\x - 3 y & \geq 0\end{aligned} \right.$

Question 71

Sketch the graph of the solution set of each system of inequalities. $2 x - 5 y < - 6$ $3 x + y < 8$

A)
B)
C)
D)
E)

Question 72

A company makes two types of telephone answering machines: the standard model and the deluxe model. Each machine passes through three processes: $P _ { 1 } ,$ $P _ { 2 }$ and $P _ { 3 }.$ One standard answering machine requires 1 hour in $P _ { 1 } ,$ 4 hours in $P _ { 2 },$ and 3 hours in $P _ { 3 }.$ One deluxe answering machine requires 3 hours in $P _ { 1 } ,$ 5 hours in $P _ { 2 },$ and 1 hour in $P _ { 3 }.$ Because of employee work schedules, $P _ { 1 }$ is available for 24 hours, $P _ { 2 }$ is available for 47 hours, and $P _ { 3 }$ is available for 27 hours. If the profit is $22 for each standard model and $23 for each deluxe model, how many units of each type should the company produce to maximize profit?

A)8 standard models and 3 deluxe models
B)9 standard models and 0 deluxe models
C)8 standard models and 0 deluxe models
D)0 standard models and 8 deluxe models
E)3 standard models and 7 deluxe models

Question 73

An investor has $750,000 to invest in two types of investments. Type A pays 4% annually and type B pays 5% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-third of the total portfolio is to be allocated to type A investments and at least one-third of the portfolio is to be allocated to type B investments. What is the optimal amount that should be invested in each investment?

A)$250,000 in type A (4%), $500,000 in type B (5%)
B)$0 in type A (4%), $750,000 in type B (5%)
C)$500,000 in type A (4%), $250,000 in type B (5%)
D)$750,000 in type A (4%), $0 in type B (5%)
E)$260,000 in type A (4%), $490,000 in type B (5%)

Question 74

An ice cream supplier has two machines that produce vanilla and chocolate ice cream. To meet one of its contractual obligations, the company must produce at least 9 gallons of vanilla ice cream and 15 gallons of chocolate ice cream per day. One machine makes 6 gallons of vanilla and 3 gallons of chocolate ice cream per hour. The second machine makes 1 gallons of vanilla and 4 gallons of chocolate ice cream per hour. It costs $28 per hour to run machine 1 and $22 per hour to run machine 2. How many hours should each machine be operated to fulfill the contract at the least expense?

A)Machine 1 for 9 hours and machine 2 for no hours.
B)Machine 1 for 3 hours and machine 2 for 1 hour.
C)Machine 1 for 1 hour and machine 2 for 3 hours.
D)Machine 1 for no hours and machine 2 for 1 hour.
E)Machine 1 for no hours and machine 2 for 3 hours.

Question 75

A company has budgeted a maximum of $1,200,000 for national advertising an allergy medication. Each minute of television time costs $100,000 and each one-page newspaper ad costs $40,000. Each television ad is expected to be viewed by 43 million viewers, and each newspaper ad is expected to be seen by 18 million readers. What is the optimal amount that should be spent on advertising for each type ad?

A)$0 on television time and $1,200,000 on newspaper ads
B)$400,000 on television time and $800,000 on newspaper ads
C)$900,000 on television time and $300,000 on newspaper ads
D)$1,100,000 on television time and $100,000 on newspaper ads
E)$200,000 on television time and $1,000,000 on newspaper ads

Question 76

Find the maximum value of the objective function and where it occurs, subject to the indicated constraints. (You should graph the feasible solutions on the grid below before you attempt to find the minimum and maximum values.) Objective function: $z = 6 x - 7 y$ Constraints: $$\left\{ \begin{array} { l } x \geq 0 \\y \geq 0 \\5 x + 4 y \leq 20 \\3 x + 2 y \leq 6\end{array} \right.$$

A) $\text { maximum } = 24 \text { at } ( 4,0 )$
B) $\text { maximum } = 12 \text { at } ( 2,0 )$
C) $\text { maximum } = 35 \text { at } ( 0,5 )$
D) $\text { The maximum is unbounded. }$
E) $\text { There is no maximum. The feasable solution set is empty. }$

Question 77

A dietitian designs a special diet supplement using two different foods. Each ounce of food X contains 15 units of calcium, 10 units of iron, and 20 units of vitamin B. Each ounce of food Y contains 20 units of calcium, 10 units of iron, and 15 units of vitamin B. The minimum daily requirements for the diet are 360 units of calcium, 320 units of iron, and 370 units of vitamin B. Which combinations of foods X and Y below, if any, can be given to the patient to meet the minimum daily requirements? Combination I : 13 ounces of food $\mathrm { X }$ and 16 ounces of food $\mathrm { Y }$
Combination II : 11 ounces of food $X$ and 20 ounces of food $Y$
Combination II : 20 ounces of food $X$ and 23 ounces of food $Y$

A)Only combinations I and II meet the minimum daily requirement.
B)Only combinations II and III meet the minimum daily requirement.
C)Only combinations I and III meet the minimum daily requirement.
D)Only combination III meets the minimum daily requirement.
E)Combinations I, II, and III meet the minimum daily requirement.

Question 78

Sketch the graph of the solution set of each system of inequalities. $y > 2 x - 1$ $y > x ^ { 2 } - 2 x + 2$

A)
B)
C)
D)
E)

Question 79

Graph the solution set of the system of inequalities below. $\left\{ \begin{array} { l } y \leq - x + 2 \\y \geq - x - 2 \\y \geq 0 \\y \leq 2\end{array} \right.$

A)
B)
C)
D)
E)

Question 80

Sketch the graph and label the vertices of the solution set of the system of inequalities. Shade the solution set. $\left\{ \begin{array} { l } x ^ { 2 } + y \leq 4 \\x \geq 1 \\y \geq 0\end{array} \right.$

A)
B)
C)
D)
E)