Question 1

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

Accepted Answer

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

Question 2

A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. $\begin{array} { l l l } 
\text { Process } & \text { Model A } & \text { Model B } \
\text { Assembling } & 2 & 2.5 \
\text { Painting } & 4 & 1 \
\text { Packaging } & 1 & 0.75
\end{array}$ The total times available for assembling, painting, and packing are 4000 hours, 4800 hours, and 1500 hours, respectively. The profit per unit are $50 for model A and $75 for model B. What is the optimal production level for each model? What is the optimal profit?

Accepted Answer

A) The optimal profit of $120,000 occurs when no units of model A and 1600 units of of model B are produced. 
B) The optimal profit of $112,000 occurs when no units of model A and 1000 units of of model B are produced. 
C) The optimal profit of $97,500 occurs when no units of model A and 600 units of of model B are produced. 
D) The optimal profit of $60,000 occurs when no units of model A and 0 units of of model B are produced. 
E) The optimal profit of $112,000 occurs when no units of model A and 750 units of of model B are produced. 
A) The optimal profit of $120,000 occurs when no units of model A and 1600 units of of model B are produced. 
B) The optimal profit of $112,000 occurs when no units of model A and 1000 units of of model B are produced. 
C) The optimal profit of $97,500 occurs when no units of model A and 600 units of of model B are produced. 
D) The optimal profit of $60,000 occurs when no units of model A and 0 units of of model B are produced. 
E) The optimal profit of $112,000 occurs when no units of model A and 750 units of of model B are produced.

Question 3

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: $$\left\{ \begin{array} { r } 
6 x - 3 y + 2 z = 19 \
3 x + y - 6 z = - 2 \
5 x + 4 y - 2 z = 15
\end{array} \right.$$

Accepted Answer

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. 
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 4

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 : $( 4,7 , - 1 )$

Accepted Answer

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. 
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 5

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.

Accepted Answer

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 
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 6

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?

Accepted Answer

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

Question 7

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?

Accepted Answer

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 
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 8

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

Accepted Answer

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 $( 4 , - 14 )$ 
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 $( 4 , - 14 )$

Question 9

Find an objective function that has a minimum value at the indicated vertex D of the constraint region shown below.

Accepted Answer

A) Answer will vary. Sample answer: $z = x + 3 y$ . 
B) Answer will vary. Sample answer: $z = 3 x + y$ . 
C) Answer will vary. Sample answer: $z = 3 ( x + y )$ . 
D) Answer will vary. Sample answer: $z = 3 ( x + 2 y )$ . 
E) Answer will vary. Sample answer: $z = 3 ( 2 x + y )$ . 
A) Answer will vary. Sample answer: $z = x + 3 y$ . 
B) Answer will vary. Sample answer: $z = 3 x + y$ . 
C) Answer will vary. Sample answer: $z = 3 ( x + y )$ . 
D) Answer will vary. Sample answer: $z = 3 ( x + 2 y )$ . 
E) Answer will vary. Sample answer: $z = 3 ( 2 x + y )$ .

Question 10

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

Accepted Answer

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)  $y = 6 x ^ { 2 } + 2 x - 1$ 
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 11

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?

Accepted Answer

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 
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 12

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

Accepted Answer

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

Question 13

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.

Accepted Answer

A)  $r = 6 , s = - 5$ 
B)  $r = 6 , s = 5$ 
C)  $r = 5 , s = 6$ 
D)  $r = 5 , s = - 6$ 
E)  $r = - 5 , s = 6$ 
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 14

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: $$\left\{ \begin{aligned}
x - 4 y - z & = 20 \
y + 3 z & = 14 \
z & = 9
\end{aligned} \right.$$

Accepted Answer

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. 
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 15

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
$p = 30 - 0.04 x \quad p = 0.8 x - 390$

Accepted Answer

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

Question 16

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

Accepted Answer

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 
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

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

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 : $( 4,7 , - 1 )$

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.

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?

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

Find an objective function that has a minimum value at the indicated vertex D of the constraint region shown below.

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

Find the consumer surplus for the pair of demand and supply equations below. p=75-0.00002x Demand p=30+0.00003x Suppply

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.

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 $p = 30 - 0.04 x \quad p = 0.8 x - 390$

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

Fundamental Concepts of Algebra

Equations and Inequalities

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Polynomial and Rational Functions

Exponential and Logarithmic Functions

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

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Derivatives of Exponential and Logarithmic Functions

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Exam 6: Systems of Equations and Inequalities

Solve the system below by method of substitution, if possible. {x2+y2=25−x+y=12\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 25 \\- x + y = 12\end{array} \right.{x2+y2=25−x+y=12​

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.

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?

Solve the system of equations below. {y=x2−3x−6y=−2x−6\left\{ \begin{array} { l } y = x ^ { 2 } - 3 x - 6 \\y = - 2 x - 6\end{array} \right.{y=x2−3x−6y=−2x−6​

Find an objective function that has a minimum value at the indicated vertex D of the constraint region shown below.

Find an equation of the form y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c whose graph passes through the points (1,7),( 1,7 ),(1,7), (−4,−18),( - 4 , - 18 ),(−4,−18), and (−1,3).( - 1,3 ).(−1,3).

Find the consumer surplus for the pair of demand and supply equations below. p=75-0.00002x Demand p=30+0.00003x Suppply

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

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=30−0.04xp=0.8x−390p = 30 - 0.04 x \quad p = 0.8 x - 390p=30−0.04xp=0.8x−390

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

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

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

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

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

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.

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 $p = 30 - 0.04 x \quad p = 0.8 x - 390$

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