Question 1

Solve the problem.
-The Family Arts Center charges $21 for adults, $13 for senior citizens, and $8 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $12,388 for 905 tickets sold. There
Were 50 more children than adults. How many children attended?
A) 291 children
B) 341 children
C) 273 children
D) 331 children

Accepted Answer

Let A be the number of adults, S be the number of senior citizens, and C be the number of children. We have the following equations:  1) A + S + C = 905  2) 21A + 13S + 8C = 12,388  3) C = A + 50  Substitute C = A + 50 into equation 1:  A + S + (A + 50) = 905  2A + S = 855  Substitute C = A + 50 into equation 2:  21A + 13S + 8(A + 50) = 12,388  21A + 13S + 8A + 400 = 12,388  29A + 13S = 11,988  Now solve the system of equations:  2A + S = 855  29A + 13S = 11,988  Multiply the first equation by 13:  26A + 13S = 11,115  Subtract from the second equation:  (29A + 13S) - (26A + 13S) = 11,988 - 11,115  3A = 873  A = 291  Substitute A = 291 into C = A + 50:  C = 291 + 50 = 341  Therefore, 341 children attended.

Question 2

Solve the problem.
-An 8-cylinder Crown Victoria gives 18 miles per gallon in city driving and 21 miles per gallon in highway driving. A 300-mile trip required 15.5 gallons of gasoline. How many whole miles were driven in the city?
A) 132 miles
B) 147 miles
C) 168 miles
D) 153 miles

Accepted Answer

D

Question 3

Verify that the values of the variables listed are solutions of the system of equations.
-$$\left\{ \begin{array} { c } 
x + y + z = - 1 \
x - y + 3 z = - 13 \
4 x + y + z = 8 \
\end{array} \right.$$

x = 3 , y = 1 , z = - 5

A) not a solution
B) solution

Accepted Answer

Substituting $x = 3$, $y = 1$, and $z = -5$ into each equation of the system, we get: 1. $3 + 1 - 5 = -1$, which is true.2. $3 - 1 + 3(-5) = 3 - 1 - 15 = -13$, which is true.3. $4(3) + 1 - 5 = 12 + 1 - 5 = 8$, which is true.Since all three equations are satisfied, the values are indeed solutions to the system.

Question 4

Solve the system.
-$$\left\{ \begin{array} { l } 
x - 4 y = - 10 \
2 x - 8 y = - 17
\end{array} \right.$$
A) dependent (many solutions)
B) (2, 4)
C) (2, 3)
D) inconsistent (no solution)

Accepted Answer

The second equation is just the first equation multiplied by 2, but the constants do not align (-10 multiplied by 2 should be -20, not -17), indicating the lines are parallel and never intersect, hence the system is inconsistent.

Question 5

Solve the problem. -Find real numbers a, b, and c such that the graph of the function y = ax² + bx + c contains the points (1, 1), (2, 4), and (-3, 29).

Accepted Answer

The answer of Solve the problem.
-Find real numbers a, b,...

Question 6

Solve the problem.
-The Family Fine Arts Center charges $24 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 538 people paid admission, the total receipts were $7592. How many who paid were
Senior citizens?
A) 158 senior citizens
B) 290 senior citizens
C) 248 senior citizens
D) 380 senior citizens

Accepted Answer

The answer of Solve the problem.
-The Family Fine Arts Center...

Question 7

Solve the problem.
-Lexie wants to have an income of $9000 per year from investments. To that end she is going to invest $90,000 in
three different accounts. These accounts pay 7%, 10%, and 14% simple interest. If she wants to have $10,000
more in the account paying 7% simple interest than she has in the account paying 14% simple interest, how
much should go into each account?

Accepted Answer

The answer of Solve the problem.
-Lexie wants to have an...

Question 8

Verify that the values of the variables listed are solutions of the system of equations.
-$$\begin{array} { l } 
\left\{ \begin{array} { l } 
2 x + y = 6 \
3 x + 2 y = 7
\end{array} \right. \
x = 5 , y = - 4
\end{array}$$
A) not a solution
B) solution

Accepted Answer

Substitute the values of x and y into the equations and check if they satisfy the equations.

Question 9

Solve the problem.
-A tour group split into two groups when waiting in line for food at a fast food counter. The first group bought 8 slices of pizza and 7 soft drinks for $36.53. The second group bought 6 slices of pizza and 5 soft drinks for $
26)93. How much does one slice of pizza cost?
A) $1.87 per slice of pizza
B) $2.37 per slice of pizza
C) $2.43 per slice of pizza
D) $2.93 per slice of pizza

Accepted Answer

The answer of Solve the problem.
-A tour group split into...

Question 10

Verify that the values of the variables listed are solutions of the system of equations.
-$$\left\{ \begin{array} { l } 
x + y = - 1 \
x - y = 11 \
\end{array} \right.$$

x = 5 , y = - 6

A) solution
B) not a solution

Accepted Answer

Substituting $x = 5$ and $y = -6$ into both equations: For the first equation, $5 + (-6) = -1$, which is true. For the second equation, $5 - (-6) = 11$, which is also true. Therefore, $x = 5$ and $y = -6$ satisfy both equations, making them a solution to the system.

Question 11

Solve the problem. -A company has sales (measured in millions of dollars) of 50, 60, and 75 during the first three consecutive years.Find a quadratic function that fits these data, and use the result to predict the sales during the fourth year. Assume that the quadratic function is of the form y = ax² + bx + c A) $y = \frac { 15 } { 2 } x ^ { 2 } - \frac { 25 } { 2 } x + \frac { 325 } { 4 }$; sales during the fourth year $= \$ 151.25$ million B) $y = 5 x ^ { 2 } + 5 x + 40$; sales during the fourth year $= \$ 180$ million C) $y = - 5 x ^ { 2 } + 40 x + 15$; sales during the fourth year $= \$ 95$ million D) $y = \frac { 5 } { 2 } x ^ { 2 } + \frac { 5 } { 2 } x + 45$; sales during the fourth year $= \$ 95$ million

Accepted Answer

The answer of Solve the problem.
-A company has sales (measured...

Question 12

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { l } 
3 x + y = 13 \
2 x + 9 y = - 8
\end{array} \right.$$
A) x = 5, y = 2
B) x = -5, y = -2
C) x = -5, y = 2
D) x = 5, y = -2

Accepted Answer

The answer of Solve the system of equations by using...

Question 13

Verify that the values of the variables listed are solutions of the system of equations.
-$$\left\{ \begin{array} { c } 
x - y + 3 z = - 20 \
4 x + z = - 5 \
x + 2 y + z = 5 \
\end{array} \right.$$

x = 0 , y = 5 , z = - 5

A) solution
B) not a solution

Accepted Answer

Substituting $x = 0$, $y = 5$, and $z = -5$ into each equation: 1) $0 - 5 + 3(-5) = -20$, 2) $4(0) + (-5) = -5$, 3) $0 + 2(5) + (-5) = 5$. All equations are satisfied, confirming the values as a solution.

Question 14

Solve the problem.
-A flat rectangular piece of aluminum has a perimeter of 64 inches. The length is 14 inches longer than the width. Find the width.
A) 32 inches
B) 9 inches
C) 37 inches
D) 23 inches

Accepted Answer

The perimeter of a rectangle is given by the formula $P = 2l + 2w$, where $l$ is the length and $w$ is the width. Given that the length is 14 inches longer than the width, we can express the length as $l = w + 14$. Substituting the given perimeter and the expression for $l$ into the perimeter formula gives $64 = 2(w + 14) + 2w$. Simplifying, we find $w = 9$ inches.

Question 15

Write the augmented matrix for the system.
-$$\left\{ \begin{array} { l } 
2 x + 8 y = 64 \
5 x + 7 y = 69
\end{array} \right.$$
A) $\left[ \begin{array} { l l | l } 2 & 8 & 64 \ 5 & 7 & 69 \end{array} \right]$
B) $\left[ \begin{array} { l l | l } 2 & 5 & 64 \ 8 & 7 & 69 \end{array} \right]$
C) $\left[ \begin{array} { l l | l } 2 & 8 & 69 \ 7 & 5 & 64 \end{array} \right]$
D) $\left[ \begin{array} { l l | l } 64 & 8 & 2 \ 69 & 5 & 7 \end{array} \right]$

Accepted Answer

The augmented matrix correctly represents the system of equations with coefficients of $x$ and $y$ in the first two columns and the constants on the right side of the bar.

Question 16

Solve the problem.
-A movie theater charges $8.00 for adults and $5.00 for children. If there were 40 people altogether and the theater collected $272.00 at the end of the day, how many of them were adults?
A) 10 adults
B) 24 adults
C) 29 adults
D) 16 adults

Accepted Answer

Let the number of adults be $x$ and the number of children be $y$. We have two equations: $x + y = 40$ (total number of people) and $8x + 5y = 272$ (total amount collected). Solving these equations simultaneously, we find that $x = 24$, meaning there were 24 adults.

Question 17

Write the augmented matrix for the system.
-$$\left\{ \begin{array} { l } 
9 x - 2 y + 6 z = 54 \
6 x + 2 y + 9 z = 51 \
5 x + 3 y + 2 z = 41
\end{array} \right.$$
A) $\left[ \begin{array} { r r r | r } 54 & 6 & - 2 & 9 \ 51 & 9 & 2 & 6 \ 41 & 2 & 3 & 5 \end{array} \right]$
B) $\left[ \begin{array} { r r r | r } 9 & 6 & 5 & 54 \ - 2 & 2 & 3 & 51 \ 6 & 9 & 2 & 41 \end{array} \right]$
C) $\left[ \begin{array} { r r | r } 9 & - 2 & 6 \ 6 & 2 & 9 \ 5 & 3 & 2 \end{array} \right]$
D) $\left[ \begin{array} { r r r | r } 9 & - 2 & 6 & 54 \ 6 & 2 & 9 & 51 \ 5 & 3 & 2 & 41 \end{array} \right]$

Accepted Answer

The augmented matrix correctly represents the system of equations with the coefficients of $x$, $y$, and $z$ in the first three columns and the constants on the right side of the equation in the fourth column.

Question 18

Solve the problem.
-A retired couple has $140,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 12% per year. How much
Should they invest in each to realize exactly $13,300 per year?
A) $90,000 at 5% and $50,000 at 12%
B) $80,000 at 5% and $60,000 at 12%
C) $90,000 at 12% and $50,000 at 5%
D) $100,000 at 12% and $40,000 at 5%

Accepted Answer

The answer of Solve the problem.
-A retired couple has $140,000...

Question 19

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { c } 
x + 6 y = 6 \
4 x - 9 y = - 9
\end{array} \right.$$
A) x = 0, y = 1
B) x = 1, y = 1
C) x = 1, y = 0
D) x = 0, y = 0

Accepted Answer

Solving the first equation for x gives x = 6 - 6y. Substituting this into the second equation gives 4(6 - 6y) - 9y = -9, which simplifies to 24 - 24y - 9y = -9. Combining like terms gives -33y = -33, so y = 1. Substituting y = 1 into x = 6 - 6y gives x = 0.

Question 20

Write the augmented matrix for the system.
-$$\left\{ \begin{array} { r } 
- 2 x + 8 y = 28 \
5 y = 20
\end{array} \right.$$
A) $\left[ \begin{array} { r | r | r } 5 & 0 & 20 \ - 2 & 8 & 8 \end{array} \right]$
B) $\left[ \begin{array} { r r | r } 28 & 8 & - 2 \ 20 & 0 & 5 \end{array} \right]$
C) $\left[ \begin{array} { r r | r } - 2 & 8 & 28 \ 0 & 5 & 20 \end{array} \right]$
D) $\left[ \begin{array} { r r | r } - 2 & 8 & 28 \ 5 & 20 & 0 \end{array} \right]$

Accepted Answer

The augmented matrix for a system of equations is formed by writing the coefficients of the variables in each equation as the rows of the matrix, with the constants on the right side of the bar. For the given system, the first row corresponds to the coefficients and constant from the first equation $-2x + 8y = 28$, and the second row corresponds to the coefficients and constant from the second equation $5y = 20$, with an implicit coefficient of 0 for $x$ in the second equation. This matches option C.

Quiz 6: Systems of Equations and Inequalities

Solve the problem. -An 8-cylinder Crown Victoria gives 18 miles per gallon in city driving and 21 miles per gallon in highway driving. A 300-mile trip required 15.5 gallons of gasoline. How many whole miles were driven in the city?

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { c } x + y + z = - 1 \\x - y + 3 z = - 13 \\4 x + y + z = 8 \\\end{array} \right.$ x = 3 , y = 1 , z = - 5

$Solve the system. - \left\{ \begin{array} { l } x - 4 y = - 10 \\ 2 x - 8 y = - 17 \end{array} \right. A) dependent (many solutions) B) (2, 4) C) (2, 3) D) inconsistent (no solution)$ Solve the system. - $\left\{ \begin{array} { l } x - 4 y = - 10 \\2 x - 8 y = - 17\end{array} \right.$

Solve the problem. -Find real numbers a, b, and c such that the graph of the function y = ax² + bx + c contains the points (1, 1), (2, 4), and (-3, 29).

Solve the problem. -The Family Fine Arts Center charges $24 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 538 people paid admission, the total receipts were $7592. How many who paid were Senior citizens?

Verify that the values of the variables listed are solutions of the system of equations. - $\begin{array} { l } \left\{ \begin{array} { l } 2 x + y = 6 \\3 x + 2 y = 7\end{array} \right. \\x = 5 , y = - 4\end{array}$

Solve the problem. -A tour group split into two groups when waiting in line for food at a fast food counter. The first group bought 8 slices of pizza and 7 soft drinks for $36.53. The second group bought 6 slices of pizza and 5 soft drinks for $ 26) 93. How much does one slice of pizza cost?

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { l } x + y = - 1 \\x - y = 11 \\\end{array} \right.$ x = 5 , y = - 6

Solve the system of equations by using substitution. - $\left\{ \begin{array} { l } 3 x + y = 13 \\2 x + 9 y = - 8\end{array} \right.$

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { c } x - y + 3 z = - 20 \\4 x + z = - 5 \\x + 2 y + z = 5 \\\end{array} \right.$ x = 0 , y = 5 , z = - 5

Solve the problem. -A flat rectangular piece of aluminum has a perimeter of 64 inches. The length is 14 inches longer than the width. Find the width.

Write the augmented matrix for the system. - $\left\{ \begin{array} { l } 2 x + 8 y = 64 \\5 x + 7 y = 69\end{array} \right.$

Solve the problem. -A movie theater charges $8.00 for adults and $5.00 for children. If there were 40 people altogether and the theater collected $272.00 at the end of the day, how many of them were adults?

Write the augmented matrix for the system. - $\left\{ \begin{array} { l } 9 x - 2 y + 6 z = 54 \\6 x + 2 y + 9 z = 51 \\5 x + 3 y + 2 z = 41\end{array} \right.$

Solve the problem. -A retired couple has $140,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 12% per year. How much Should they invest in each to realize exactly $13,300 per year?

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.$

Write the augmented matrix for the system. - $\left\{ \begin{array} { r } - 2 x + 8 y = 28 \\5 y = 20\end{array} \right.$

Quiz 6: Systems of Equations and Inequalities

Solve the problem. -An 8-cylinder Crown Victoria gives 18 miles per gallon in city driving and 21 miles per gallon in highway driving. A 300-mile trip required 15.5 gallons of gasoline. How many whole miles were driven in the city?

Verify that the values of the variables listed are solutions of the system of equations. - {x+y+z=−1x−y+3z=−134x+y+z=8\left\{ \begin{array} { c } x + y + z = - 1 \\x - y + 3 z = - 13 \\4 x + y + z = 8 \\\end{array} \right.⎩⎨⎧​x+y+z=−1x−y+3z=−134x+y+z=8​ x = 3 , y = 1 , z = - 5

Solve the system. - {x−4y=−102x−8y=−17\left\{ \begin{array} { l } x - 4 y = - 10 \\2 x - 8 y = - 17\end{array} \right.{x−4y=−102x−8y=−17​

Solve the problem. -Find real numbers a, b, and c such that the graph of the function y = ax2 + bx + c contains the points (1, 1), (2, 4), and (-3, 29).

Solve the problem. -The Family Fine Arts Center charges $24 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 538 people paid admission, the total receipts were $7592. How many who paid were Senior citizens?

Verify that the values of the variables listed are solutions of the system of equations. - {2x+y=63x+2y=7x=5,y=−4\begin{array} { l } \left\{ \begin{array} { l } 2 x + y = 6 \\3 x + 2 y = 7\end{array} \right. \\x = 5 , y = - 4\end{array}{2x+y=63x+2y=7​x=5,y=−4​

Solve the problem. -A tour group split into two groups when waiting in line for food at a fast food counter. The first group bought 8 slices of pizza and 7 soft drinks for $36.53. The second group bought 6 slices of pizza and 5 soft drinks for $ 26) 93. How much does one slice of pizza cost?

Verify that the values of the variables listed are solutions of the system of equations. - {x+y=−1x−y=11\left\{ \begin{array} { l } x + y = - 1 \\x - y = 11 \\\end{array} \right.{x+y=−1x−y=11​ x = 5 , y = - 6

Solve the system of equations by using substitution. - {3x+y=132x+9y=−8\left\{ \begin{array} { l } 3 x + y = 13 \\2 x + 9 y = - 8\end{array} \right.{3x+y=132x+9y=−8​

Verify that the values of the variables listed are solutions of the system of equations. - {x−y+3z=−204x+z=−5x+2y+z=5\left\{ \begin{array} { c } x - y + 3 z = - 20 \\4 x + z = - 5 \\x + 2 y + z = 5 \\\end{array} \right.⎩⎨⎧​x−y+3z=−204x+z=−5x+2y+z=5​ x = 0 , y = 5 , z = - 5

Solve the problem. -A flat rectangular piece of aluminum has a perimeter of 64 inches. The length is 14 inches longer than the width. Find the width.

Write the augmented matrix for the system. - {2x+8y=645x+7y=69\left\{ \begin{array} { l } 2 x + 8 y = 64 \\5 x + 7 y = 69\end{array} \right.{2x+8y=645x+7y=69​

Solve the problem. -A movie theater charges $8.00 for adults and $5.00 for children. If there were 40 people altogether and the theater collected $272.00 at the end of the day, how many of them were adults?

Write the augmented matrix for the system. - {9x−2y+6z=546x+2y+9z=515x+3y+2z=41\left\{ \begin{array} { l } 9 x - 2 y + 6 z = 54 \\6 x + 2 y + 9 z = 51 \\5 x + 3 y + 2 z = 41\end{array} \right.⎩⎨⎧​9x−2y+6z=546x+2y+9z=515x+3y+2z=41​

Solve the problem. -A retired couple has $140,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 12% per year. How much Should they invest in each to realize exactly $13,300 per year?

Solve the system of equations by using substitution. - {x+6y=64x−9y=−9\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.{x+6y=64x−9y=−9​

Write the augmented matrix for the system. - {−2x+8y=285y=20\left\{ \begin{array} { r } - 2 x + 8 y = 28 \\5 y = 20\end{array} \right.{−2x+8y=285y=20​

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { c } x + y + z = - 1 \\x - y + 3 z = - 13 \\4 x + y + z = 8 \\\end{array} \right.$ x = 3 , y = 1 , z = - 5

$Solve the system. - \left\{ \begin{array} { l } x - 4 y = - 10 \\ 2 x - 8 y = - 17 \end{array} \right. A) dependent (many solutions) B) (2, 4) C) (2, 3) D) inconsistent (no solution)$ Solve the system. - $\left\{ \begin{array} { l } x - 4 y = - 10 \\2 x - 8 y = - 17\end{array} \right.$

Solve the problem. -Find real numbers a, b, and c such that the graph of the function y = ax² + bx + c contains the points (1, 1), (2, 4), and (-3, 29).

Verify that the values of the variables listed are solutions of the system of equations. - $\begin{array} { l } \left\{ \begin{array} { l } 2 x + y = 6 \\3 x + 2 y = 7\end{array} \right. \\x = 5 , y = - 4\end{array}$

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { l } x + y = - 1 \\x - y = 11 \\\end{array} \right.$ x = 5 , y = - 6

Solve the system of equations by using substitution. - $\left\{ \begin{array} { l } 3 x + y = 13 \\2 x + 9 y = - 8\end{array} \right.$

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { c } x - y + 3 z = - 20 \\4 x + z = - 5 \\x + 2 y + z = 5 \\\end{array} \right.$ x = 0 , y = 5 , z = - 5

Write the augmented matrix for the system. - $\left\{ \begin{array} { l } 2 x + 8 y = 64 \\5 x + 7 y = 69\end{array} \right.$

Write the augmented matrix for the system. - $\left\{ \begin{array} { l } 9 x - 2 y + 6 z = 54 \\6 x + 2 y + 9 z = 51 \\5 x + 3 y + 2 z = 41\end{array} \right.$

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.$

Write the augmented matrix for the system. - $\left\{ \begin{array} { r } - 2 x + 8 y = 28 \\5 y = 20\end{array} \right.$