Question 1

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { l } 
5 x + 3 y = 80 \
2 x + y = 30
\end{array} \right.$$
A) $x = 0 , y = 0$
B) $x = 10 , y = 10$
C) $x = 0 , y = 10$
D) $x = 10 , y = 0$

Accepted Answer

First, solve the second equation for $y$: $y = 30 - 2x$. Then substitute this expression for $y$ in the first equation: $5x + 3(30 - 2x) = 80$. Simplifying gives $5x + 90 - 6x = 80$, which simplifies to $-x = -10$, so $x = 10$. Substituting $x = 10$ back into $y = 30 - 2x$ gives $y = 30 - 20 = 10$.

Question 2

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

Accepted Answer

The second equation is not a multiple of the first, suggesting a unique solution, but when attempting to solve, the system becomes inconsistent, indicating no solution.

Question 3

Solve the problem.
-A retired couple has $170,000 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 11% per year. How much
Should they invest in each to realize exactly $16,300 per year?
A) $120,000 at 5% and $70,000 at 12%
B) $130,000 at 5% and $60,000 at 12%
C) $130,000 at 12% and $60,000 at 5%
D) $140,000 at 12% and $50,000 at 5%

Accepted Answer

To solve, let x be the amount invested at 5% and y be the amount invested at 11%. We have two equations: x + y = 170,000 and 0.05x + 0.11y = 16,300. Solving these, we find x = 60,000 and y = 110,000.

Question 4

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { r } 
x + 7 y = - 2 \
3 x + y = 34
\end{array} \right.$$
A) $x = - 2 , y = 3$
B) $x = 7 , y = 12$
C) $x = 12 , y = - 2$
D) $x = 3 , y = 7$

Accepted Answer

From the first equation, solve for $x$: $x = -2 - 7y$. Substitute this expression for $x$ in the second equation: $3(-2 - 7y) + y = 34$. Simplify and solve for $y$, then substitute $y$ back to find $x$. The solution is $x = 12, y = -2$.

Question 5

Solve the system.
-$$\left\{ \begin{array} { l } 
x + y = 1 \
x + y = - 3
\end{array} \right.$$
A) inconsistent (no solution)
B) $( 0 , - 2 )$
C) dependent (many solutions)
D) $( 1 , - 3 )$

Accepted Answer

The two equations are parallel lines (same x and y coefficients but different constants), meaning they never intersect. This indicates the system is inconsistent with no solution.

Question 6

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) 168 miles
B) 153 miles
C) 147 miles
D) 132 miles

Accepted Answer

Let x be the miles driven in the city and (300 - x) be the miles driven on the highway. The total gasoline used for both city and highway driving is 15.5 gallons. The equation can be set up as x/18 + (300-x)/21 = 15.5. Solving this equation for x gives the miles driven in the city as 153 miles.

Question 7

Solve the problem.
-The Family Fine Arts Center charges $20 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 560 people paid admission, the total receipts were $7,390. How many who paid were
Senior citizens?
A) 282 senior citizens
B) 115 senior citizens
C) 205 senior citizens
D) 372 senior citizens

Accepted Answer

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

Question 8

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

First, solve the first equation for $y$: $y = 13 - 3x$. Then substitute this expression for $y$ in the second equation: $2x + 9(13 - 3x) = -8$. Simplify and solve for $x$, which gives $x = 5$. Substitute $x = 5$ back into $y = 13 - 3x$ to find $y = -2$.

Question 9

Use the elimination method to solve the system.
-$$\left\{ \begin{array} { c } 
2 x + 24 y = - 154 \
9 x + 4 y = 35
\end{array} \right.$$
A) $x = 9 , y = - 9$
B) $x = 7 , y = - 7$
C) $x = - 4 , y = 7$
D) $x = - 7 , y = 7$

Accepted Answer

The answer of Use the elimination method to solve the...

Question 10

Use the elimination method to solve the system.
-$$\left\{ \begin{array} { l } 
x + y = - 8 \
x - y = 12
\end{array} \right.$$
A) $x = 5 , y = 14$
B) $x = 8 , y = - 10$
C) $x = 2 , y = - 10$
D) $x = 9 , y = 5$

Accepted Answer

Adding the two equations $x + y = -8$ and $x - y = 12$ eliminates $y$, giving $2x = 4$, so $x = 2$. Substituting $x = 2$ into the first equation gives $2 + y = -8$, so $y = -10$.

Question 11

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { r } 
5 x - 2 y = - 1 \
x + 4 y = 35
\end{array} \right.$$
A) $x = 3 , y = 9$
B) $x = 2 , y = 9$
C) $x = 3 , y = 8$
D) $x = 2 , y = 8$

Accepted Answer

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

Question 12

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 5 soft drinks for $40.29. The second group bought 6 slices of pizza and 7 soft drinks for $
35)71. How much does one slice of pizza cost?
A) $1.94 per slice of pizza
B) $2.44 per slice of pizza
C) $3.27 per slice of pizza
D) $2.77 per slice of pizza

Accepted Answer

By setting up a system of equations based on the given information and solving for the cost of one slice of pizza, we find that it costs $3.27 per slice.

Question 13

Verify that the values of the variables listed are solutions of the system of equations.
-$$\left\{ \begin{array} { r } 
x - y + 4 z = - 3 \
3 x + z = - 2 \
x + 2 y + z = - 12 \
\end{array} \right.$$
x = 0 , y = - 5 , z = - 2

A) solution
B) not a solution

Accepted Answer

Substituting $x = 0$, $y = -5$, and $z = -2$ into each equation: 1. $0 - (-5) + 4(-2) = 5 - 8 = -3$, 2. $3(0) + (-2) = -2$, 3. $0 + 2(-5) + (-2) = -10 - 2 = -12$. All equations are satisfied, thus the values are a solution.

Question 14

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) 29 adults
C) 16 adults
D) 24 adults

Accepted Answer

The answer of Solve the problem.
-A movie theater charges $8.00...

Question 15

Use the elimination method to solve the system.
-$\left\{ \begin{array} { c } 3 x + 48 y = 48 \ 3 x - 6 y = - 6 \ \end{array} \right.$

A) x = 0 , y = 1 
B) $x = 1 , y = 1$
C) $x = 0 , y = 0$
D) $x = 1 , y = 0$

Accepted Answer

Subtracting the second equation from the first eliminates $x$, leading to $54y = 54$, so $y = 1$. Substituting $y = 1$ into the second equation gives $3x - 6(1) = -6$, so $3x = 0$, and $x = 0$.

Question 16

Solve the problem.
-A flat rectangular piece of aluminum has a perimeter of 68 inches. The length is 14 inches longer than the width. Find the width.
A) 28 inches
B) 26 inches
C) 8 inches
D) 18 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 ($l = w + 14$), and the perimeter is 68 inches, we can set up the equation $68 = 2(w + 14) + 2w$. Simplifying, we get $68 = 4w + 28$, and solving for $w$, we find $w = 10$. Thus, the width is 10 inches, which is not listed among the options provided, indicating a mistake in my calculation. Let's correct the calculation: Given $68 = 2(w + 14) + 2w$, simplifying gives $68 = 4w + 28$. Subtracting 28 from both sides gives $40 = 4w$, and dividing by 4 gives $w = 10$. My initial explanation correctly identifies the process but incorrectly concludes the answer based on the options provided. The correct process indeed leads to $w = 10$, which is not listed among the options, suggesting a misunderstanding in the final step of my explanation or a misinterpretation of the options. Let's accurately address the problem with the correct approach:Given the perimeter formula $P = 2l + 2w$ and the relationship $l = w + 14$, substituting the values correctly, we should solve for $w$ accurately. The correct setup is $68 = 2(w + 14) + 2w$, leading to $68 = 2w + 28 + 2w$, which simplifies to $68 = 4w + 28$. Subtracting 28 from both sides gives $40 = 4w$, and dividing by 4 gives $w = 10$. It appears there was a mistake in matching the calculated width to the provided options. The correct calculation process indeed indicates that the width should be 10 inches, which was not an option provided. This indicates an error in my initial response in identifying the correct option based on the calculation. Given the context and the options, the correct approach to solving the problem was demonstrated, but the conclusion was inaccurately matched to the available choices, leading to confusion. The correct answer, based on the calculation, should be 10 inches, which necessitates a reevaluation of the options or an acknowledgment of an error in the initial problem setup or in the interpretation of the options.

Question 17

Verify that the values of the variables listed are solutions of the system of equations.
-$$\begin{array} { l } 
\left\{ \begin{array} { l } 
4 x + y = 13 \
2 x + 4 y = 24
\end{array} \right. \
\end{array}$$
x = 2 , y = 5

A) not a solution
B) solution

Accepted Answer

Substituting x = 2 and y = 5 into both equations: For the first equation, 4(2) + 5 = 8 + 5 = 13, which matches the right side. For the second equation, 2(2) + 4(5) = 4 + 20 = 24, which also matches the right side. Therefore, x = 2 and y = 5 are solutions to the system of equations.

Question 18

Verify that the values of the variables listed are solutions of the system of equations.
-$$\left\{ \begin{array} { c } 
x + y + z = 6 \
x - y + 4 z = 14 \
5 x + y + z = 26 \
\end{array} \right.$$
x = 5 , y = - 1 , z = 2
A) not a solution
B) solution

Accepted Answer

To verify if $x = 5$, $y = -1$, and $z = 2$ are solutions to the system of equations, we substitute these values into each equation and check if they satisfy all the equations.1. For the first equation $x + y + z = 6$:$$5 + (-1) + 2 = 6$$$$6 = 6$$ (True)2. For the second equation $x - y + 4z = 14$:$$5 - (-1) + 4(2) = 5 + 1 + 8 = 14$$$$14 = 14$$ (True)3. For the third equation $5x + y + z = 26$:$$5(5) + (-1) + 2 = 25 - 1 + 2 = 26$$$$26 = 26$$ (True)Since the values satisfy all three equations, they are indeed solutions to the system of equations.

Question 19

Solve the system of equations by using substitution.
-$$\left\{ \begin{array} { c } 
x + 7 y = 7 \
3 x - 8 y = - 8
\end{array} \right.$$
A) $x = 1 , y = 0$
B) $x = 0 , y = 0$
C) $x = 0 , y = 1$
D) $x = 1 , y = 1$

Accepted Answer

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

Question 20

Verify that the values of the variables listed are solutions of the system of equations.
-$$\left\{ \begin{array} { l } 
x + y = 8 \
x - y = 4 \
\end{array} \right.$$
x = 6 , y = 2
A) not a solution
B) solution

Accepted Answer

Substituting x = 6 and y = 2 into both equations: For the first equation, 6 + 2 = 8, which is true. For the second equation, 6 - 2 = 4, which is also true. Therefore, x = 6 and y = 2 are solutions to the system of equations.

Quiz 10: Systems of Equations and Inequalities

Solve the system of equations by using substitution. - $\left\{ \begin{array} { l } 5 x + 3 y = 80 \\2 x + y = 30\end{array} \right.$

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

Solve the problem. -A retired couple has $170,000 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 11% per year. How much Should they invest in each to realize exactly $16,300 per year?

Solve the system of equations by using substitution. - $\left\{ \begin{array} { r } x + 7 y = - 2 \\3 x + y = 34\end{array} \right.$

Solve the system. - $\left\{ \begin{array} { l } x + y = 1 \\x + y = - 3\end{array} \right.$

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?

Solve the problem. -The Family Fine Arts Center charges $20 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 560 people paid admission, the total receipts were $7,390. How many who paid were Senior citizens?

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

Use the elimination method to solve the system. - $\left\{ \begin{array} { c } 2 x + 24 y = - 154 \\9 x + 4 y = 35\end{array} \right.$

Use the elimination method to solve the system. - $\left\{ \begin{array} { l } x + y = - 8 \\x - y = 12\end{array} \right.$

Solve the system of equations by using substitution. - $\left\{ \begin{array} { r } 5 x - 2 y = - 1 \\x + 4 y = 35\end{array} \right.$

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 5 soft drinks for $40.29. The second group bought 6 slices of pizza and 7 soft drinks for $ 35) 71. How much does one slice of pizza cost?

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { r } x - y + 4 z = - 3 \\3 x + z = - 2 \\x + 2 y + z = - 12 \\\end{array} \right.$ x = 0 , y = - 5 , z = - 2

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?

Use the elimination method to solve the system. - $\left\{ \begin{array} { c } 3 x + 48 y = 48 \\ 3 x - 6 y = - 6 \\ \end{array} \right.$

Solve the problem. -A flat rectangular piece of aluminum has a perimeter of 68 inches. The length is 14 inches longer than the width. Find the width.

Verify that the values of the variables listed are solutions of the system of equations. - $\begin{array} { l } \left\{ \begin{array} { l } 4 x + y = 13 \\2 x + 4 y = 24\end{array} \right. \\\end{array}$ x = 2 , y = 5

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { c } x + y + z = 6 \\x - y + 4 z = 14 \\5 x + y + z = 26 \\\end{array} \right.$ x = 5 , y = - 1 , z = 2

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 7 y = 7 \\3 x - 8 y = - 8\end{array} \right.$

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { l } x + y = 8 \\x - y = 4 \\\end{array} \right.$ x = 6 , y = 2

Quiz 10: Systems of Equations and Inequalities

Solve the system of equations by using substitution. - {5x+3y=802x+y=30\left\{ \begin{array} { l } 5 x + 3 y = 80 \\2 x + y = 30\end{array} \right.{5x+3y=802x+y=30​

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. -A retired couple has $170,000 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 11% per year. How much Should they invest in each to realize exactly $16,300 per year?

Solve the system of equations by using substitution. - {x+7y=−23x+y=34\left\{ \begin{array} { r } x + 7 y = - 2 \\3 x + y = 34\end{array} \right.{x+7y=−23x+y=34​

Solve the system. - {x+y=1x+y=−3\left\{ \begin{array} { l } x + y = 1 \\x + y = - 3\end{array} \right.{x+y=1x+y=−3​

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?

Solve the problem. -The Family Fine Arts Center charges $20 per adult and $10 per senior citizen for its performances. On a recent weekend evening when 560 people paid admission, the total receipts were $7,390. How many who paid were Senior citizens?

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​

Use the elimination method to solve the system. - {2x+24y=−1549x+4y=35\left\{ \begin{array} { c } 2 x + 24 y = - 154 \\9 x + 4 y = 35\end{array} \right.{2x+24y=−1549x+4y=35​

Use the elimination method to solve the system. - {x+y=−8x−y=12\left\{ \begin{array} { l } x + y = - 8 \\x - y = 12\end{array} \right.{x+y=−8x−y=12​

Solve the system of equations by using substitution. - {5x−2y=−1x+4y=35\left\{ \begin{array} { r } 5 x - 2 y = - 1 \\x + 4 y = 35\end{array} \right.{5x−2y=−1x+4y=35​