Question 1

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 2

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.

Accepted Answer

This system is consistent because the second equation is just the first equation multiplied by -4. Therefore, the two equations are really just stating the same thing and are not independent. This means that the system has infinitely many solutions, which makes it consistent.

Question 3

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.

Accepted Answer

The answer of An airplane flying into a headwind travels...

Question 4

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.

Accepted Answer

Let x be the price of adult tickets and y be the price of children's tickets.
Then we have the system of equations:
160x + 237y = 3478.25
193x + 314y = 4399.5
Multiplying the first equation by 193 and the second equation by -160 and adding them, we get:
267y = 3191
y = 11.97
Substituting y = 11.97 into the first equation, we get:
160x + 237(11.97) = 3478.25
160x = 2833.76
x = 17.71
Therefore, the price of adult tickets is $11 and the price of children's tickets is $7.25.

Question 5

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 6

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 7

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.

Accepted Answer

To solve this system, we can multiply the first equation by 3 to get:

\begin{align*}
15x - 18y &= 12 \
-15x + 18y &= 14
\end{align*}

Adding the two equations together, we get:

\begin{align*}
0 &= 26
\end{align*}

This is a contradiction, so the system is inconsistent. The answer is A.

Question 8

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 9

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 .

Accepted Answer

The answer of Find two positive integers that satisfy the...

Question 10

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

Accepted Answer

To solve this, we can use the equation for mixture problems: (amount of first solution * its percentage + amount of second solution * its percentage) / total amount = desired percentage. Let x be the amount of 34% solution, then (16-x) is the amount of 84% solution. The equation is: (0.34x + 0.84(16-x)) / 16 = 0.715. Solving for x gives x = 4, so 4 liters of 34% solution and 12 liters of 84% solution are needed.

Question 11

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

Accepted Answer

Multiply the second equation by 2 and add the two equations to eliminate $x$:
\begin{align*}
2x-2y&=2 \
-4x+2y&=-4 \
\end{align*}
Adding yields $-2x=-2$, or $x=1$. Substituting into the second equation gives $y=0$. Therefore, the solution is $(1,0)$, which is choice $\boxed{	extbf{(C)}}$.

Question 12

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

Accepted Answer

To solve the system of equations by elimination, we aim to eliminate one variable by adding or subtracting the equations. First, we can multiply the second equation by 2 to make the coefficients of $y$ in both equations equal:Original equations:1. $9x + 4y = -108$2. $-7x + 2y = 38$Multiply the second equation by 2:1. $9x + 4y = -108$2. $-14x + 4y = 76$Now, subtract the second equation from the first to eliminate $y$:$$9x - (-14x) = -108 - 76$$$$23x = -184$$$$x = -184 / 23$$$$x = -8$$Substitute $x = -8$ into one of the original equations to find $y$. Using the first equation:$$9(-8) + 4y = -108$$$$-72 + 4y = -108$$$$4y = -108 + 72$$$$4y = -36$$$$y = -9$$Therefore, the solution to the system of equations is $x = -8, y = -9$, which corresponds to choice B.

Question 13

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.

Accepted Answer

To break even, the revenue needs to cover the costs, meaning:

$9,700x = $8,850x + $150,000

$850x = $150,000

x = 176.47

Rounding up to the nearest week, it will take 177 weeks to break even. Therefore, the best choice is C.

Question 14

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

Accepted Answer

Multiplying the second equation by $4$ we get, $36u+8v=8$. Adding this to the first equation which is $-9u-8v=-8$ gives $27u=0$ which implies $u=0$. Substituting the value of $u$ in any one equation above we get $v=1$. Thus the solution is $(u,v)=(0,1)$. Therefore, the answer is D.

Question 15

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

Accepted Answer

The second equation is a multiple of the first (multiplying the first equation by 2 gives the second equation), indicating that both equations represent the same line. Therefore, there are infinitely many solutions, as every point on the line is a solution to the system of equations.

Question 16

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 17

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

The answer of Solve the system of linear equations below...

Question 20

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

Accepted Answer

The answer of Determine the value of k such that...

Quiz 4: Systems of Equations and Inequalities

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

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 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.

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

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

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.

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

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 .

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

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

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

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.

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

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

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

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

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

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

Quiz 4: Systems of Equations and Inequalities

Solve the system of linear equations below by the method of elimination. {4r−5s=52r−5s=15\left\{ \begin{array} { c } 4 r - 5 s = 5 \\2 r - 5 s = 15\end{array} \right.{4r−5s=52r−5s=15​

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

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.

Solve the system of linear equations below by the method of elimination, if a single solution exists. {−2x+y=72x−5y=−27\left\{ \begin{array} { c } - 2 x + y = 7 \\2 x - 5 y = - 27\end{array} \right.{−2x+y=72x−5y=−27​

Solve the system of linear equations below by the method of elimination, if a single solution exists. {−4x+y=−412x−3y=3\left\{ \begin{array} { l } - 4 x + y = - 4 \\12 x - 3 y = 3\end{array} \right.{−4x+y=−412x−3y=3​

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

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

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 .

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

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

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

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.

Solve the system of linear equations below by the method of elimination, if a single solution exists. {−9u−8v=−89u+2v=2\left\{ \begin{array} { c } - 9 u - 8 v = - 8 \\9 u + 2 v = 2\end{array} \right.{−9u−8v=−89u+2v=2​

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

Solve the system of linear equations below by the method of elimination. {12x+13y=193x−4y=−133\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.⎩⎨⎧​21​x+31​y=91​3x−4y=−313​​

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

Determine the value of k such that the system of linear equations below is inconsistent. {7x+18y=−35x+ky=3\left\{ \begin{array} { c } 7 x + 18 y = - 3 \\5 x + k y = 3\end{array} \right.{7x+18y=−35x+ky=3​

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

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.

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

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

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.

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

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

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

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

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

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

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