Question 1

Solve the system by using the substitution method. $$\left( \begin{array} { c } 
5 x - 4 y = 1 \
x = 4 y - 11
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 2

Use the elimination-by-addition method to solve the system. $$\left( \begin{array} { c } 
4 x - 6 y = 10 \
4 x + 2 y = - 14
\end{array} \right)$$

Accepted Answer

If we add the two equations together, the $4x$ terms will cancel out, leaving us with $-4y=-4.$ Dividing by $-4$ gives us $y=1.$ Plugging this value of $y$ into either of the original equations, we get $4x-6=10$ or $4x+2=-14.$ Solving for $x$ in each case gives us $x=-2$ in both cases. Therefore, the solution is $(x,y)=(-2,1).$ This corresponds to choice C.

Question 3

If an equivalent system for an original system is $$\left( \begin{array} { c } 
5 x + 2 y = - 3 \
0 = 0
\end{array} \right)$$ , then the original system is dependent and has infinitely many solutions.

Accepted Answer

The second equation, 0=0, indicates that there is no restriction on the values of x and y, meaning that they can be any values. Therefore, the system is dependent and has infinitely many solutions.

Question 4

Cindy has 30 coins, consisting of dimes and quarters, that total $5.10. How many coins of each kind does she have? Enter your answer as two numbers, listed from least to greatest, separated by a comma.

Accepted Answer

The answer of Cindy has 30 coins, consisting of dimes...

Question 5

Any equation of the system can be replaced by the sum of the equation and a multiple of another equation.

Accepted Answer

The answer of Any equation of the system can be...

Question 6

Solve the system by using the substitution method. $$\left( \begin{array} { c } 
5 x - y = 71 \
x = \frac { 3 } { 2 } y - 4
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 7

Solve the system by using the substitution method. $$\left( \begin{array} { c } 
2 x - 6 y = - 8 \
3 x - 9 y = - 12
\end{array} \right)$$

Accepted Answer

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

Question 8

The system $$\left( \begin{array} { l } 
2 x + 5 y = 6 \
2 x + 5 y = - 6
\end{array} \right)$$ has the null set for its solution.

Accepted Answer

Since the two equations have the same coefficients for $x$ and $y$, subtracting one equation from the other yields $0=-12$, which is impossible. Therefore, the system has no solution, or a null set.

Question 9

Solve the system by using the substitution method. $$\left( \begin{array} { c } 
x + 2 y = 5 \
2 x + 4 y = - 1
\end{array} \right)$$

Accepted Answer

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

Question 10

The elimination-by-addition method involves replacing systems of equations with equivalent systems.

Accepted Answer

The answer of The elimination-by-addition method involves replacing systems of...

Question 11

Doris invested some money at 7% and some money at 8%. She invested $6,000 more at 8% than she did at 7%. Her total yearly interest from the two investments was $930. How much did Doris invest at each rate? Enter your answer as two numbers: amount invested at 7%, amount invested at 8%. Separate them by a semicolon. $__________; $__________

Accepted Answer

The answer of Doris invested some money at 7% and...

Question 12

Solve the system by using the substitution method. $$\left( \begin{array} { l } 
x = - 3 y \
7 x - 5 y = - 52
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 13

Use the elimination-by-addition method to solve the system. $$\left( \begin{array} { c } 
5 x + 6 y = - 1 \
8 x - 6 y = 92
\end{array} \right)$$

Accepted Answer

Adding the two equations directly eliminates $y$, resulting in $13x = 91$, which simplifies to $x = 7$. Substituting $x = 7$ into one of the original equations allows solving for $y$, yielding $y = -6$.

Question 14

Solve the system by using the substitution method. $$\left( \begin{array} { l } 
4 x - 5 y = - 5 \
8 x + 15 y = 25
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 15

The income from a student production was $13,250. The price of a student ticket was $7, and nonstudent tickets were sold at $12 each. One thousand five hundred tickets were sold. How many nonstudent tickets were sold?

Accepted Answer

The answer of The income from a student production was...

Question 16

Solve the system by using the substitution method. $$\left( \begin{array} { l } 
x - 6 y = 28 \
- 5 x + 8 y = - 30
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 17

The perimeter of a rectangle is 62 inches. The length of the rectangle is 9 inches more than the width. Find the width of the rectangle.

Accepted Answer

The answer of The perimeter of a rectangle is 62...

Question 18

Solve the system by using the substitution method. $$\left( \begin{array} { c } 
4 x + y = 40 \
5 x - y = 41
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 19

Solve the system by using the substitution method. $$\left( \begin{array} { c } 
y = \frac { 1 } { 5 } x - 3 \
4 x + 5 y = - 9
\end{array} \right)$$ Please enter your answer as an ordered pair in the form ( x , y ).

Accepted Answer

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

Question 20

Use the elimination-by-addition method to solve the system. $$\left( \begin{array} { c } 
x - 8 y = 65 \
4 x + 4 y = - 28
\end{array} \right)$$

Accepted Answer

To solve the system using the elimination-by-addition method, we first align the equations and then manipulate them to eliminate one of the variables. The given system is:$$x - 8y = 65$$$$4x + 4y = -28$$To eliminate $y$, we can multiply the first equation by 4 to make the coefficients of $y$ opposites:$$4(x - 8y) = 4(65)$$$$4x + 4y = -28$$This simplifies to:$$4x - 32y = 260$$$$4x + 4y = -28$$Adding these equations together to eliminate $y$, we get:$$4x + 4x - 32y + 4y = 260 - 28$$$$8x - 28y = 232$$Since we made an error in the addition step (we should have correctly eliminated $y$ and solved for $x$ directly), let's correctly eliminate $y$ and solve the system properly:The correct step is to add the original equations directly without multiplying the first equation by 4 (that step was unnecessary and led to an incorrect elimination process). The correct process is:$$x - 8y = 65$$$$4x + 4y = -28$$We can multiply the first equation by 4 to match the coefficient of $x$ in the second equation:$$4(x - 8y) = 4(65)$$$$4x + 4y = -28$$This gives us:$$4x - 32y = 260$$$$4x + 4y = -28$$Now, subtract the second equation from the first to eliminate $x$:$$4x - 32y - (4x + 4y) = 260 - (-28)$$$$-36y = 288$$Solving for $y$, we get:$$y = 288 / -36$$$$y = -8$$Substitute $y = -8$ into one of the original equations to solve for $x$:$$x - 8(-8) = 65$$$$x + 64 = 65$$$$x = 1$$Therefore, the solution is $x = 1$, $y = -8$, which corresponds to choice (C).

Quiz 5: Coordinate Geometry and Linear Systems

Solve the system by using the substitution method. $\left( \begin{array} { c } 5 x - 4 y = 1 \\x = 4 y - 11\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. $\left( \begin{array} { c } 4 x - 6 y = 10 \\4 x + 2 y = - 14\end{array} \right)$

If an equivalent system for an original system is $\left( \begin{array} { c } 5 x + 2 y = - 3 \\0 = 0\end{array} \right)$ , then the original system is dependent and has infinitely many solutions.

Cindy has 30 coins, consisting of dimes and quarters, that total $5.10. How many coins of each kind does she have? Enter your answer as two numbers, listed from least to greatest, separated by a comma.

Any equation of the system can be replaced by the sum of the equation and a multiple of another equation.

Solve the system by using the substitution method. $\left( \begin{array} { c } 5 x - y = 71 \\x = \frac { 3 } { 2 } y - 4\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. $\left( \begin{array} { c } 2 x - 6 y = - 8 \\3 x - 9 y = - 12\end{array} \right)$

The system $\left( \begin{array} { l } 2 x + 5 y = 6 \\2 x + 5 y = - 6\end{array} \right)$ has the null set for its solution.

Solve the system by using the substitution method. $\left( \begin{array} { c } x + 2 y = 5 \\2 x + 4 y = - 1\end{array} \right)$

The elimination-by-addition method involves replacing systems of equations with equivalent systems.

Solve the system by using the substitution method. $\left( \begin{array} { l } x = - 3 y \\7 x - 5 y = - 52\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. $\left( \begin{array} { c } 5 x + 6 y = - 1 \\8 x - 6 y = 92\end{array} \right)$

Solve the system by using the substitution method. $\left( \begin{array} { l } 4 x - 5 y = - 5 \\8 x + 15 y = 25\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

The income from a student production was $13,250. The price of a student ticket was $7, and nonstudent tickets were sold at $12 each. One thousand five hundred tickets were sold. How many nonstudent tickets were sold?

Solve the system by using the substitution method. $\left( \begin{array} { l } x - 6 y = 28 \\- 5 x + 8 y = - 30\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

The perimeter of a rectangle is 62 inches. The length of the rectangle is 9 inches more than the width. Find the width of the rectangle.

Solve the system by using the substitution method. $\left( \begin{array} { c } 4 x + y = 40 \\5 x - y = 41\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. $\left( \begin{array} { c } y = \frac { 1 } { 5 } x - 3 \\4 x + 5 y = - 9\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. $\left( \begin{array} { c } x - 8 y = 65 \\4 x + 4 y = - 28\end{array} \right)$

Quiz 5: Coordinate Geometry and Linear Systems

Solve the system by using the substitution method. (5x−4y=1x=4y−11)\left( \begin{array} { c } 5 x - 4 y = 1 \\x = 4 y - 11\end{array} \right)(5x−4y=1x=4y−11​) Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. (4x−6y=104x+2y=−14) \left( \begin{array} { c } 4 x - 6 y = 10 \\4 x + 2 y = - 14\end{array} \right) (4x−6y=104x+2y=−14​)

If an equivalent system for an original system is (5x+2y=−30=0)\left( \begin{array} { c } 5 x + 2 y = - 3 \\0 = 0\end{array} \right)(5x+2y=−30=0​) , then the original system is dependent and has infinitely many solutions.

Cindy has 30 coins, consisting of dimes and quarters, that total $5.10. How many coins of each kind does she have? Enter your answer as two numbers, listed from least to greatest, separated by a comma.

Any equation of the system can be replaced by the sum of the equation and a multiple of another equation.

Solve the system by using the substitution method. (5x−y=71x=32y−4)\left( \begin{array} { c } 5 x - y = 71 \\x = \frac { 3 } { 2 } y - 4\end{array} \right)(5x−y=71x=23​y−4​) Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. (2x−6y=−83x−9y=−12)\left( \begin{array} { c } 2 x - 6 y = - 8 \\3 x - 9 y = - 12\end{array} \right)(2x−6y=−83x−9y=−12​)

The system (2x+5y=62x+5y=−6)\left( \begin{array} { l } 2 x + 5 y = 6 \\2 x + 5 y = - 6\end{array} \right)(2x+5y=62x+5y=−6​) has the null set for its solution.

Solve the system by using the substitution method. (x+2y=52x+4y=−1)\left( \begin{array} { c } x + 2 y = 5 \\2 x + 4 y = - 1\end{array} \right)(x+2y=52x+4y=−1​)

The elimination-by-addition method involves replacing systems of equations with equivalent systems.

Solve the system by using the substitution method. (x=−3y7x−5y=−52)\left( \begin{array} { l } x = - 3 y \\7 x - 5 y = - 52\end{array} \right)(x=−3y7x−5y=−52​) Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. (5x+6y=−18x−6y=92) \left( \begin{array} { c } 5 x + 6 y = - 1 \\8 x - 6 y = 92\end{array} \right) (5x+6y=−18x−6y=92​)

Solve the system by using the substitution method. (4x−5y=−58x+15y=25)\left( \begin{array} { l } 4 x - 5 y = - 5 \\8 x + 15 y = 25\end{array} \right)(4x−5y=−58x+15y=25​) Please enter your answer as an ordered pair in the form ( x , y ).

The income from a student production was $13,250. The price of a student ticket was $7, and nonstudent tickets were sold at $12 each. One thousand five hundred tickets were sold. How many nonstudent tickets were sold?

Solve the system by using the substitution method. (x−6y=28−5x+8y=−30)\left( \begin{array} { l } x - 6 y = 28 \\- 5 x + 8 y = - 30\end{array} \right)(x−6y=28−5x+8y=−30​) Please enter your answer as an ordered pair in the form ( x , y ).

The perimeter of a rectangle is 62 inches. The length of the rectangle is 9 inches more than the width. Find the width of the rectangle.

Solve the system by using the substitution method. (4x+y=405x−y=41)\left( \begin{array} { c } 4 x + y = 40 \\5 x - y = 41\end{array} \right)(4x+y=405x−y=41​) Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. (y=15x−34x+5y=−9)\left( \begin{array} { c } y = \frac { 1 } { 5 } x - 3 \\4 x + 5 y = - 9\end{array} \right)(y=51​x−34x+5y=−9​) Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. (x−8y=654x+4y=−28) \left( \begin{array} { c } x - 8 y = 65 \\4 x + 4 y = - 28\end{array} \right) (x−8y=654x+4y=−28​)

Solve the system by using the substitution method. $\left( \begin{array} { c } 5 x - 4 y = 1 \\x = 4 y - 11\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. $\left( \begin{array} { c } 4 x - 6 y = 10 \\4 x + 2 y = - 14\end{array} \right)$

If an equivalent system for an original system is $\left( \begin{array} { c } 5 x + 2 y = - 3 \\0 = 0\end{array} \right)$ , then the original system is dependent and has infinitely many solutions.

Solve the system by using the substitution method. $\left( \begin{array} { c } 5 x - y = 71 \\x = \frac { 3 } { 2 } y - 4\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. $\left( \begin{array} { c } 2 x - 6 y = - 8 \\3 x - 9 y = - 12\end{array} \right)$

The system $\left( \begin{array} { l } 2 x + 5 y = 6 \\2 x + 5 y = - 6\end{array} \right)$ has the null set for its solution.

Solve the system by using the substitution method. $\left( \begin{array} { c } x + 2 y = 5 \\2 x + 4 y = - 1\end{array} \right)$

Solve the system by using the substitution method. $\left( \begin{array} { l } x = - 3 y \\7 x - 5 y = - 52\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. $\left( \begin{array} { c } 5 x + 6 y = - 1 \\8 x - 6 y = 92\end{array} \right)$

Solve the system by using the substitution method. $\left( \begin{array} { l } 4 x - 5 y = - 5 \\8 x + 15 y = 25\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. $\left( \begin{array} { l } x - 6 y = 28 \\- 5 x + 8 y = - 30\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. $\left( \begin{array} { c } 4 x + y = 40 \\5 x - y = 41\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Solve the system by using the substitution method. $\left( \begin{array} { c } y = \frac { 1 } { 5 } x - 3 \\4 x + 5 y = - 9\end{array} \right)$ Please enter your answer as an ordered pair in the form ( x , y ).

Use the elimination-by-addition method to solve the system. $\left( \begin{array} { c } x - 8 y = 65 \\4 x + 4 y = - 28\end{array} \right)$