Solved on Jan 25, 2024

Solve the system of linear equations $-4x - 8y = 8$ and $3x - 5y = 16$ using the elimination method.

STEP 1

Assumptions
1. We have a system of two linear equations with two variables $x$ and $y$ .
2. We will use the elimination method to solve the system.

STEP 2

Write down the system of equations.
$\begin{aligned} -4x - 8y &= 8 \\ 3x - 5y &= 16 \end{aligned}$

STEP 3

To use the elimination method, we need to eliminate one of the variables. We can do this by making the coefficients of either $x$ or $y$ the same in both equations. Let's choose to eliminate $x$ .

STEP 4

Find a common multiple for the coefficients of $x$ in both equations. The coefficients are $-4$ and $3$ . The least common multiple (LCM) of $4$ and $3$ is $12$ .

STEP 5

Multiply the first equation by $3$ and the second equation by $4$ to get the coefficients of $x$ to be $12$ and $-12$ , respectively.
$\begin{aligned} 3(-4x - 8y) &= 3(8) \\ 4(3x - 5y) &= 4(16) \end{aligned}$

STEP 6

Perform the multiplication for both equations.
$\begin{aligned} -12x - 24y &= 24 \\ 12x - 20y &= 64 \end{aligned}$

STEP 7

Now, add the two equations together to eliminate $x$ .
$(-12x - 24y) + (12x - 20y) = 24 + 64$

STEP 8

Perform the addition.
$-44y = 88$

STEP 9

Solve for $y$ by dividing both sides of the equation by $-44$ .
$y = \frac{88}{-44}$

STEP 10

Calculate the value of $y$ .
$y = -2$

STEP 11

Now that we have the value of $y$ , we can substitute it back into one of the original equations to find $x$ . Let's use the first equation.
$-4x - 8(-2) = 8$

STEP 12

Simplify the equation by performing the multiplication.
$-4x + 16 = 8$

STEP 13

Subtract $16$ from both sides of the equation to isolate the term with $x$ .
$-4x = 8 - 16$

STEP 14

Calculate the right side of the equation.
$-4x = -8$

STEP 15

Divide both sides by $-4$ to solve for $x$ .
$x = \frac{-8}{-4}$

STEP 16

Calculate the value of $x$ .
$x = 2$
The solution to the system of equations is $x = 2$ and $y = -2$ .

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Solve the system of linear equations $-4x - 8y = 8$ and $3x - 5y = 16$ using the elimination method.

STEP 1

Assumptions
1. We have a system of two linear equations with two variables $x$ and $y$ .
2. We will use the elimination method to solve the system.

STEP 2

Write down the system of equations.
$\begin{aligned} -4x - 8y &= 8 \\ 3x - 5y &= 16 \end{aligned}$

STEP 3

To use the elimination method, we need to eliminate one of the variables. We can do this by making the coefficients of either $x$ or $y$ the same in both equations. Let's choose to eliminate $x$ .

STEP 4

Find a common multiple for the coefficients of $x$ in both equations. The coefficients are $-4$ and $3$ . The least common multiple (LCM) of $4$ and $3$ is $12$ .

STEP 5

Multiply the first equation by $3$ and the second equation by $4$ to get the coefficients of $x$ to be $12$ and $-12$ , respectively.
$\begin{aligned} 3(-4x - 8y) &= 3(8) \\ 4(3x - 5y) &= 4(16) \end{aligned}$

STEP 6

Perform the multiplication for both equations.
$\begin{aligned} -12x - 24y &= 24 \\ 12x - 20y &= 64 \end{aligned}$

STEP 7

Now, add the two equations together to eliminate $x$ .
$(-12x - 24y) + (12x - 20y) = 24 + 64$

STEP 8

Perform the addition.
$-44y = 88$

STEP 9

Solve for $y$ by dividing both sides of the equation by $-44$ .
$y = \frac{88}{-44}$

STEP 10

Calculate the value of $y$ .
$y = -2$

STEP 11

Now that we have the value of $y$ , we can substitute it back into one of the original equations to find $x$ . Let's use the first equation.
$-4x - 8(-2) = 8$

STEP 12

Simplify the equation by performing the multiplication.
$-4x + 16 = 8$

STEP 13

Subtract $16$ from both sides of the equation to isolate the term with $x$ .
$-4x = 8 - 16$

STEP 14

Calculate the right side of the equation.
$-4x = -8$

STEP 15

Divide both sides by $-4$ to solve for $x$ .
$x = \frac{-8}{-4}$

STEP 16

Calculate the value of $x$ .
$x = 2$
The solution to the system of equations is $x = 2$ and $y = -2$ .

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Solve the system of linear equations −4x−8y=8-4x - 8y = 8−4x−8y=8 and 3x−5y=163x - 5y = 163x−5y=16 using the elimination method.

STEP 1

Assumptions1. We have a system of two linear equations with two variables xxx and yyy.2. We will use the elimination method to solve the system.

STEP 2

Write down the system of equations.−4x−8y=83x−5y=16 \begin{aligned} -4x - 8y &= 8 \\ 3x - 5y &= 16 \end{aligned} −4x−8y3x−5y​=8=16​

STEP 3

To use the elimination method, we need to eliminate one of the variables. We can do this by making the coefficients of either xxx or yyy the same in both equations. Let's choose to eliminate xxx.

STEP 4

Find a common multiple for the coefficients of xxx in both equations. The coefficients are −4-4−4 and 333. The least common multiple (LCM) of 444 and 333 is 121212.

STEP 5

Multiply the first equation by 333 and the second equation by 444 to get the coefficients of xxx to be 121212 and −12-12−12, respectively.3(−4x−8y)=3(8)4(3x−5y)=4(16) \begin{aligned} 3(-4x - 8y) &= 3(8) \\ 4(3x - 5y) &= 4(16) \end{aligned} 3(−4x−8y)4(3x−5y)​=3(8)=4(16)​

STEP 6

Perform the multiplication for both equations.−12x−24y=2412x−20y=64 \begin{aligned} -12x - 24y &= 24 \\ 12x - 20y &= 64 \end{aligned} −12x−24y12x−20y​=24=64​

STEP 7

Now, add the two equations together to eliminate xxx.(−12x−24y)+(12x−20y)=24+64 (-12x - 24y) + (12x - 20y) = 24 + 64 (−12x−24y)+(12x−20y)=24+64

STEP 8

Perform the addition.−44y=88 -44y = 88 −44y=88

STEP 9

Solve for yyy by dividing both sides of the equation by −44-44−44.y=88−44 y = \frac{88}{-44} y=−4488​

STEP 10

Calculate the value of yyy.y=−2 y = -2 y=−2

STEP 11

Now that we have the value of yyy, we can substitute it back into one of the original equations to find xxx. Let's use the first equation.−4x−8(−2)=8 -4x - 8(-2) = 8 −4x−8(−2)=8

STEP 12

Simplify the equation by performing the multiplication.−4x+16=8 -4x + 16 = 8 −4x+16=8

STEP 13

Subtract 161616 from both sides of the equation to isolate the term with xxx.−4x=8−16 -4x = 8 - 16 −4x=8−16

STEP 14

Calculate the right side of the equation.−4x=−8 -4x = -8 −4x=−8

STEP 15

Divide both sides by −4-4−4 to solve for xxx.x=−8−4 x = \frac{-8}{-4} x=−4−8​

STEP 16

Calculate the value of xxx.x=2 x = 2 x=2The solution to the system of equations is x=2x = 2x=2 and y=−2y = -2y=−2.

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Solve the system of linear equations $-4x - 8y = 8$ and $3x - 5y = 16$ using the elimination method.

Assumptions
1. We have a system of two linear equations with two variables $x$ and $y$ .
2. We will use the elimination method to solve the system.

Write down the system of equations.
$\begin{aligned} -4x - 8y &= 8 \\ 3x - 5y &= 16 \end{aligned}$

To use the elimination method, we need to eliminate one of the variables. We can do this by making the coefficients of either $x$ or $y$ the same in both equations. Let's choose to eliminate $x$ .

Find a common multiple for the coefficients of $x$ in both equations. The coefficients are $-4$ and $3$ . The least common multiple (LCM) of $4$ and $3$ is $12$ .

Multiply the first equation by $3$ and the second equation by $4$ to get the coefficients of $x$ to be $12$ and $-12$ , respectively.
$\begin{aligned} 3(-4x - 8y) &= 3(8) \\ 4(3x - 5y) &= 4(16) \end{aligned}$

Perform the multiplication for both equations.
$\begin{aligned} -12x - 24y &= 24 \\ 12x - 20y &= 64 \end{aligned}$

Now, add the two equations together to eliminate $x$ .
$(-12x - 24y) + (12x - 20y) = 24 + 64$

Perform the addition.
$-44y = 88$

Solve for $y$ by dividing both sides of the equation by $-44$ .
$y = \frac{88}{-44}$

Calculate the value of $y$ .
$y = -2$

Now that we have the value of $y$ , we can substitute it back into one of the original equations to find $x$ . Let's use the first equation.
$-4x - 8(-2) = 8$

Simplify the equation by performing the multiplication.
$-4x + 16 = 8$

Subtract $16$ from both sides of the equation to isolate the term with $x$ .
$-4x = 8 - 16$

Calculate the right side of the equation.
$-4x = -8$

Divide both sides by $-4$ to solve for $x$ .
$x = \frac{-8}{-4}$

Calculate the value of $x$ .
$x = 2$
The solution to the system of equations is $x = 2$ and $y = -2$ .