Solve the system of equations. 4x-8y=-36, -6x+y=32

Answer 1

To solve the system of equations:

\[ 4x - 8y = -36 \]
\[ -6x + y = 32 \]

we can use either the substitution method or the elimination method. Here, we will use the elimination method.

First, let's align the equations for clarity:

\[ 4x - 8y = -36 \]
\[ -6x + y = 32 \]

To eliminate one of the variables, we will make the coefficients of \( y \) the same in both equations. Let's multiply the second equation by 8:

\[ 8(-6x + y) = 8(32) \]
\[ -48x + 8y = 256 \]

Now the system of equations becomes:

\[ 4x - 8y = -36 \]
\[ -48x + 8y = 256 \]

Next, we add these two equations to eliminate \( y \):

\[ (4x - 8y) + (-48x + 8y) = -36 + 256 \]
\[ 4x - 48x = 220 \]
\[ -44x = 220 \]

Divide both sides by \(-44\):

\[ x = \frac{220}{-44} \]
\[ x = -5 \]

Now that we have \( x = -5 \), we can substitute \( x \) back into the second equation to find \( y \):

\[ -6x + y = 32 \]
\[ -6(-5) + y = 32 \]
\[ 30 + y = 32 \]

Subtract 30 from both sides:

\[ y = 2 \]

So, the solution to the system of equations is \( x = -5 \) and \( y = 2 \).

To verify, substitute \( x = -5 \) and \( y = 2 \) back into the original equations:

1. \( 4(-5) - 8(2) = -20 - 16 = -36 \) (true)
2. \( -6(-5) + 2 = 30 + 2 = 32 \) (true)

Both equations are satisfied, confirming that the solution is correct. Therefore, the solution is:

\[ x = -5 \]
\[ y = 2 \]