To solve the system of equations using the elimination method, we'll follow these steps:
1. Write down the given system of equations:
[tex]\[
\begin{cases}
5x + 8y = -88 \\
x - y = -2
\end{cases}
\][/tex]
2. Eliminate one of the variables:
To eliminate the variable [tex]\(x\)[/tex], we can first ensure that the coefficient of [tex]\(x\)[/tex] in the second equation is the same as in the first. We'll make the coefficient of [tex]\(x\)[/tex] in the second equation equal to 5 by multiplying the entire second equation by 5:
[tex]\[
5(x - y) = 5(-2)
\][/tex]
This gives us the new system:
[tex]\[
\begin{cases}
5x + 8y = -88 \\
5x - 5y = -10
\end{cases}
\][/tex]
3. Subtract the second modified equation from the first to eliminate [tex]\(x\)[/tex]:
[tex]\[
\begin{aligned}
(5x + 8y) - (5x - 5y) &= -88 - (-10) \\
5x + 8y - 5x + 5y &= -88 + 10 \\
13y &= -78
\end{aligned}
\][/tex]
Simplifying this, we find:
[tex]\[
y = \frac{-78}{13} = -6
\][/tex]
4. Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
We'll use the second equation for substitution:
[tex]\[
x - y = -2
\][/tex]
Substitute [tex]\(y = -6\)[/tex]:
[tex]\[
x - (-6) = -2 \\
x + 6 = -2 \\
x = -2 - 6 \\
x = -8
\][/tex]
5. Write the solution as an ordered pair:
[tex]\[
(x, y) = (-8, -6)
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(-8, -6)
\][/tex]
