To solve the given system of equations:
[tex]\[
\begin{cases}
-x + 3y = 32 \\
7x + 3y = -8
\end{cases}
\][/tex]
we can use the method of elimination to find [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Eliminate variable [tex]\( y \)[/tex]:
To eliminate [tex]\( y \)[/tex], we subtract the first equation from the second equation. This gives us a system of equations with only [tex]\( x \)[/tex] as the variable. Let's write down both equations:
[tex]\[
\begin{cases}
-x + 3y = 32 \quad \text{(Equation 1)} \\
7x + 3y = -8 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
Subtract Equation 1 from Equation 2:
[tex]\[
(7x + 3y) - (-x + 3y) = -8 - 32
\][/tex]
Simplify the resulting equation:
[tex]\[
7x + 3y + x - 3y = -8 - 32
\][/tex]
[tex]\[
8x = -40
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{40}{8} = -5
\][/tex]
So, [tex]\( x = -5 \)[/tex].
2. Solve for [tex]\( y \)[/tex]:
Now that we have [tex]\( x = -5 \)[/tex], substitute this value into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use Equation 1:
[tex]\[
-x + 3y = 32
\][/tex]
Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[
-(-5) + 3y = 32
\][/tex]
Simplify:
[tex]\[
5 + 3y = 32
\][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[
3y = 32 - 5
\][/tex]
[tex]\[
3y = 27
\][/tex]
[tex]\[
y = \frac{27}{3} = 9
\][/tex]
So, [tex]\( y = 9 \)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (-5, 9)
\][/tex]
