To solve the system of equations:
[tex]\[
\begin{cases}
2x + 3y = 2 \\
x - 2y = 8
\end{cases}
\][/tex]
we can use the method of substitution or elimination. In this case, we'll use the elimination method. Here are the steps:
1. Rewrite both equations:
[tex]\[
\begin{cases}
2x + 3y = 2 \quad \text{(Equation 1)} \\
x - 2y = 8 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
2. Align the equations for elimination:
We notice that the coefficients of \(x\) and \(y\) are not the same in both equations, so we need to manipulate the equations to make it possible to eliminate one variable. To do that, it is helpful to eliminate \(x\) first.
3. Multiply Equation 2 by 2 to align the coefficients of \(x\):
[tex]\[
2 \cdot (x - 2y = 8) \implies 2x - 4y = 16 \quad \text{(Equation 3)}
\][/tex]
Now we have:
[tex]\[
\begin{cases}
2x + 3y = 2 \quad \text{(Equation 1)} \\
2x - 4y = 16 \quad \text{(Equation 3)}
\end{cases}
\][/tex]
4. Subtract Equation 1 from Equation 3 to eliminate \(x\):
[tex]\[
(2x - 4y) - (2x + 3y) = 16 - 2
\][/tex]
Simplify:
[tex]\[
2x - 4y - 2x - 3y = 14 \implies -7y = 14
\][/tex]
5. Solve for \(y\):
[tex]\[
y = \frac{14}{-7} = -2
\][/tex]
Now we know:
[tex]\[
y = -2
\][/tex]
6. Substitute \(y = -2\) back into the original Equation 2 to solve for \(x\):
[tex]\[
x - 2(-2) = 8
\][/tex]
Simplify:
[tex]\[
x + 4 = 8
\][/tex]
Solve for \(x\):
[tex]\[
x = 8 - 4 = 4
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = 4, \quad y = -2
\][/tex]
Hence, the solution is [tex]\( \boxed{(4, -2)} \)[/tex].
