To solve the given system of linear equations step-by-step, we have:
[tex]\[
\begin{cases}
2x + y = 4 \quad \text{(1)} \\
x + y = -2 \quad \text{(2)} \\
\end{cases}
\][/tex]
Step 1: Subtract Equation (2) from Equation (1). This will eliminate [tex]\( y \)[/tex] and allow us to solve for [tex]\( x \)[/tex].
[tex]\[
(2x + y) - (x + y) = 4 - (-2)
\][/tex]
Simplify the left-hand side:
[tex]\[
2x + y - x - y = 4 + 2
\][/tex]
This reduces to:
[tex]\[
x = 6
\][/tex]
So, we have found that:
[tex]\[
x = 6
\][/tex]
Step 2: Substitute [tex]\( x = 6 \)[/tex] into Equation (2) to solve for [tex]\( y \)[/tex].
Recall Equation (2):
[tex]\[
x + y = -2
\][/tex]
Substitute [tex]\( x = 6 \)[/tex]:
[tex]\[
6 + y = -2
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = -2 - 6
\][/tex]
[tex]\[
y = -8
\][/tex]
So, we have found that:
[tex]\[
y = -8
\][/tex]
Conclusion: The solution to the system of equations is:
[tex]\[
(x, y) = (6, -8)
\][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are:
[tex]\[
x = 6 \quad \text{and} \quad y = -8
\][/tex]
