Certainly! Let's solve the system of equations using the elimination method step-by-step.
Given equations:
[tex]\[
\begin{cases}
3x + y = 5 \\
2x - 2y = -2
\end{cases}
\][/tex]
1. Simplify the second equation:
Divide the second equation by 2 to simplify it:
[tex]\[
\frac{2x - 2y}{2} = \frac{-2}{2}
\][/tex]
Simplifying, we get:
[tex]\[
x - y = -1
\][/tex]
2. Rewrite the system:
Now, we have a simpler system of equations:
[tex]\[
\begin{cases}
3x + y = 5 \\
x - y = -1
\end{cases}
\][/tex]
3. Add the equations to eliminate [tex]\( y \)[/tex]:
[tex]\[
\begin{aligned}
&3x + y + x - y = 5 + (-1) \\
&4x = 4 \\
&x = 1
\end{aligned}
\][/tex]
4. Substitute [tex]\( x = 1 \)[/tex] back into the simpler equation [tex]\( x - y = -1 \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
\begin{aligned}
1 - y &= -1 \\
-y &= -1 - 1 \\
-y &= -2 \\
y &= 2
\end{aligned}
\][/tex]
Hence, the solution to the system of equations is:
[tex]\[
(x, y) = (1, 2)
\][/tex]
Therefore, the correct answer is:
[tex]\[
(1, 2)
\][/tex]
