Solve the following system of equations using elimination:

[tex]\[
\begin{array}{l}
3x + y = 5 \\
2x - 2y = -2
\end{array}
\][/tex]

A. [tex]\((-1, -2)\)[/tex]
B. [tex]\((2, 1)\)[/tex]
C. [tex]\((1, 2)\)[/tex]
D. [tex]\((-2, -1)\)[/tex]
E. No Solution
F. Infinite Solutions



Answer :

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]