To find the solution to the given system of equations, let's go through the solution step-by-step:
The system of equations is:
[tex]\[
\begin{cases}
-2x + 2y = -4 \\
3x + 3y = -18
\end{cases}
\][/tex]
1. Equation Simplification:
Let's start by simplifying the second equation. Since every term in the second equation is divisible by 3, we can simplify it by dividing each term by 3:
[tex]\[
\frac{3x + 3y}{3} = \frac{-18}{3} \implies x + y = -6
\][/tex]
2. Solve one equation for one variable:
Next, we solve the simplified equation for one variable, for example, [tex]\(x\)[/tex]:
[tex]\[
x + y = -6 \implies x = -6 - y
\][/tex]
3. Substitute into the other equation:
Substitute the expression for [tex]\(x\)[/tex] into the first equation:
[tex]\[
-2(-6 - y) + 2y = -4
\][/tex]
Simplify and solve the substituted equation:
[tex]\[
12 + 2y + 2y = -4 \implies 12 + 4y = -4
\][/tex]
Subtract 12 from both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
4y = -4 - 12 \implies 4y = -16
\][/tex]
Divide both sides by 4 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -4
\][/tex]
4. Substitute back to find [tex]\(x\)[/tex]:
Substitute [tex]\(y = -4\)[/tex] back into the equation [tex]\(x = -6 - y\)[/tex]:
[tex]\[
x = -6 - (-4) \implies x = -6 + 4 \implies x = -2
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -2, \quad y = -4
\][/tex]
