To solve the system of equations:
[tex]\[
\begin{aligned}
-3x + 6y &= 9 \quad \text{(Equation 1)} \\
5x + 7y &= -49 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
we can use the method of elimination or substitution. Here, I'll demonstrate using the substitution method.
1. Solve one of the equations for one variable:
Let's solve Equation 1 for [tex]\(x\)[/tex]:
[tex]\[
-3x + 6y = 9
\][/tex]
Divide both sides by 3:
[tex]\[
-x + 2y = 3
\][/tex]
Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 2y - 3 \quad \text{(Equation 3)}
\][/tex]
2. Substitute the expression for [tex]\(x\)[/tex] in the second equation:
Substitute Equation 3 into Equation 2:
[tex]\[
5(2y - 3) + 7y = -49
\][/tex]
Expand and simplify:
[tex]\[
10y - 15 + 7y = -49
\][/tex]
Combine like terms:
[tex]\[
17y - 15 = -49
\][/tex]
Add 15 to both sides:
[tex]\[
17y = -34
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
y = -2
\][/tex]
3. Substitute [tex]\(y = -2\)[/tex] back into the expression we found for [tex]\(x\)[/tex]:
Substitute [tex]\(y = -2\)[/tex] into Equation 3:
[tex]\[
x = 2(-2) - 3
\][/tex]
Simplify:
[tex]\[
x = -4 - 3
\][/tex]
[tex]\[
x = -7
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (-7, -2)
\][/tex]
The correct answer is:
[tex]\[
\boxed{(-7, -2)}
\][/tex]
