To solve the system of equations
[tex]\[
\begin{aligned}
-3x + 6y & = 9 \\
5x + 7y & = -49
\end{aligned}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously.
Step 1: Simplify the first equation
[tex]\[
-3x + 6y = 9
\][/tex]
Divide every term by 3:
[tex]\[
-x + 2y = 3
\][/tex]
This is the simplified form of the first equation:
[tex]\[
x - 2y = -3 \quad \text{(Equation 1)}
\][/tex]
Step 2: Multiply the first equation to make the coefficients of [tex]\( x \)[/tex] equal in both equations. We multiply Equation 1 by 5:
[tex]\[
5(x - 2y) = 5(-3)
\][/tex]
[tex]\[
5x - 10y = -15 \quad \text{(Equation 3)}
\][/tex]
Step 3: Write down the second original equation:
[tex]\[
5x + 7y = -49 \quad \text{(Equation 2)}
\][/tex]
Step 4: Subtract Equation 2 from Equation 3 to eliminate [tex]\( x \)[/tex]:
[tex]\[
(5x - 10y) - (5x + 7y) = -15 - (-49)
\][/tex]
[tex]\[
5x - 10y - 5x - 7y = 34
\][/tex]
[tex]\[
-17y = 34
\][/tex]
Step 5: Solve for [tex]\( y \)[/tex]:
[tex]\[
y = -2
\][/tex]
Step 6: Substitute [tex]\( y = -2 \)[/tex] back into Equation 1:
[tex]\[
x - 2(-2) = -3
\][/tex]
[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].
