To solve the system of equations
[tex]\[
\begin{aligned}
-3x + 6y &= 9 \\
5x + 7y &= -49
\end{aligned}
\][/tex]
we aim to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Here's a step-by-step solution:
1. Rewrite the first equation:
[tex]\[
-3x + 6y = 9
\][/tex]
This can be divided by 3 for simplicity:
[tex]\[
-x + 2y = 3 \quad \text{(Equation 1')}
\][/tex]
2. Rewrite the second equation:
[tex]\[
5x + 7y = -49 \quad \text{(Equation 2)}
\][/tex]
3. Solve Equation 1' for [tex]\(x\)[/tex]:
[tex]\[
-x + 2y = 3 \implies x = 2y - 3
\][/tex]
4. Substitute [tex]\(x = 2y - 3\)[/tex] into Equation 2:
[tex]\[
5(2y - 3) + 7y = -49
\][/tex]
Simplify this expression:
[tex]\[
10y - 15 + 7y = -49
\][/tex]
Combine like terms:
[tex]\[
17y - 15 = -49
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
17y = -49 + 15
\][/tex]
[tex]\[
17y = -34
\][/tex]
[tex]\[
y = -2
\][/tex]
5. Use [tex]\(y = -2\)[/tex] to solve for [tex]\(x\)[/tex]:
Substitute [tex]\(y = -2\)[/tex] back into Equation 1':
[tex]\[
x = 2(-2) - 3
\][/tex]
Simplify the expression:
[tex]\[
x = -4 - 3
\][/tex]
[tex]\[
x = -7
\][/tex]
So, the solution to the system of equations is [tex]\(x = -7\)[/tex] and [tex]\(y = -2\)[/tex].
Hence, the correct answer is:
D. [tex]\((-7, -2)\)[/tex]
