To solve the given system of equations by elimination, we will follow these detailed steps:
Given system of equations:
[tex]\[
\begin{cases}
x - 9y = -13 \\
2x + y = -7
\end{cases}
\][/tex]
### Step 1: Align the Equations for Elimination
We first multiply the first equation by -2 so that the coefficients of [tex]\( x \)[/tex] in both equations have opposite signs. This will allow us to eliminate [tex]\( x \)[/tex] when we add the two equations.
### Step 2: Multiply the First Equation by -2
[tex]\[
-2(x - 9y) = -2(-13)
\][/tex]
[tex]\[
-2x + 18y = 26
\][/tex]
So now, the modified system of equations becomes:
[tex]\[
\begin{cases}
-2x + 18y = 26 \\
2x + y = -7
\end{cases}
\][/tex]
### Step 3: Add the Equations Together
Adding the two equations results in:
[tex]\[
(-2x + 18y) + (2x + y) = 26 + (-7)
\][/tex]
[tex]\[
-2x + 2x + 18y + y = 26 - 7
\][/tex]
[tex]\[
19y = 19
\][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
[tex]\[
y = \frac{19}{19} = 1
\][/tex]
### Step 5: Substitute [tex]\( y = 1 \)[/tex] into One of the Original Equations
Now we substitute [tex]\( y = 1 \)[/tex] into the second equation [tex]\( 2x + y = -7 \)[/tex]:
[tex]\[
2x + 1 = -7
\][/tex]
[tex]\[
2x = -7 - 1
\][/tex]
[tex]\[
2x = -8
\][/tex]
[tex]\[
x = \frac{-8}{2} = -4
\][/tex]
### Final Answer
By substituting back and solving, the solutions are [tex]\( x = -4 \)[/tex] and [tex]\( y = 1 \)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{(-4, 1)}
\][/tex]
