Certainly! Let's solve the system of equations using the elimination method:
[tex]\[
\begin{cases}
3x + 2y = -1 \\
x - 2y = 11
\end{cases}
\][/tex]
### Step 1: Align the Coefficients of [tex]\(y\)[/tex]
First, we want to align the coefficients of [tex]\(y\)[/tex] in both equations so that they can be eliminated.
Starting with the second equation [tex]\(x - 2y = 11\)[/tex], we can multiply it by 1:
[tex]\[
1 \cdot (x - 2y) = 1 \cdot 11 \\
x - 2y = 11
\][/tex]
The first equation remains:
[tex]\[
3x + 2y = -1
\][/tex]
### Step 2: Add the Equations to Eliminate [tex]\(y\)[/tex]
Now we add the two equations together to eliminate [tex]\(y\)[/tex]:
[tex]\[
(3x + 2y) + (x - 2y) = -1 + 11
\][/tex]
[tex]\[
3x + 2y + x - 2y = 10
\][/tex]
[tex]\[
4x = 10
\][/tex]
[tex]\[
x = \frac{10}{4}
\][/tex]
[tex]\[
x = \frac{5}{2}
\][/tex]
### Step 3: Substitute [tex]\(x\)[/tex] Back to Find [tex]\(y\)[/tex]
Next, we substitute the value of [tex]\(x\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. We'll use the second equation:
[tex]\[
x - 2y = 11
\][/tex]
[tex]\[
\frac{5}{2} - 2y = 11
\][/tex]
Subtract [tex]\(\frac{5}{2}\)[/tex] from both sides:
[tex]\[
-2y = 11 - \frac{5}{2}
\][/tex]
[tex]\[
-2y = \frac{22}{2} - \frac{5}{2}
\][/tex]
[tex]\[
-2y = \frac{17}{2}
\][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
y = -\frac{17}{4}
\][/tex]
### Conclusion
Therefore, the solution to the system is:
[tex]\[
\left( x, y \right) = \left( \frac{5}{2}, -\frac{17}{4} \right)
\][/tex]
So the correct answer from the given options is:
[tex]\[
\left( \frac{5}{2}, -\frac{17}{4} \right)
\][/tex]
