To solve the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
2x - 3y = -27 \\
-3x + 2y = 23
\end{array}
\right.
\][/tex]
we need to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously. Let's solve this step-by-step.
### Step 1: Write the system in matrix form
The system can be written in the form [tex]\( Ax = B \)[/tex] where [tex]\( A \)[/tex] is the coefficient matrix and [tex]\( B \)[/tex] is the constant matrix:
[tex]\[
A = \begin{pmatrix}
2 & -3 \\
-3 & 2
\end{pmatrix}
, \quad
B = \begin{pmatrix}
-27 \\
23
\end{pmatrix}
\][/tex]
### Step 2: Solve the system using matrix methods
We can solve this system using the method of linear algebra (though this is often done with computational tools for convenience). The solution provided is:
[tex]\[
(x, y) = (-3, 7)
\][/tex]
### Step 3: Verify the solution against the potential answers
Given the potential answers:
- (3, 11)
- (7, -3)
- (-24, -7)
- (-3, 7)
The correct answer must match our solved values. Comparing:
- (3, 11): This doesn't match.
- (7, -3): This doesn't match.
- (-24, -7): This doesn't match.
- (-3, 7): This matches the solution.
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (-3, 7)
\][/tex]
and the correct answer among the provided options is:
[tex]\[
\boxed{4}
\][/tex]
