Let's solve the given system of linear equations using Cramer's Rule. The system is:
[tex]\[
\begin{cases}
2x + 5y = -13 \\
-3x - 2y = 3
\end{cases}
\][/tex]
To apply Cramer's Rule, we first need to find the determinant of the coefficient matrix [tex]\(A\)[/tex]:
[tex]\[
A = \begin{pmatrix}
2 & 5 \\
-3 & -2
\end{pmatrix}
\][/tex]
The determinant [tex]\(\det(A)\)[/tex] is given by:
[tex]\[
\det(A) = (2 \times -2) - (5 \times -3)
\][/tex]
Calculating the terms within the determinant:
[tex]\[
\det(A) = -4 + 15 = 11
\][/tex]
Next, we need to find the determinant of matrix [tex]\(A_y\)[/tex], which is obtained by replacing the [tex]\(y\)[/tex]-coefficients in the original matrix with the constants from the right-hand side of the equations:
[tex]\[
A_y = \begin{pmatrix}
2 & -13 \\
-3 & 3
\end{pmatrix}
\][/tex]
We calculate the determinant [tex]\(\det(A_y)\)[/tex]:
[tex]\[
\det(A_y) = (2 \times 3) - (-13 \times -3)
\][/tex]
Calculating the terms within the determinant:
[tex]\[
\det(A_y) = 6 - 39 = -33
\][/tex]
Using Cramer's Rule, the value of [tex]\(y\)[/tex] is given by:
[tex]\[
y = \frac{\det(A_y)}{\det(A)}
\][/tex]
Substituting the values of the determinants:
[tex]\[
y = \frac{-33}{11} = -3
\][/tex]
Therefore, the value of [tex]\(y\)[/tex] in the solution to the system of linear equations is [tex]\(-3\)[/tex]. The correct answer is:
[tex]\[
\boxed{-3}
\][/tex]
