To solve the system of equations:
[tex]\[
\begin{array}{l}
5x + 2y = 9 \\
2x - 3y = 15
\end{array}
\][/tex]
we can use the method of Cramer's Rule, which is applicable because we have a system of linear equations with different slopes.
First, let's identify the coefficients and constants in the equations:
[tex]\[
\begin{array}{l}
a_1 = 5, \, b_1 = 2, \, c_1 = 9 \\
a_2 = 2, \, b_2 = -3, \, c_2 = 15
\end{array}
\][/tex]
According to Cramer's Rule, we first need to find the determinant [tex]\( D \)[/tex] of the coefficient matrix:
[tex]\[
D =
\begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2 \\
\end{vmatrix}
= (5 \cdot -3) - (2 \cdot 2) = -15 - 4 = -19
\][/tex]
Next, we find the determinant [tex]\( D_x \)[/tex] for the numerator of [tex]\( x \)[/tex] by replacing the [tex]\( x \)[/tex]-column with the constants:
[tex]\[
D_x =
\begin{vmatrix}
c_1 & b_1 \\
c_2 & b_2 \\
\end{vmatrix}
= (9 \cdot -3) - (15 \cdot 2) = -27 - 30 = -57
\][/tex]
Similarly, we find the determinant [tex]\( D_y \)[/tex] for the numerator of [tex]\( y \)[/tex] by replacing the [tex]\( y \)[/tex]-column with the constants:
[tex]\[
D_y =
\begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2 \\
\end{vmatrix}
= (5 \cdot 15) - (2 \cdot 9) = 75 - 18 = 57
\][/tex]
The solutions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be found using:
[tex]\[
x = \frac{D_x}{D} = \frac{-57}{-19} = 3
\][/tex]
[tex]\[
y = \frac{D_y}{D} = \frac{57}{-19} = -3
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{(3, -3)}
\][/tex]
So, the correct answer is:
[tex]\[
\text{B. } (3, -3)
\][/tex]
