Perform matrix multiplication:
\[
\begin{bmatrix}
2 & 3 \\
5 & 7
\end{bmatrix}
\begin{bmatrix}
1 & -3 \\
-2 & 4
\end{bmatrix}
=
\begin{bmatrix}
2 \times 1 + 3 \times (-2) & 2 \times (-3) + 3 \times 4 \\
5 \times 1 + 7 \times (-2) & 5 \times (-3) + 7 \times 4
\end{bmatrix}
\]
Calculate each element of the resulting matrix:
\[
a_{11} = 2 - 6 = -4
\]
\[
a_{12} = -6 + 12 = 6
\]
\[
a_{21} = 5 - 14 = -9
\]
\[
a_{22} = -15 + 28 = 13
\]
The resulting matrix is:
\[
\begin{bmatrix} -4 & 6 \\ -9 & x \end{bmatrix}
\]
By equating the calculated elements with the unknown matrix, we find:
\[
x = 13
\]
This result, \(x = 13\), corresponds to option (A). However, the provided options also include -13. A review of the calculation for \(a_{22}\) confirms:
\[
a_{22} = 5 \times (-3) + 7 \times 4 = -15 + 28 = 13
\]
Therefore, the value of \(x\) is definitively 13.