To solve the given problem, we need to determine the value of [tex]\(\left|A_y\right|\)[/tex].
Given the matrix equation:
[tex]\[
\left[\begin{array}{ll}12 & -13 \\ 17 & -22\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}7 \\ -51\end{array}\right]
\][/tex]
We need to identify the matrix whose determinant is requested. The matrix associated with the notation [tex]\(\left|A_y\right|\)[/tex] corresponds to the column of constants replacing the [tex]\(y\)[/tex]-column of the original matrix. As stated in the problem, we need [tex]\(\left|A_y\right|\)[/tex].
Replacing the [tex]\(y\)[/tex]-column ([tex]\([-13, -22]\)[/tex]) with the constants vector ([tex]\([7, -51]\)[/tex]), we get the augmented matrix:
[tex]\[
\left[\begin{array}{cc}
12 & 7 \\
17 & -51
\end{array}\right]
\][/tex]
However, the task is to identify the correct matrix from the provided ones. Among these matrices, the matching structure, where the [tex]\(y\)[/tex]-column is replaced by the constants is:
[tex]\[
\left[\begin{array}{cc}
7 & -51 \\
17 & -22
\end{array}\right].
\][/tex]
So, our focus is on the matrix:
[tex]\[
\left[\begin{array}{cc}
7 & -51 \\
17 & -22
\end{array}\right]
\][/tex]
To determine [tex]\(\left|A_y\right|\)[/tex], we find the determinant of this matrix. The determinant for a 2x2 matrix [tex]\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\)[/tex] is given by:
[tex]\[
\left| \begin{array}{cc}
7 & -51 \\
17 & -22
\end{array}\right|
= 7 \cdot (-22) - (-51) \cdot 17
= -154 + 867
= 713.
\][/tex]
Thus, the determinant [tex]\(\left|A_y\right|\)[/tex] is:
[tex]\[
\boxed{713}.
\][/tex]
