To find [tex]\(\left|A_y\right|\)[/tex] in the context of the given matrix equation and options, we need to understand that [tex]\(\left|A_y\right|\)[/tex] represents the determinant of the matrix formed by replacing the second column of the coefficient matrix [tex]\(A\)[/tex] with the constants from the right-hand side of the equation.
The system of equations given is:
[tex]\[
\begin{cases}
12x - 13y = 7 \\
17x - 22y = -51
\end{cases}
\][/tex]
The coefficient matrix [tex]\(A\)[/tex] is:
[tex]\[
A = \begin{pmatrix}
12 & -13 \\
17 & -22
\end{pmatrix}
\][/tex]
The constants matrix is:
[tex]\[
\mathbf{b} = \begin{pmatrix}
7 \\
-51
\end{pmatrix}
\][/tex]
To find [tex]\(\left|A_y\right|\)[/tex], we replace the second column of [tex]\(A\)[/tex] with [tex]\(\mathbf{b}\)[/tex]:
1. Start with possible replacements from the given options to construct the matrices.
[tex]\(\begin{pmatrix} 7 & -13 \\ -51 & -22 \end{pmatrix}\)[/tex]
2. Calculate the determinant:
[tex]\[
\left| \begin{pmatrix} 7 & -13 \\ -51 & -22 \end{pmatrix} \right| = (7 \cdot -22) - (-13 \cdot -51) = -154 - 663 = -817
\][/tex]
Following this result, let's analyze the determinants from the remaining options:
[tex]\[
\left| \begin{pmatrix} 12 & 7 \\ 17 & -51 \end{pmatrix} \right| = (12 \cdot -51) - (7 \cdot 17) = -612 - 119 = -731
\][/tex]
[tex]\[
\left| \begin{pmatrix} 7 & -51 \\ 17 & -22 \end{pmatrix} \right| = (7 \cdot -22) - (-51 \cdot 17) = -154 + 867 = 713
\][/tex]
[tex]\[
\left| \begin{pmatrix} 12 & -13 \\ 7 & -51 \end{pmatrix} \right| = (12 \cdot -51) - (-13 \cdot 7) = -612 + 91 = -521
\][/tex]
From these calculations, the correct determinant matching [tex]\(\left|A_y\right|\)[/tex] is the one from:
[tex]\[
\left| \begin{pmatrix} 7 & -51 \\ 17 & -22 \end{pmatrix} \right| = 713
\][/tex]
So, [tex]\(\left|A_y\right|\)[/tex] is:
[tex]\[
713
\][/tex]
