Let's determine [tex]\(\left|A_y\right|\)[/tex] for the given system of linear equations:
Given system:
[tex]\[
\begin{array}{l}
-8x + y = -6 \\
3x - 2y = -1
\end{array}
\][/tex]
We want to find the determinant [tex]\(\left|A_y\right|\)[/tex].
First, represent the system in matrix form as [tex]\(Ax = b\)[/tex]:
[tex]\[
\begin{array}{cc}
\begin{pmatrix}
-8 & 1 \\
3 & -2
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
-6 \\
-1
\end{pmatrix}
\end{array}
\][/tex]
To find [tex]\(\left|A_y\right|\)[/tex], we replace the column of the variable [tex]\(y\)[/tex] in the coefficient matrix [tex]\(A\)[/tex] with the right-hand side vector [tex]\(b\)[/tex]:
Matrix [tex]\(A_y\)[/tex]:
[tex]\[
A_y = \begin{pmatrix}
-8 & -6 \\
3 & -1
\end{pmatrix}
\][/tex]
We now need to find the determinant of [tex]\(A_y\)[/tex]:
[tex]\[
\left|A_y\right| = \begin{vmatrix}
-8 & -6 \\
3 & -1
\end{vmatrix}
\][/tex]
The determinant [tex]\(\left|A_y\right|\)[/tex] is calculated using the formula for a 2x2 matrix [tex]\(\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc\)[/tex]:
[tex]\[
\left|A_y\right| = (-8)(-1) - (3)(-6)
\][/tex]
Carrying out the multiplication and subtraction:
[tex]\[
\left|A_y\right| = 8 + 18 = 26
\][/tex]
Therefore, the determinant [tex]\(\left|A_y\right|\)[/tex] for the given system of equations is:
[tex]\[
\left|A_y\right| = 25.99999999999999
\][/tex]
Thus, [tex]\(\left|A_y\right|\)[/tex] is approximately [tex]\(26\)[/tex].
