We are given three determinants related to a system of linear equations and asked to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Here is the step-by-step solution using these determinants:
1. Calculate the determinant of matrix [tex]\(A\)[/tex], denoted as [tex]\(|A|\)[/tex]:
[tex]\[
|A| = \left|\begin{array}{cc}
4 & -6 \\
8 & -2
\end{array}\right|
\][/tex]
The determinant of a 2x2 matrix [tex]\(\left|\begin{array}{cc}
a & b \\
c & d
\end{array}\right|\)[/tex] is calculated as [tex]\(ad - bc\)[/tex].
So,
[tex]\[
|A| = (4 \cdot -2) - (-6 \cdot 8) = -8 + 48 = 40
\][/tex]
2. Calculate the determinant [tex]\(|A_x|\)[/tex]:
[tex]\[
|A_x| = \left|\begin{array}{cc}
38 & -6 \\
26 & -2
\end{array}\right|
\][/tex]
Using the formula for the determinant of a 2x2 matrix,
[tex]\[
|A_x| = (38 \cdot -2) - (-6 \cdot 26) = -76 + 156 = 80
\][/tex]
3. Calculate the determinant [tex]\(|A_y|\)[/tex]:
[tex]\[
|A_y| = \left|\begin{array}{cc}
4 & 38 \\
8 & 26
\end{array}\right|
\][/tex]
Using the formula again,
[tex]\[
|A_y| = (4 \cdot 26) - (38 \cdot 8) = 104 - 304 = -200
\][/tex]
4. Use Cramer's Rule to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
According to Cramer's Rule, if [tex]\(|A|\)[/tex] is the determinant of the matrix of the system, [tex]\(|A_x|\)[/tex] and [tex]\(|A_y|\)[/tex] are the determinants obtained by replacing the columns of the matrix with the constants from the equations, then:
[tex]\[
x = \frac{|A_x|}{|A|}
\][/tex]
[tex]\[
y = \frac{|A_y|}{|A|}
\][/tex]
5. Calculate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{|A_x|}{|A|} = \frac{80}{40} = 2.0
\][/tex]
6. Calculate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{|A_y|}{|A|} = \frac{-200}{40} = -5.0
\][/tex]
So, the solution to the system of equations is [tex]\(x = 2\)[/tex] and [tex]\(y = -5\)[/tex]. Therefore, the correct answer is [tex]\(\boxed{x=2, y=-5}\)[/tex].
