To solve the given system of linear equations using determinants, we can employ Cramer's Rule. The system of equations is:
[tex]\[
\begin{array}{l}
-3x + 2y = -9 \\
4x - 15y = -25
\end{array}
\][/tex]
According to Cramer's Rule, we solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the following determinants:
1. Determinant of the Coefficient Matrix ([tex]\(|A|\)[/tex]):
[tex]\[
|A| = \left|\begin{array}{cc}
-3 & 2 \\
4 & -15
\end{array}\right|
\][/tex]
2. Determinant of the Matrix for [tex]\(x\)[/tex] ([tex]\(|A_x|\)[/tex]):
[tex]\[
|A_x| = \left|\begin{array}{cc}
-9 & 2 \\
-25 & -15
\end{array}\right|
\][/tex]
3. Determinant of the Matrix for [tex]\(y\)[/tex] ([tex]\(|A_y|\)[/tex]):
[tex]\[
|A_y| = \left|\begin{array}{cc}
-3 & -9 \\
4 & -25
\end{array}\right|
\][/tex]
Using the determinant values you provided:
[tex]\[
|A| = 37.000000000000014
\][/tex]
[tex]\[
|A_x| = 184.99999999999991
\][/tex]
[tex]\[
|A_y| = 110.99999999999997
\][/tex]
We can now form the equations for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using Cramer's Rule:
[tex]\[
x = \frac{|A_x|}{|A|}
\][/tex]
[tex]\[
y = \frac{|A_y|}{|A|}
\][/tex]
Substituting the calculated determinants:
[tex]\[
x = \frac{184.99999999999991}{37.000000000000014}
\][/tex]
[tex]\[
y = \frac{110.99999999999997}{37.000000000000014}
\][/tex]
Upon simplifying these fractions, we get:
[tex]\[
x \approx 5
\][/tex]
[tex]\[
y \approx 3
\][/tex]
Therefore, the determinants that can be used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in this system of linear equations are:
[tex]\[
|A| = \left|\begin{array}{cc}
-3 & 2 \\
4 & -15
\end{array}\right|, \quad |A_x| = \left|\begin{array}{cc}
-9 & 2 \\
-25 & -15
\end{array}\right|, \quad |A_y| = \left|\begin{array}{cc}
-3 & -9 \\
4 & -25
\end{array}\right|
\][/tex]
