To solve the given system of linear equations using determinants, we can apply Cramer's Rule. Here's the step-by-step process:
Given the system of linear equations:
[tex]\[
\begin{array}{l}
-3x + 2y = -9 \\
4x - 15y = -25
\end{array}
\][/tex]
We can rewrite this system in matrix form [tex]\(AX = B\)[/tex], where:
[tex]\[
A = \begin{pmatrix}
-3 & 2 \\
4 & -15
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y
\end{pmatrix}, \quad
B = \begin{pmatrix}
-9 \\
-25
\end{pmatrix}
\][/tex]
According to Cramer's Rule, the solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be found using the determinants of matrices derived from [tex]\(A\)[/tex] by replacing the respective columns with the column matrix [tex]\(B\)[/tex]. The solution is given as:
[tex]\[
x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}
\][/tex]
Where:
- [tex]\(\text{det}(A)\)[/tex] is the determinant of the matrix [tex]\(A\)[/tex].
- [tex]\(\text{det}(A_x)\)[/tex] is the determinant of the matrix formed by replacing the first column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex].
- [tex]\(\text{det}(A_y)\)[/tex] is the determinant of the matrix formed by replacing the second column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex].
Let's identify each determinant:
1. The determinant of [tex]\(A\)[/tex]:
[tex]\[
A = \begin{pmatrix}
-3 & 2 \\
4 & -15
\end{pmatrix}
\][/tex]
2. The determinant of [tex]\(A_x\)[/tex]:
[tex]\[
A_x = \begin{pmatrix}
-9 & 2 \\
-25 & -15
\end{pmatrix}
\][/tex]
3. The determinant of [tex]\(A_y\)[/tex]:
[tex]\[
A_y = \begin{pmatrix}
-3 & -9 \\
4 & -25
\end{pmatrix}
\][/tex]
Given the determinants:
[tex]\[
\text{det}(A) = 37.000000000000014, \quad \text{det}(A_x) = 184.99999999999991, \quad \text{det}(A_y) = 110.99999999999997
\][/tex]
These determinants can be used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using Cramer's Rule:
[tex]\[
x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{184.99999999999991}{37.000000000000014}
\][/tex]
[tex]\[
y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{110.99999999999997}{37.000000000000014}
\][/tex]
Thus, the determinants used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the system of linear equations are:
- [tex]\(\text{det}(A) = 37.000000000000014\)[/tex]
- [tex]\(\text{det}(A_x) = 184.99999999999991\)[/tex]
- [tex]\(\text{det}(A_y) = 110.99999999999997\)[/tex]
