Let's use Cramer's Rule to find the value of [tex]\( x \)[/tex] for the given system of equations:
[tex]\[
\begin{cases}
-x - 3y = -3 \\
-2x - 5y = -8
\end{cases}
\][/tex]
Cramer's Rule states that for a system of linear equations [tex]\( A \mathbf{x} = \mathbf{b} \)[/tex], the solution for [tex]\( x \)[/tex] can be found using determinants:
[tex]\[
x = \frac{\det(A_x)}{\det(A)}
\][/tex]
where [tex]\( \det(A) \)[/tex] is the determinant of the coefficient matrix [tex]\( A \)[/tex], and [tex]\( \det(A_x) \)[/tex] is the determinant of the matrix [tex]\( A \)[/tex] with its [tex]\( x \)[/tex]-column replaced by the constants [tex]\( \mathbf{b} \)[/tex] from the right-hand side.
First, let's identify the coefficient matrix [tex]\( A \)[/tex] and the constants vector [tex]\( \mathbf{b} \)[/tex]:
[tex]\[
A = \begin{pmatrix}
-1 & -3 \\
-2 & -5
\end{pmatrix}
\][/tex]
[tex]\[
\mathbf{b} = \begin{pmatrix}
-3 \\
-8
\end{pmatrix}
\][/tex]
Next, we need to find the determinant of matrix [tex]\( A \)[/tex]:
[tex]\[
\det(A) = \begin{vmatrix}
-1 & -3 \\
-2 & -5
\end{vmatrix}
= (-1)(-5) - (-3)(-2)
= 5 - 6
= -1
\][/tex]
Now, we form the matrix [tex]\( A_x \)[/tex] by replacing the first column of [tex]\( A \)[/tex] with the constants [tex]\( \mathbf{b} \)[/tex]:
[tex]\[
A_x = \begin{pmatrix}
-3 & -3 \\
-8 & -5
\end{pmatrix}
\][/tex]
Next, we calculate the determinant of [tex]\( A_x \)[/tex]:
[tex]\[
\det(A_x) = \begin{vmatrix}
-3 & -3 \\
-8 & -5
\end{vmatrix}
= (-3)(-5) - (-3)(-8)
= 15 - 24
= -9
\][/tex]
Now, using Cramer's Rule, we can find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\det(A_x)}{\det(A)} = \frac{-9}{-1} = 9
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 9
\][/tex]
