Certainly! Let's solve the system of linear equations using Cramer's rule.
The given system of equations is:
[tex]\[
\begin{align*}
-\frac{1}{2} x + 3y &= -4 \\
-x - y &= -1
\end{align*}
\][/tex]
### Step-by-Step Solution
1. Form the coefficient matrix [tex]\( A \)[/tex]:
The coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are taken to form matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
-\frac{1}{2} & 3 \\
-1 & -1
\end{pmatrix}
\][/tex]
2. Form the constant matrix [tex]\( B \)[/tex]:
The constants on the right side of the equations make up matrix [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix}
-4 \\
-1
\end{pmatrix}
\][/tex]
3. Calculate the determinant of the coefficient matrix [tex]\( A \)[/tex]:
The determinant [tex]\( \text{det}(A) \)[/tex] is calculated as follows:
[tex]\[
\text{det}(A) = \left(-\frac{1}{2}\right) \cdot (-1) - 3 \cdot (-1) = \frac{1}{2} + 3 = 3.5
\][/tex]
4. Form the matrix [tex]\( A_x \)[/tex] by replacing the first column of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
[tex]\[
A_x = \begin{pmatrix}
-4 & 3 \\
-1 & -1
\end{pmatrix}
\][/tex]
5. Calculate the determinant of [tex]\( A_x \)[/tex]:
The determinant [tex]\( \text{det}(A_x) \)[/tex] is calculated as follows:
[tex]\[
\text{det}(A_x) = (-4) \cdot (-1) - 3 \cdot (-1) = 4 + 3 = 7
\][/tex]
6. Solve for [tex]\( x \)[/tex] using Cramer's rule:
According to Cramer's rule, the solution for [tex]\( x \)[/tex] can be found using:
[tex]\[
x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{7}{3.5} = 2
\][/tex]
Thus, the value of [tex]\( x \)[/tex] in the solution to the system of linear equations is [tex]\( \boxed{2} \)[/tex].
