Using Cramer's rule, what is the value of [tex] x [/tex] in the solution to the system of linear equations below?

[tex]\[
\begin{array}{l}
-\frac{1}{2} x + 3 y = -4 \\
-x - y = -1
\end{array}
\][/tex]

A. -2
B. -1
C. [tex]\(\frac{1}{2}\)[/tex]
D. 2



Answer :

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].