To solve for [tex]\( x \)[/tex] using Cramer's rule, we first need to establish the coefficient matrix [tex]\( A \)[/tex] and the constant vector [tex]\( B \)[/tex].
Our system of equations is:
[tex]\[
\begin{array}{rcl}
x + 4y - z &=& -14 \\
5x + 6y + 3z &=& 4 \\
-2x + 7y + 2z &=& -17
\end{array}
\][/tex]
### Step 1: Define the coefficient matrix [tex]\( A \)[/tex] and vector [tex]\( B \)[/tex]
The coefficient matrix [tex]\(A\)[/tex] is:
[tex]\[
A = \begin{pmatrix}
1 & 4 & -1 \\
5 & 6 & 3 \\
-2 & 7 & 2
\end{pmatrix}
\][/tex]
The constant vector [tex]\( B \)[/tex] is:
[tex]\[
B = \begin{pmatrix}
-14 \\
4 \\
-17
\end{pmatrix}
\][/tex]
### Step 2: Calculate the determinant of matrix [tex]\( A \)[/tex]
The determinant of matrix [tex]\( A \)[/tex], denoted as [tex]\(|A|\)[/tex], is:
[tex]\[
|A| = -120
\][/tex]
### Step 3: Form matrix [tex]\( A_x \)[/tex]
Matrix [tex]\( A_x \)[/tex] is formed by replacing the first column of [tex]\(A\)[/tex] (corresponding to the [tex]\(x\)[/tex] variable) with vector [tex]\( B \)[/tex].
[tex]\[
A_x = \begin{pmatrix}
-14 & 4 & -1 \\
4 & 6 & 3 \\
-17 & 7 & 2
\end{pmatrix}
\][/tex]
### Step 4: Calculate the determinant of matrix [tex]\( A_x \)[/tex]
The determinant of matrix [tex]\( A_x \)[/tex], denoted as [tex]\(|A_x|\)[/tex], is:
[tex]\[
\left|A_x\right| = -240
\][/tex]
### Step 5: Solve for [tex]\( x \)[/tex] using Cramer's rule
Cramer's rule states that for a system of linear equations [tex]\( AX = B \)[/tex], the solution for [tex]\( x \)[/tex] is given by:
[tex]\[
x = \frac{|A_x|}{|A|}
\][/tex]
Substituting the values of [tex]\(|A|\)[/tex] and [tex]\(|A_x|\)[/tex]:
[tex]\[
x = \frac{-240}{-120} = 2
\][/tex]
### Summary
[tex]\[
\begin{array}{l}
|A| = -120 \\
\left|A_x\right| = -240 \\
x = 2
\end{array}
\][/tex]
