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Use Cramer's rule to solve for [tex]x[/tex]:

[tex]\[
\begin{cases}
x + 4y - z = -14 \\
5x + 6y + 3z = 4 \\
-2x + 7y + 2z = -17
\end{cases}
\][/tex]

[tex]\[
|A| = \square
\][/tex]

[tex]\[
|A_x| = \square
\][/tex]

[tex]\[
x = \square
\][/tex]


Sagot :

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]