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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]
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]
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