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To solve the system of linear equations using determinants, we will use Cramer's Rule. Let's break down the process of finding the solutions [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] step by step. The given system of linear equations is:
[tex]\[ \begin{cases} x + 4y - z = -14 \\ 5x + 6y + 3z = 4 \\ -2x + 7y + 2z = -17 \end{cases} \][/tex]
First, organize the system into matrix form [tex]\(AX = B\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(X\)[/tex] is the vector of variables, and [tex]\(B\)[/tex] is the constant vector:
[tex]\[ A = \begin{pmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} -14 \\ 4 \\ -17 \end{pmatrix} \][/tex]
Cramer's Rule states that each variable can be found by the ratio of the determinant of a modified matrix to the determinant of the original coefficient matrix [tex]\(A\)[/tex]. Let's calculate the determinant of [tex]\(A\)[/tex]:
[tex]\[ |A| = \begin{vmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{vmatrix} \][/tex]
The determinant [tex]\(|A|\)[/tex] is found to be [tex]\(-119.99999999999997\)[/tex].
Next, we need to calculate the determinants of the matrices obtained by replacing each column of [tex]\(A\)[/tex] with the constant vector [tex]\(B\)[/tex].
1. Replace the first column with [tex]\(B\)[/tex] to get [tex]\(A_x\)[/tex]:
[tex]\[ A_x = \begin{pmatrix} -14 & 4 & -1 \\ 4 & 6 & 3 \\ -17 & 7 & 2 \end{pmatrix} \][/tex]
The determinant [tex]\(|A_x|\)[/tex] is found to be [tex]\(-240.0000000000002\)[/tex].
2. Replace the second column with [tex]\(B\)[/tex] to get [tex]\(A_y\)[/tex]:
[tex]\[ A_y = \begin{pmatrix} 1 & -14 & -1 \\ 5 & 4 & 3 \\ -2 & -17 & 2 \end{pmatrix} \][/tex]
The determinant [tex]\(|A_y|\)[/tex] is found to be [tex]\(360.00000000000006\)[/tex].
3. Replace the third column with [tex]\(B\)[/tex] to get [tex]\(A_z\)[/tex]:
[tex]\[ A_z = \begin{pmatrix} 1 & 4 & -14 \\ 5 & 6 & 4 \\ -2 & 7 & -17 \end{pmatrix} \][/tex]
The determinant [tex]\(|A_z|\)[/tex] is found to be [tex]\(-480.0\)[/tex].
Summarizing the results:
[tex]\[ |A| = -119.99999999999997 \][/tex]
[tex]\[ |A_x| = -240.0000000000002 \][/tex]
[tex]\[ |A_y| = 360.00000000000006 \][/tex]
[tex]\[ |A_z| = -480.0 \][/tex]
These determinants are validated true for this system of equations.
[tex]\[ \begin{cases} x + 4y - z = -14 \\ 5x + 6y + 3z = 4 \\ -2x + 7y + 2z = -17 \end{cases} \][/tex]
First, organize the system into matrix form [tex]\(AX = B\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(X\)[/tex] is the vector of variables, and [tex]\(B\)[/tex] is the constant vector:
[tex]\[ A = \begin{pmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} -14 \\ 4 \\ -17 \end{pmatrix} \][/tex]
Cramer's Rule states that each variable can be found by the ratio of the determinant of a modified matrix to the determinant of the original coefficient matrix [tex]\(A\)[/tex]. Let's calculate the determinant of [tex]\(A\)[/tex]:
[tex]\[ |A| = \begin{vmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{vmatrix} \][/tex]
The determinant [tex]\(|A|\)[/tex] is found to be [tex]\(-119.99999999999997\)[/tex].
Next, we need to calculate the determinants of the matrices obtained by replacing each column of [tex]\(A\)[/tex] with the constant vector [tex]\(B\)[/tex].
1. Replace the first column with [tex]\(B\)[/tex] to get [tex]\(A_x\)[/tex]:
[tex]\[ A_x = \begin{pmatrix} -14 & 4 & -1 \\ 4 & 6 & 3 \\ -17 & 7 & 2 \end{pmatrix} \][/tex]
The determinant [tex]\(|A_x|\)[/tex] is found to be [tex]\(-240.0000000000002\)[/tex].
2. Replace the second column with [tex]\(B\)[/tex] to get [tex]\(A_y\)[/tex]:
[tex]\[ A_y = \begin{pmatrix} 1 & -14 & -1 \\ 5 & 4 & 3 \\ -2 & -17 & 2 \end{pmatrix} \][/tex]
The determinant [tex]\(|A_y|\)[/tex] is found to be [tex]\(360.00000000000006\)[/tex].
3. Replace the third column with [tex]\(B\)[/tex] to get [tex]\(A_z\)[/tex]:
[tex]\[ A_z = \begin{pmatrix} 1 & 4 & -14 \\ 5 & 6 & 4 \\ -2 & 7 & -17 \end{pmatrix} \][/tex]
The determinant [tex]\(|A_z|\)[/tex] is found to be [tex]\(-480.0\)[/tex].
Summarizing the results:
[tex]\[ |A| = -119.99999999999997 \][/tex]
[tex]\[ |A_x| = -240.0000000000002 \][/tex]
[tex]\[ |A_y| = 360.00000000000006 \][/tex]
[tex]\[ |A_z| = -480.0 \][/tex]
These determinants are validated true for this system of equations.
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