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.
