To solve the system of linear equations using determinants, we can use Cramer's Rule. Let’s denote the coefficients matrix as [tex]\( A \)[/tex] and the constants vector as [tex]\( B \)[/tex].
The system of equations can be expressed in matrix form as follows:
[tex]$
A \mathbf{x} = B
$[/tex]
where:
[tex]\[
A = \begin{pmatrix}
1 & 4 & -1 \\
5 & 6 & 3 \\
-2 & 7 & 2
\end{pmatrix},
\quad
B = \begin{pmatrix}
-14 \\
4 \\
-17
\end{pmatrix}
\][/tex]
First, we need to find the determinant of matrix [tex]\( A \)[/tex], denoted as [tex]\( |A| \)[/tex].
Next, we create three matrices [tex]\( A_x \)[/tex], [tex]\( A_y \)[/tex], and [tex]\( A_z \)[/tex] by replacing the columns of [tex]\( A \)[/tex] with the vector [tex]\( B \)[/tex]:
1. [tex]\( A_x \)[/tex] is obtained by replacing the first column of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
[tex]\[
A_x = \begin{pmatrix}
-14 & 4 & -1 \\
4 & 6 & 3 \\
-17 & 7 & 2
\end{pmatrix}
\][/tex]
2. [tex]\( A_y \)[/tex] is obtained by replacing the second column of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
[tex]\[
A_y = \begin{pmatrix}
1 & -14 & -1 \\
5 & 4 & 3 \\
-2 & -17 & 2
\end{pmatrix}
\][/tex]
3. [tex]\( A_z \)[/tex] is obtained by replacing the third column of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
[tex]\[
A_z = \begin{pmatrix}
1 & 4 & -14 \\
5 & 6 & 4 \\
-2 & 7 & -17
\end{pmatrix}
\][/tex]
The determinants we need to compute are [tex]\( |A| \)[/tex], [tex]\( |A_x| \)[/tex], [tex]\( |A_y| \)[/tex], and [tex]\( |A_z| \)[/tex]. The solutions for the determinants are as follows:
[tex]\[
|A| = -119.99999999999997
\][/tex]
[tex]\[
|A_x| = -240.0000000000002
\][/tex]
[tex]\[
|A_y| = 360.00000000000006
\][/tex]
[tex]\[
|A_z| = -480.0
\][/tex]
By using these determinants, you can solve for the unknowns [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] as follows:
[tex]\[
x = \frac{|A_x|}{|A|}, \quad y = \frac{|A_y|}{|A|}, \quad z = \frac{|A_z|}{|A|}
\][/tex]
In conclusion:
[tex]\[
|A| = -119.99999999999997
\][/tex]
[tex]\[
|A_x| = -240.0000000000002
\][/tex]
[tex]\[
|A_y| = 360.00000000000006
\][/tex]
[tex]\[
|A_z| = -480.0
\][/tex]
