Sure, let's solve the given system of linear equations using determinants. The system of equations is:
[tex]\[
\begin{array}{l}
x + 4y - z = -14 \\
5x + 6y + 3z = 4 \\
-2x + 7y + 2z = -17
\end{array}
\][/tex]
To solve this system using determinants, we'll use Cramer's rule, which requires calculating the determinants of several matrices.
1. Construct the coefficient matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
1 & 4 & -1 \\
5 & 6 & 3 \\
-2 & 7 & 2
\end{pmatrix}
\][/tex]
2. Construct the right-hand side vector [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix}
-14 \\
4 \\
-17
\end{pmatrix}
\][/tex]
3. Calculate the determinant of the matrix [tex]\( A \)[/tex], denoted as [tex]\( |A| \)[/tex]:
Given is:
[tex]\[
|A| = -119.99999999999997
\][/tex]
4. Construct matrices [tex]\( A_x \)[/tex], [tex]\( A_y \)[/tex], and [tex]\( A_z \)[/tex] by replacing the respective columns of [tex]\( A \)[/tex] with the vector [tex]\( B \)[/tex]:
- Matrix [tex]\( A_x \)[/tex] (replace 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]
Given is:
[tex]\[
|A_x| = -240.0000000000002
\][/tex]
- Matrix [tex]\( A_y \)[/tex] (replace 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]
Given is:
[tex]\[
|A_y| = 360.00000000000006
\][/tex]
- Matrix [tex]\( A_z \)[/tex] (replace 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]
Given is:
[tex]\[
|A_z| = -480.0
\][/tex]
5. Apply Cramer's rule to find [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex]:
According to Cramer's rule:
[tex]\[
x = \frac{|A_x|}{|A|}, \quad y = \frac{|A_y|}{|A|}, \quad z = \frac{|A_z|}{|A|}
\][/tex]
Substitute the values of the determinants:
[tex]\[
x = \frac{-240.0000000000002}{-119.99999999999997} = 2.000000000000002 \approx 2
\][/tex]
[tex]\[
y = \frac{360.00000000000006}{-119.99999999999997} = -3.0000000000000004 \approx -3
\][/tex]
[tex]\[
z = \frac{-480.0}{-119.99999999999997} = 4.000000000000001 \approx 4
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = 2, \quad y = -3, \quad z = 4
\][/tex]
