To find the solution to the system of linear equations using Cramer's Rule, we need to follow these steps:
The given system of equations is:
[tex]\[
\begin{array}{l}
-5x + 2y - 2z = 26 \\
3x + 5y + z = -22 \\
-3x - 5y - 2z = 21
\end{array}
\][/tex]
Step 1: Write the system of equations in matrix form [tex]\(AX = B\)[/tex].
Here,
[tex]\[ A = \begin{pmatrix}
-5 & 2 & -2 \\
3 & 5 & 1 \\
-3 & -5 & -2
\end{pmatrix}, \quad X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad B = \begin{pmatrix}
26 \\
-22 \\
21
\end{pmatrix} \][/tex]
Step 2: Calculate the determinant of matrix [tex]\(A\)[/tex] ([tex]\(\det(A)\)[/tex]).
The determinant of [tex]\(A\)[/tex] is:
[tex]\[
\det(A) = 31.0
\][/tex]
Since [tex]\(\det(A) \neq 0\)[/tex], we know that a unique solution exists.
Step 3: Calculate the determinants of matrices formed by replacing each column of [tex]\(A\)[/tex] with vector [tex]\(B\)[/tex].
1. For [tex]\(A_1\)[/tex] (Replace the first column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex]):
[tex]\[
A_1 = \begin{pmatrix}
26 & 2 & -2 \\
-22 & 5 & 1 \\
21 & -5 & -2
\end{pmatrix}
\][/tex]
The determinant of [tex]\(A_1\)[/tex] is:
[tex]\[
\det(A_1) = -186.0
\][/tex]
2. For [tex]\(A_2\)[/tex] (Replace the second column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex]):
[tex]\[
A_2 = \begin{pmatrix}
-5 & 26 & -2 \\
3 & -22 & 1 \\
-3 & 21 & -2
\end{pmatrix}
\][/tex]
The determinant of [tex]\(A_2\)[/tex] is:
[tex]\[
\det(A_2) = -31.0
\][/tex]
3. For [tex]\(A_3\)[/tex] (Replace the third column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex]):
[tex]\[
A_3 = \begin{pmatrix}
-5 & 2 & 26 \\
3 & 5 & -22 \\
-3 & -5 & 21
\end{pmatrix}
\][/tex]
The determinant of [tex]\(A_3\)[/tex] is:
[tex]\[
\det(A_3) = 31.0
\][/tex]
Step 4: Use Cramer's Rule to solve for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
[tex]\[ x = \frac{\det(A_1)}{\det(A)} = \frac{-186.0}{31.0} = -6.0 \][/tex]
[tex]\[ y = \frac{\det(A_2)}{\det(A)} = \frac{-31.0}{31.0} = -1.0 \][/tex]
[tex]\[ z = \frac{\det(A_3)}{\det(A)} = \frac{31.0}{31.0} = 1.0 \][/tex]
Thus, the unique solution to the system of equations is:
[tex]\[ (x, y, z) = (-6, -1, 1) \][/tex]
