To solve the given system of linear equations, follow these detailed steps:
1. Write down the system of equations:
[tex]\[
\begin{cases}
x + 3y + 2z = 26 \\
x - 3y + 4z = 2 \\
2x + y + z = 8
\end{cases}
\][/tex]
2. Construct the corresponding augmented matrix:
The coefficient matrix [tex]\(A\)[/tex] and the constant matrix [tex]\(B\)[/tex] for the system can be written as:
[tex]\[
A = \begin{pmatrix}
1 & 3 & 2 \\
1 & -3 & 4 \\
2 & 1 & 1
\end{pmatrix}
\quad \text{and} \quad
B = \begin{pmatrix}
26 \\
2 \\
8
\end{pmatrix}
\][/tex]
3. Solve the linear system:
Using appropriate methods (like Gaussian elimination, matrix inversion, or any other reliable method), we will solve for the vector [tex]\( \begin{pmatrix} x \\ y \\ z \end{pmatrix} \)[/tex].
4. Find the values of the variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
Solving the system correctly, the solution is found to be:
[tex]\[
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} =
\begin{pmatrix}
-1.4285714285714297 \\
5.714285714285715 \\
5.142857142857144
\end{pmatrix}
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -1.4285714285714297, \quad y = 5.714285714285715, \quad z = 5.142857142857144
\][/tex]
This means:
[tex]\[
\boxed{\left( -1.4285714285714297, 5.714285714285715, 5.142857142857144 \right)}
\][/tex]
