We start with the given system of equations:
\[
\begin{aligned}
2 x_{1} + 3 x_{2} + 8 x_{3} & = -2 \quad (1) \\
x_{1} - x_{2} + 5 x_{3} & = 11 \quad (2)
\end{aligned}
\]
We can represent the system of equations in augmented matrix form:
\[
\begin{bmatrix}
2 & 3 & 8 & | & -2 \\
1 & -1 & 5 & | & 11
\end{bmatrix}
\]
To solve the system, we will perform row operations to bring the matrix to row-echelon form.
- We can multiply the second row by 2 and subtract it from the first row to eliminate \(x_1\) from the first equation:
\[
R_1 \leftarrow R_1 - 2R_2
\]
This gives us:
\[
\begin{bmatrix}
0 & 5 & -2 & | & -24 \\
1 & -1 & 5 & | & 11
\end{bmatrix}
\]
- Next, we can simplify the first row by dividing it by 5:
\[
R_1 \leftarrow \frac{1}{5} R_1
\]
This results in:
\[
\begin{bmatrix}
0 & 1 & -\frac{2}{5} & | & -\frac{24}{5} \\
1 & -1 & 5 & | & 11
\end{bmatrix}
\]
Now we can express \(x_2\) in terms of \(x_3\) from the first row:
\[
x_{2} = -\frac{24}{5} + \frac{2}{5} x_{3}
\]
Substituting \(x_2\) into the second equation:
\[
x_{1} - \left(-\frac{24}{5} + \frac{2}{5} x_{3}\right) + 5 x_{3} = 11
\]
This simplifies to:
\[
x_{1} + \frac{24}{5} - \frac{2}{5} x_{3} + 5 x_{3} = 11
\]
Combining like terms gives:
\[
x_{1} + \frac{24}{5} + \frac{23}{5} x_{3} = 11
\]
Rearranging the equation for \(x_1\):
\[
x_{1} = 11 - \frac{24}{5} - \frac{23}{5} x_{3}
\]
This can be simplified to:
\[
x_{1} = \frac{55}{5} - \frac{24}{5} - \frac{23}{5} x_{3} = \frac{31}{5} - \frac{23}{5} x_{3}
\]
Since \(x_3\) is a free variable, we can assign it any constant value. For simplicity, let’s set \(x_3 = 0\).
Substituting \(x_3 = 0\) into the equations for \(x_1\) and \(x_2\):
\[
x_{1} = \frac{31}{5} - \frac{23}{5}(0) = \frac{31}{5}
\]
\[
x_{2} = -\frac{24}{5} + \frac{2}{5}(0) = -\frac{24}{5}
\]
We can verify the solution by substituting \(x_1\), \(x_2\), and \(x_3\) back into the original equations:
- For the first equation:
\[
2\left(\frac{31}{5}\right) + 3\left(-\frac{24}{5}\right) + 8(0) = \frac{62}{5} - \frac{72}{5} = -\frac{10}{5} = -2
\]
- For the second equation:
\[
\frac{31}{5} - \left(-\frac{24}{5}\right) + 5(0) = \frac{31}{5} + \frac{24}{5} = \frac{55}{5} = 11
\]
Both equations are satisfied, confirming that the solution is correct.
\(\boxed{x_{1} = \frac{31}{5}, \, x_{2} = -\frac{24}{5}, \, x_{3} = 0}\)
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