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To solve the given system of equations using the Gaussian elimination method, we need to follow these steps:

1. **Write the augmented matrix for the system of equations:**
\[
\begin{align*}
x - y + 3z &= -5 \\
6x + y + z &= 2 \\
x + 2y - z &= 2
\end{align*}
\]

The augmented matrix is:
\[
\left[\begin{array}{ccc|c}
1 & -1 & 3 & -5 \\
6 & 1 & 1 & 2 \\
1 & 2 & -1 & 2
\end{array}\right]
\]

2. **Perform row operations to transform the matrix into an upper triangular form:**

- **Step 1: Make the element in the first row, first column a leading 1 (already done).**
- **Step 2: Eliminate the elements below the leading 1 in the first column.**

Subtract \(6\) times the first row from the second row (\(R2 \rightarrow R2 - 6R1\)):
\[
\left[\begin{array}{ccc|c}
1 & -1 & 3 & -5 \\
0 & 7 & -17 & 32 \\
1 & 2 & -1 & 2
\end{array}\right]
\]

Subtract the first row from the third row (\(R3 \rightarrow R3 - R1\)):
\[
\left[\begin{array}{ccc|c}
1 & -1 & 3 & -5 \\
0 & 7 & -17 & 32 \\
0 & 3 & -4 & 7
\end{array}\right]
\]

- **Step 3: Make the element in the second row, second column a leading 1.**

Divide the second row by 7 (\(R2 \rightarrow \frac{1}{7}R2\)):
\[
\left[\begin{array}{ccc|c}
1 & -1 & 3 & -5 \\
0 & 1 & -\frac{17}{7} & \frac{32}{7} \\
0 & 3 & -4 & 7
\end{array}\right]
\]

- **Step 4: Eliminate the elements below and above the leading 1 in the second column.**

Subtract 3 times the second row from the third row (\(R3 \rightarrow R3 - 3R2\)):
\[
\left[\begin{array}{ccc|c}
1 & -1 & 3 & -5 \\
0 & 1 & -\frac{17}{7} & \frac{32}{7} \\
0 & 0 & \frac{37}{7} & -\frac{5}{7}
\end{array}\right]
\]

Add the second row to the first row (\(R1 \rightarrow R1 + R2\)):
\[
\left[\begin{array}{ccc|c}
1 & 0 & \frac{4}{7} & -\frac{3}{7} \\
0 & 1 & -\frac{17}{7} & \frac{32}{7} \\
0 & 0 & \frac{37}{7} & -\frac{5}{7}
\end{array}\right]
\]

- **Step 5: Make the element in the third row, third column a leading 1.**

Multiply the third row by \(\frac{7}{37}\) (\(R3 \rightarrow \frac{7}{37}R3\)):
\[
\left[\begin{array}{ccc|c}
1 & 0 & \frac{4}{7} & -\frac{3}{7} \\
0 & 1 & -\frac{17}{7} & \frac{32}{7} \\
0 & 0 & 1 & -\frac{5}{37} \cdot \frac{7}{37}
\end{array}\right]
\]

- **Step 6: Eliminate the elements above the leading 1 in the third column.**

Subtract \(\frac{4}{7}\) times the third row from the first row (\(R1 \rightarrow R1 - \frac{4}{7}R3\)):
\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & -1 \\
0 & 1 & -\frac{17}{7} & \frac{32}{7} \\
0 & 0 & 1 & -\frac{5}{37}
\end{array}\right]
\]

Add \(\frac{17}{7}\) times the third row to the second row (\(R2 \rightarrow R2 + \frac{17}{7}R3\)):
\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & -\frac{5}{37}
\end{array}\right]
\]

3. **Interpret the final matrix:**

The final augmented matrix represents the following system of equations:
\[
\begin{align*}
x &= -1 \\
y &= 1 \\
z &= -\frac{5}{37}
\end{align*}
\]

So, the solution to the system of equations is:
\[
(x, y, z) = \left( -1, 1, -\frac{5}{37} \right)
\]

Would you like further details or have any questions about this solution?

Here are some additional questions to expand on this topic:
1. What are the steps for solving a system of equations using Gaussian elimination?
2. How can you verify the solution to a system of linear equations?
3. What are the differences between Gaussian elimination and Gauss-Jordan elimination?
4. How do you handle systems of equations with no solutions or infinite solutions?
5. Can Gaussian elimination be used for non-square matrices?
6. How does the determinant of a matrix relate to the solvability of the system?
7. What are the computational complexities of Gaussian elimination?
8. How can Gaussian elimination be implemented in a computer program?

**Tip:** Always check the consistency of the system of equations before applying Gaussian elimination to ensure that there are solutions.

Learn how to solve a system of linear equations using the Gaussian elimination method with detailed steps and examples. This problem covers linear algebra concepts, including augmented matrix representation and row operations. Suitable for college-level students and mathematics enthusiasts.

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