Solve the following system of equations using Gaussian elimination.
𝑥 + 3𝑦 − 5𝑧 = 2
3𝑥 + 11𝑦 − 9𝑧 = 4
−𝑥 + 𝑦 + 6𝑧 = 5

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

1. **Write the system in augmented matrix form:**

\[
\begin{bmatrix}
1 & 3 & -5 & | & 2 \\
3 & 11 & -9 & | & 4 \\
-1 & 1 & 6 & | & 5
\end{bmatrix}
\]

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

3. **Back-substitute to find the values of \( x \), \( y \), and \( z \).**

### Step 1: Augmented Matrix
\[
\begin{bmatrix}
1 & 3 & -5 & | & 2 \\
3 & 11 & -9 & | & 4 \\
-1 & 1 & 6 & | & 5
\end{bmatrix}
\]

### Step 2: Row Operations

**Operation 1: Make the first element of the second row zero.**

\[ R_2 \leftarrow R_2 - 3R_1 \]

\[
\begin{bmatrix}
1 & 3 & -5 & | & 2 \\
0 & 2 & 6 & | & -2 \\
-1 & 1 & 6 & | & 5
\end{bmatrix}
\]

**Operation 2: Make the first element of the third row zero.**

\[ R_3 \leftarrow R_3 + R_1 \]

\[
\begin{bmatrix}
1 & 3 & -5 & | & 2 \\
0 & 2 & 6 & | & -2 \\
0 & 4 & 1 & | & 7
\end{bmatrix}
\]

**Operation 3: Make the second element of the third row zero.**

\[ R_3 \leftarrow R_3 - 2R_2 \]

\[
\begin{bmatrix}
1 & 3 & -5 & | & 2 \\
0 & 2 & 6 & | & -2 \\
0 & 0 & -11 & | & 11
\end{bmatrix}
\]

### Step 3: Back-Substitution

1. From the third row: \( -11z = 11 \Rightarrow z = -1 \).

2. Substitute \( z = -1 \) into the second row:

\[
2y + 6(-1) = -2 \Rightarrow 2y - 6 = -2 \Rightarrow 2y = 4 \Rightarrow y = 2
\]

3. Substitute \( y = 2 \) and \( z = -1 \) into the first row:

\[
x + 3(2) - 5(-1) = 2 \Rightarrow x + 6 + 5 = 2 \Rightarrow x = 2 - 11 = -9
\]

### Solution

\[
x = -9, \quad y = 2, \quad z = -1
\]

This is the solution to the system of equations.

Would you like any further details or have any questions? Here are some related questions you can explore:

1. How does Gaussian elimination compare to other methods like Cramer's rule?
2. What are the conditions for a system to have no solution or infinitely many solutions?
3. Can you explain the significance of the augmented matrix in solving systems of equations?
4. How does row echelon form differ from reduced row echelon form?
5. Can Gaussian elimination be used for non-square matrices?

**Tip:** Always double-check your row operations to avoid calculation errors in Gaussian elimination.

This page explains how to solve a system of linear equations using Gaussian elimination. The system involves three variables (x, y, z) with a step-by-step guide on forming an augmented matrix, performing row operations, and using back-substitution. Suitable for high school students in Grades 10-12.

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