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To solve this system of equations using the Gauss Elimination method, we proceed as follows:

### System of equations:
1. \( 3x_1 + 6x_2 - 9x_3 = 15 \)
2. \( 2x_1 + 4x_2 - 6x_3 = 10 \)
3. \( -2x_1 - 3x_2 + 4x_3 = -6 \)

### Step-by-step Gauss Elimination:

1. **Write the augmented matrix** for the system:

\[
\begin{bmatrix}
3 & 6 & -9 & | & 15 \\
2 & 4 & -6 & | & 10 \\
-2 & -3 & 4 & | & -6
\end{bmatrix}
\]

2. **Perform row operations to get an upper triangular matrix**:

   - **Row 1 (R1):** Divide by 3 to normalize the leading coefficient:
   
     \[
     R1 \leftarrow \frac{1}{3}R1
     \]
     \[
     \begin{bmatrix}
     1 & 2 & -3 & | & 5 \\
     2 & 4 & -6 & | & 10 \\
     -2 & -3 & 4 & | & -6
     \end{bmatrix}
     \]
   
   - **Row 2 (R2):** Eliminate the first entry in row 2 by subtracting \( 2R1 \) from row 2:
   
     \[
     R2 \leftarrow R2 - 2R1
     \]
     \[
     \begin{bmatrix}
     1 & 2 & -3 & | & 5 \\
     0 & 0 & 0 & | & 0 \\
     -2 & -3 & 4 & | & -6
     \end{bmatrix}
     \]
   
   - **Row 3 (R3):** Eliminate the first entry in row 3 by adding \( 2R1 \) to row 3:
   
     \[
     R3 \leftarrow R3 + 2R1
     \]
     \[
     \begin{bmatrix}
     1 & 2 & -3 & | & 5 \\
     0 & 0 & 0 & | & 0 \\
     0 & 1 & -2 & | & 4
     \end{bmatrix}
     \]
   
3. **Back-substitution**:
   The second row in the matrix is all zeros, which means the system has dependent equations, leading to infinitely many solutions. Let's write the solution in parametric form.

   From the third row:
   \[
   x_2 - 2x_3 = 4 \implies x_2 = 4 + 2x_3
   \]
   From the first row:
   \[
   x_1 + 2x_2 - 3x_3 = 5
   \]
   Substitute \( x_2 = 4 + 2x_3 \):
   \[
   x_1 + 2(4 + 2x_3) - 3x_3 = 5 \implies x_1 + 8 + 4x_3 - 3x_3 = 5
   \]
   Simplify:
   \[
   x_1 + x_3 = -3 \implies x_1 = -3 - x_3
   \]

### Final Parametric Solution:
- \( x_1 = -3 - x_3 \)
- \( x_2 = 4 + 2x_3 \)
- \( x_3 = x_3 \) (free variable)

The system has infinitely many solutions parameterized by \( x_3 \).

---

Would you like further details or clarifications? Here are some related questions to deepen your understanding:

1. How do you identify when a system has infinitely many solutions during Gaussian elimination?
2. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
3. Can Gaussian elimination always be used to solve any system of equations?
4. How can the parametric solution be interpreted geometrically?
5. How would you solve the same system using matrix inversion?

**Tip**: Always check for dependent or inconsistent equations when performing Gaussian elimination to understand the nature of the solution.

Solve using Gauss Elimination method: 3x₁ + 6x₂ - 9x₃ = 15, 2x₁ + 4x₂ - 6x₃ = 10, -2x₁ - 3x₂ + 4x₃ = -6.

Learn how to solve a system of three linear equations using the Gauss Elimination method. The problem includes detailed steps, matrix row operations, and the final parametric solution. This is suitable for advanced high school or university-level students.

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