https://file.aiquickdraw.com/mathbot/file/172545102013388a7pevo.jpg

The system of linear equations provided in the image is:

\[
\begin{aligned}
3x_1 + 6x_2 - 9x_3 &= -65.925 \quad \text{(Equation 1)}\\
x_1 + 4x_2 - x_3 &= -19.875 \quad \text{(Equation 2)}\\
2x_1 + 5x_2 - 7x_3 &= -28.875 \quad \text{(Equation 3)}
\end{aligned}
\]

We will solve this system using the Gauss elimination method. The steps involved are as follows:

1. **Form the augmented matrix** corresponding to the system of equations.
2. **Apply Gaussian elimination** to transform the matrix into an upper triangular form.
3. **Back-substitute** to find the solution for the variables.

### Step 1: Form the Augmented Matrix

The augmented matrix for the system is:

\[
\begin{pmatrix}
3 & 6 & -9 & \vert & -65.925 \\
1 & 4 & -1 & \vert & -19.875 \\
2 & 5 & -7 & \vert & -28.875
\end{pmatrix}
\]

### Step 2: Apply Gaussian Elimination

We'll start by eliminating the first column below the pivot (first element of the first row).

1. **First Row Operation**: 
   - Subtract \( \frac{1}{3} \times \) (Row 1) from Row 2.
   - Subtract \( \frac{2}{3} \times \) (Row 1) from Row 3.

After applying these operations, the new matrix is:

\[
\begin{pmatrix}
3 & 6 & -9 & \vert & -65.925 \\
0 & 2 & 2 & \vert & 1.4 \\
0 & 1 & -1 & \vert & 15.75
\end{pmatrix}
\]

2. **Second Row Operation**:
   - Subtract \( \frac{1}{2} \times \) (Row 2) from Row 3.

After this operation, the new matrix is:

\[
\begin{pmatrix}
3 & 6 & -9 & \vert & -65.925 \\
0 & 2 & 2 & \vert & 1.4 \\
0 & 0 & -2 & \vert & 15.05
\end{pmatrix}
\]

### Step 3: Back-Substitution

Now that the matrix is in upper triangular form, we can solve for \(x_3\), \(x_2\), and \(x_1\).

1. From the third row: 
   \[
   -2x_3 = 15.05 \quad \Rightarrow \quad x_3 = -7.525
   \]

2. Substitute \( x_3 \) into the second row:
   \[
   2x_2 + 2(-7.525) = 1.4 \quad \Rightarrow \quad 2x_2 - 15.05 = 1.4 \quad \Rightarrow \quad 2x_2 = 16.45 \quad \Rightarrow \quad x_2 = 8.225
   \]

3. Substitute \( x_2 \) and \( x_3 \) into the first row:
   \[
   3x_1 + 6(8.225) - 9(-7.525) = -65.925
   \]
   Simplifying:
   \[
   3x_1 + 49.35 + 67.725 = -65.925 \quad \Rightarrow \quad 3x_1 + 117.075 = -65.925 \quad \Rightarrow \quad 3x_1 = -183 \quad \Rightarrow \quad x_1 = -61
   \]

### Solution

Thus, the solution to the system is:

\[
x_1 = -61, \quad x_2 = 8.225, \quad x_3 = -7.525
\]

Would you like more details on any step, or do you have any questions?

Here are five related questions for further practice:

1. How can the Gauss-Jordan elimination method simplify solving systems of equations compared to Gaussian elimination?
2. What are some potential pitfalls when using Gaussian elimination, such as dealing with zero pivots?
3. How would the presence of a non-zero determinant of the coefficient matrix affect the system's solutions?
4. Can Gaussian elimination be applied to systems with more variables than equations? Why or why not?
5. How can row operations be used to solve a system of linear equations using an inverse matrix?

**Tip:** When solving systems of equations using elimination methods, it's important to check each step carefully to avoid computational errors that can lead to incorrect results.

Learn how to solve a system of linear equations using Gaussian elimination with a step-by-step guide. This example covers concepts in linear algebra, including forming augmented matrices, performing row operations, and back-substitution. Suitable for high school and early college students.

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