https://file.aiquickdraw.com/mathbot/file/1733921968000z5tpgd1p.jpg

The given image contains a system of three linear equations:

1. \( x_1 - x_2 - x_3 + x_4 = 3 \)
2. \( -3x_1 + x_2 - x_3 + 2x_4 = -3 \)
3. \( 2x_1 - x_2 + x_3 - 3x_4 = 2 \)

### Solution:
To solve this system of equations, we can use the **Gaussian elimination method** or any matrix-based solver. Let’s solve step by step.

#### Step 1: Write the augmented matrix
\[
\begin{bmatrix}
1 & -1 & -1 & 1 & | & 3 \\
-3 & 1 & -1 & 2 & | & -3 \\
2 & -1 & 1 & -3 & | & 2
\end{bmatrix}
\]

#### Step 2: Solve using Gaussian elimination
We will convert this matrix to row echelon form and back-substitute to find the solution.

1. Start with the first pivot (row 1, column 1):
   Divide row 1 by 1 (pivot remains unchanged):
   \[
   \begin{bmatrix}
   1 & -1 & -1 & 1 & | & 3 \\
   -3 & 1 & -1 & 2 & | & -3 \\
   2 & -1 & 1 & -3 & | & 2
   \end{bmatrix}
   \]

2. Eliminate column 1 for rows 2 and 3:
   - Row 2: Add \( 3 \times \text{row 1} \) to row 2.
   - Row 3: Subtract \( 2 \times \text{row 1} \) from row 3.

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

3. Use row 2 as the new pivot to eliminate below it:
   Divide row 2 by -2:
   \[
   \begin{bmatrix}
   1 & -1 & -1 &

Learn how to solve a system of three linear equations using Gaussian elimination. This example covers key concepts from linear algebra, including matrix representation and row reduction techniques. Suitable for Grades 10-12 students.

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