https://file.aiquickdraw.com/mathbot/file/1729362405819v5zyu3zm.jpg

The system of linear equations displayed in the image is as follows:

\[
\begin{aligned}
2x_1 + 3x_2 - x_3 &= 5 \quad (1) \\
4x_1 + x_2 + 2x_3 &= 11 \quad (2) \\
-2x_1 + 5x_2 + 3x_3 &= 0 \quad (3)
\end{aligned}
\]

The task is to solve this system using **Gaussian elimination**. Let's walk through the steps:

### Step 1: Form the augmented matrix
The augmented matrix for this system is:

\[
\begin{pmatrix}
2 & 3 & -1 & | & 5 \\
4 & 1 & 2 & | & 11 \\
-2 & 5 & 3 & | & 0
\end{pmatrix}
\]

### Step 2: Perform row operations to get the matrix in upper triangular form

#### First, make the leading coefficient of the first row a 1 by dividing Row 1 by 2:
\[
\text{Row 1} \rightarrow \frac{1}{2} \text{Row 1}
\]

\[
\begin{pmatrix}
1 & 1.5 & -0.5 & | & 2.5 \\
4 & 1 & 2 & | & 11 \\
-2 & 5 & 3 & | & 0
\end{pmatrix}
\]

#### Second, eliminate the first element in Row 2 and Row 3:
- Row 2: \( \text{Row 2} \rightarrow \text{Row 2} - 4 \times \text{Row 1} \)
- Row 3: \( \text{Row 3} \rightarrow \text{Row 3} + 2 \times \text{Row 1} \)

\[
\begin{pmatrix}
1 & 1.5 & -0.5 & | & 2.5 \\
0 & -5 & 4 & | & 1 \\
0 & 8 & 2 & | & 5
\end{pmatrix}
\]

### Step 3: Eliminate the second element in Row 3:
- Row 3: \( \text{Row 3} \rightarrow \text{Row 3} + \frac{8}{-5} \times \text{Row 2} \)

\[
\begin{pmatrix}
1 & 1.5 & -0.5 & | & 2.5 \\
0 & -5 & 4 & | & 1 \\
0 & 0 & \frac{34}{5} & | & \frac{21}{5}
\end{pmatrix}
\]

### Step 4: Back-substitution
Now that we have an upper triangular matrix, we can solve the system by back-substitution:

1. From the third row: 
   \[
   \frac{34}{5} x_3 = \frac{21}{5} \implies x_3 = \frac{21}{34} = 0.618
   \]
   
2. From the second row: 
   \[
   -5x_2 + 4x_3 = 1 \implies -5x_2 + 4(0.618) = 1 \implies -5x_2 = 1 - 2.472 = -1.472 \implies x_2 = 0.294
   \]

3. From the first row:
   \[
   x_1 + 1.5x_2 - 0.5x_3 = 2.5 \implies x_1 + 1.5(0.294) - 0.5(0.618) = 2.5
   \]
   Simplifying:
   \[
   x_1 + 0.441 - 0.309 = 2.5 \implies x_1 = 2.5 - 0.132 = 2.368
   \]

### Final Solution:
\[
x_1 = 2.368, \quad x_2 = 0.294, \quad x_3 = 0.618
\]

Would you like further details on any step? Here are some follow-up questions:

1. How can you verify that the solution is correct?
2. What is Gaussian elimination used for in general?
3. What if the system had infinitely many solutions? How would we detect that?
4. Can this method be applied to larger systems of equations?
5. How would the process differ for a 2x2 system?

**Tip:** When solving systems of linear equations using Gaussian elimination, it's important to keep track of your row operations to avoid calculation errors.

Selesaikan sistem persamaan linear berikut menggunakan metode eliminasi Gauss:

1. 2x₁ + 3x₂ - x₃ = 5
2. 4x₁ + x₂ + 2x₃ = 11
3. -2x₁ + 5x₂ + 3x₃ = 0

Learn how to solve a system of 3x3 linear equations using the Gaussian elimination method. This example walks through row operations, back-substitution, and provides a detailed solution. Suitable for high school students in Grades 10-12.

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