https://file.aiquickdraw.com/mathbot/file/1729362177265dzuvjt4v.jpg

The image shows a system of linear equations that needs to be solved using Gaussian elimination. Here's the system of equations:

\[
\text{Equation (1)}: 2x_1 + 3x_2 - x_3 = 5
\]
\[
\text{Equation (2)}: 4x_1 + x_2 + 2x_3 = 11
\]
\[
\text{Equation (3)}: -2x_1 + 5x_2 + 3x_3 = 0
\]

The steps for Gaussian elimination are outlined in the image as follows:

### PART 1: \( R_2 \to R_2 - 2 \times R_1 \)
- Here, you subtract 2 times row 1 from row 2.

### PART 2: \( R_3 \to R_3 + R_1 \)
- In this step, you add row 1 to row 3.

### PART 3: \( R_3 \to R_3 - \frac{8}{5} \times R_2 \)
- In this step, you subtract \( \frac{8}{5} \) times row 2 from row 3.

### Let's perform the steps one by one:

#### Step 1: Apply \( R_2 \to R_2 - 2 \times R_1 \)

From equation (1):
\[
R_1 = (2x_1 + 3x_2 - x_3 = 5)
\]

Multiply by 2:
\[
2R_1 = (4x_1 + 6x_2 - 2x_3 = 10)
\]

Now subtract from \( R_2 \):
\[
R_2 = (4x_1 + x_2 + 2x_3 = 11) - (4x_1 + 6x_2 - 2x_3 = 10)
\]

This gives:
\[
0x_1 - 5x_2 + 4x_3 = 1
\]
So the new \( R_2 \) is:
\[
-5x_2 + 4x_3 = 1
\]

#### Step 2: Apply \( R_3 \to R_3 + R_1 \)

From equation (1):
\[
R_1 = (2x_1 + 3x_2 - x_3 = 5)
\]

Now add \( R_1 \) to \( R_3 \):
\[
R_3 = (-2x_1 + 5x_2 + 3x_3 = 0) + (2x_1 + 3x_2 - x_3 = 5)
\]

This gives:
\[
0x_1 + 8x_2 + 2x_3 = 5
\]
So the new \( R_3 \) is:
\[
8x_2 + 2x_3 = 5
\]

#### Step 3: Apply \( R_3 \to R_3 - \frac{8}{5} \times R_2 \)

We have:
\[
R_3 = (8x_2 + 2x_3 = 5)
\]
Multiply \( R_2 \) by \( \frac{8}{5} \):
\[
\frac{8}{5} R_2 = \frac{8}{5} (-5x_2 + 4x_3) = (-8x_2 + \frac{32}{5}x_3)
\]

Now subtract this from \( R_3 \):
\[
R_3 = (8x_2 + 2x_3 = 5) - (-8x_2 + \frac{32}{5}x_3)
\]

This simplifies to:
\[
0x_2 + \left(2x_3 - \frac{32}{5}x_3 \right) = 5
\]
\[
\Rightarrow -\frac{22}{5}x_3 = 5
\]
Solving for \( x_3 \):
\[
x_3 = \frac{5 \times -5}{22} = -\frac{25}{22}
\]

### Now back-substitute to find \( x_2 \) and \( x_1 \):

From \( R_2 \):
\[
-5x_2 + 4\left(-\frac{25}{22}\right) = 1
\]
Simplifying:
\[
-5x_2 - \frac{100}{22} = 1 \quad \Rightarrow \quad -5x_2 = 1 + \frac{100}{22} = \frac{122}{22}
\]
\[
x_2 = \frac{-122}{22 \times 5} = -\frac{122}{110} = -\frac{61}{55}
\]

Finally, from \( R_1 \):
\[
2x_1 + 3\left(-\frac{61}{55}\right) - \left(-\frac{25}{22}\right) = 5
\]
Simplifying and solving for \( x_1 \) will give the final result.

Would you like a detailed step-by-step continuation of the solution for \( x_1 \)? Feel free to ask for more details!

Here are 5 related questions:
1. How can Gaussian elimination be extended to larger systems of equations?
2. What are the conditions for a system to have a unique solution?
3. How do you determine if a system of linear equations is inconsistent?
4. What is the significance of back-substitution in Gaussian elimination?
5. How does Gaussian elimination differ from Gauss-Jordan elimination?

**Tip:** Always simplify each row fully during Gaussian elimination to avoid mistakes in back-substitution.

Selesaikan sistem persamaan linear berikut menggunakan metode eliminasi Gauss: 
(1) 2x_1 + 3x_2 - x_3 = 5 
(2) 4x_1 + x_2 + 2x_3 = 11 
(3) -2x_1 + 5x_2 + 3x_3 = 0

This problem involves solving a system of linear equations using Gaussian elimination. The system includes three equations and three unknowns: 2x_1 + 3x_2 - x_3 = 5, 4x_1 + x_2 + 2x_3 = 11, and -2x_1 + 5x_2 + 3x_3 = 0. Learn step-by-step how to apply Gaussian elimination and back substitution to find the solution.

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