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.