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work using the gauss jordan elimination method 

I can see the system of equations you provided in the image:

\[
\begin{align*}
x_1 + x_2 + 2x_3 &= 9 \\
2x_1 + 4x_2 - 3x_3 &= 1 \\
3x_1 + 6x_2 - 5x_3 &= 0
\end{align*}
\]

Now, I'll solve this using the Gauss-Jordan elimination method. Let's express the system as an augmented matrix and proceed step by step to find the solution.

The augmented matrix is:

\[
\begin{pmatrix}
1 & 1 & 2 & | & 9 \\
2 & 4 & -3 & | & 1 \\
3 & 6 & -5 & | & 0
\end{pmatrix}
\]

I'll now perform Gauss-Jordan elimination to convert this matrix into reduced row-echelon form.After performing Gauss-Jordan elimination, the augmented matrix is reduced to:

\[
\begin{pmatrix}
1 & 0 & 0 & | & 1 \\
0 & 1 & 0 & | & 2 \\
0 & 0 & 1 & | & 3
\end{pmatrix}
\]

This corresponds to the solution:

\[
x_1 = 1, \quad x_2 = 2, \quad x_3 = 3
\]

Let me know if you would like more details or have any questions!

Here are 5 related questions to further explore:

1. What is the significance of using the Gauss-Jordan elimination method?
2. Can every system of linear equations be solved using Gauss-Jordan elimination?
3. How does Gauss-Jordan elimination differ from Gaussian elimination?
4. What are the potential numerical issues when solving systems with large coefficients using Gauss-Jordan elimination?
5. How can matrix operations be optimized for solving large systems?

**Tip:** Gauss-Jordan elimination not only finds the solution but also gives insights into whether a system has no solution or infinitely many solutions by reducing it to its row-reduced form.

Solve the system of equations using the Gauss-Jordan elimination method.

Learn how to solve a system of linear equations using the Gauss-Jordan elimination method. This example demonstrates step-by-step matrix operations to convert the system into reduced row-echelon form, resulting in a solution. Suitable for students in advanced high school or college-level linear algebra.

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