https://file.aiquickdraw.com/mathbot/file/1726478823429gcbsl75s.jpg

The system of equations provided is:

\[
\begin{aligned}
1. \quad x - 2y &= -4 \\
2. \quad -5y + z &= -9 \\
3. \quad 4x - 3z &= -10
\end{aligned}
\]

We will solve this system using the **Gauss-Jordan Method**. This method transforms the system into reduced row echelon form (RREF) using elementary row operations.

First, I will set up the augmented matrix for the system:

\[
\begin{bmatrix}
1 & -2 & 0 & | & -4 \\
0 & -5 & 1 & | & -9 \\
4 & 0 & -3 & | & -10
\end{bmatrix}
\]

Let's perform the row operations to solve the system step by step.After performing Gauss-Jordan elimination, the augmented matrix is transformed into:

\[
\begin{bmatrix}
1 & 0 & 0 & | & 2 \\
0 & 1 & 0 & | & 3 \\
0 & 0 & 1 & | & 6
\end{bmatrix}
\]

This gives the solution to the system:

\[
x = 2, \quad y = 3, \quad z = 6
\]

Let me know if you would like further clarification or details on the steps. Here are some related questions:

1. What are the advantages of using the Gauss-Jordan method over Gaussian elimination?
2. How do elementary row operations affect the determinant of a matrix?
3. Can the Gauss-Jordan method be applied to non-square systems of equations?
4. How does the Gauss-Jordan method handle inconsistent systems?
5. What is the role of pivoting in improving numerical stability in this method?

**Tip:** When performing the Gauss-Jordan method manually, always aim to simplify fractions at each step to make computations easier.

Solve the system of equations using Gauss Jordan Method:
x - 2y = -4,
-5y + z = -9,
4x - 3z = -10.

Learn how to solve a system of equations using the Gauss-Jordan Method. We solve the system x - 2y = -4, -5y + z = -9, and 4x - 3z = -10 step by step, transforming the equations into reduced row echelon form.

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