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Entregar: Debe entregar paso a paso, el desarrollo pertinente, para cada uno de los ejercicios propuestos, descrito con sus propias palabras, debe utilizar terminología exacta y precisa, que describa el procedimiento en un mínimo de 15 pasos, todo debe estar entregado en el documento.2. Sistemas de Ecuaciones Lineales
•	Problema: Resuelve el sistema de ecuaciones lineales utilizando el método de Gauss-Jordan
 
•	Paso a Paso:
1.	Formar la matriz aumentada del sistema.
2.	Aplicar el método de eliminación Gaussiana.
3.	Reducir la matriz a su forma escalonada.
4.	Sustituir los valores obtenidos en las ecuaciones originales para verificar la solución.


### Resolving the System of Linear Equations Using the Gauss-Jordan Method

Given the system of linear equations:

\[
\begin{aligned}
2x + 3y - z &= 5 \quad \text{(Equation 1)} \\
4x - y + 5z &= 10 \quad \text{(Equation 2)} \\
-6x + 2y + 3z &= -3 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 1: Form the Augmented Matrix
First, we convert the system of equations into an augmented matrix:

\[
\begin{pmatrix}
2 & 3 & -1 & \vert & 5 \\
4 & -1 & 5 & \vert & 10 \\
-6 & 2 & 3 & \vert & -3 \\
\end{pmatrix}
\]

### Step 2: Apply the Gaussian Elimination Method
We aim to convert the augmented matrix to row echelon form (upper triangular form) by performing row operations.

#### a. Make the pivot in the first column of the first row equal to 1:
Since the first element is 2, we can divide the entire first row by 2:

\[
R1 \to \frac{1}{2} R1
\]

This gives:

\[
\begin{pmatrix}
1 & 1.5 & -0.5 & \vert & 2.5 \\
4 & -1 & 5 & \vert & 10 \\
-6 & 2 & 3 & \vert & -3 \\
\end{pmatrix}
\]

#### b. Eliminate the elements below the pivot in the first column:
Subtract \(4 \times R1\) from \(R2\) and add \(6 \times R1\) to \(R3\):

\[
R2 \to R2 - 4R1 \quad \text{and} \quad R3 \to R3 + 6R1
\]

\[
\begin{pmatrix}
1 & 1.5 & -0.5 & \vert & 2.5 \\
0 & -7 & 7 & \vert & 0 \\
0 & 11 & 0 & \vert & 12 \\
\end{pmatrix}
\]

### Step 3: Reduce the Matrix to Row Echelon Form
Now we continue to work on the second row by making the pivot in the second column of the second row equal to 1.

#### a. Make the pivot in the second column of the second row equal to 1:
Divide the second row by \(-7\):

\[
R2 \to \frac{1}{-7} R2
\]

\[
\begin{pmatrix}
1 & 1.5 & -0.5 & \vert & 2.5 \\
0 & 1 & -1 & \vert & 0 \\
0 & 11 & 0 & \vert & 12 \\
\end{pmatrix}
\]

#### b. Eliminate the elements below and above the pivot in the second column:
To make the elements above and below this pivot (in the second column) zero, we perform the following operations:

Subtract \(1.5 \times R2\) from \(R1\) and subtract \(11 \times R2\) from \(R3\):

\[
R1 \to R1 - 1.5R2 \quad \text{and} \quad R3 \to R3 - 11R2
\]

\[
\begin{pmatrix}
1 & 0 & 1 & \vert & 2.5 \\
0 & 1 & -1 & \vert & 0 \\
0 & 0 & 11 & \vert & 12 \\
\end{pmatrix}
\]

### Step 4: Reduce the Matrix to Reduced Row Echelon Form
To make the third column's pivot equal to 1, divide \(R3\) by 11:

\[
R3 \to \frac{1}{11} R3
\]

\[
\begin{pmatrix}
1 & 0 & 1 & \vert & 2.5 \\
0 & 1 & -1 & \vert & 0 \\
0 & 0 & 1 & \vert & \frac{12}{11} \\
\end{pmatrix}
\]

#### a. Eliminate the elements above the pivot in the third column:
Subtract \(R3\) from \(R1\) and add \(R3\) to \(R2\):

\[
R1 \to R1 - R3 \quad \text{and} \quad R2 \to R2 + R3
\]

\[
\begin{pmatrix}
1 & 0 & 0 & \vert & \frac{15.5}{11} \\
0 & 1 & 0 & \vert & \frac{12}{11} \\
0 & 0 & 1 & \vert & \frac{12}{11} \\
\end{pmatrix}
\]

### Step 5: Interpret the Result
The final augmented matrix represents the solutions to the system:

\[
\begin{aligned}
x &= \frac{15.5}{11} \approx 1.41, \\
y &= \frac{12}{11} \approx 1.09, \\
z &= \frac{12}{11} \approx 1.09.
\end{aligned}
\]

These are the solutions to the system of linear equations.

---

**Would you like any further details or clarification?**  
Here are five questions to expand your understanding:
1. What are the key differences between the Gauss-Jordan method and Gaussian elimination?
2. How do you determine the number of solutions a system of linear equations has?
3. What are the implications of a system being consistent or inconsistent?
4. How can matrix operations be used in computer algorithms to solve systems of equations?
5. What are some applications of the Gauss-Jordan method in real-world problems?

**Tip:** Always check your final solutions by substituting them back into the original equations to ensure they satisfy all the equations in the system.

Learn how to solve a system of linear equations using the Gauss-Jordan method step-by-step. Includes forming an augmented matrix, applying Gaussian elimination, reducing to row echelon form, and verifying solutions. Ideal for advanced high school students studying linear algebra.

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