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resuelve con gauss jordan y determina si es consistente o inconsistente

Let's solve the given system of linear equations using the Gauss-Jordan elimination method. The system of equations is:

\[
\begin{aligned}
\frac{1}{2}x_1 - \frac{3}{7}x_2 + \frac{2}{9}x_3 &= \frac{349}{630}, \\
\frac{2}{3}x_1 + \frac{4}{9}x_2 - \frac{2}{5}x_3 &= \frac{-19}{45}, \\
\frac{4}{5}x_1 - \frac{1}{8}x_2 + \frac{4}{3}x_3 &= \frac{139}{150}.
\end{aligned}
\]

### Step 1: Convert to Augmented Matrix

First, let's write the augmented matrix corresponding to the system:

\[
\begin{pmatrix}
\frac{1}{2} & -\frac{3}{7} & \frac{2}{9} & \Big| & \frac{349}{630} \\
\frac{2}{3} & \frac{4}{9} & -\frac{2}{5} & \Big| & \frac{-19}{45} \\
\frac{4}{5} & -\frac{1}{8} & \frac{4}{3} & \Big| & \frac{139}{150}
\end{pmatrix}
\]

### Step 2: Simplify Fractions (if possible)
We'll start by simplifying the fractions in the matrix.

1. \(\frac{349}{630} = \frac{349 \div 7}{630 \div 7} = \frac{349}{630} = \frac{49}{90}\).
2. \(\frac{-19}{45}\) is already simplified.
3. \(\frac{139}{150}\) is already simplified.

The matrix becomes:

\[
\begin{pmatrix}
\frac{1}{2} & -\frac{3}{7} & \frac{2}{9} & \Big| & \frac{49}{90} \\
\frac{2}{3} & \frac{4}{9} & -\frac{2}{5} & \Big| & \frac{-19}{45} \\
\frac{4}{5} & -\frac{1}{8} & \frac{4}{3} & \Big| & \frac{139}{150}
\end{pmatrix}
\]

### Step 3: Perform Gauss-Jordan Elimination

I will now perform the row operations to transform this matrix into the reduced row echelon form.

Let's carry out the row operations and computations.

\[
\text{R}_1 \text{ becomes:} \quad R_1 \rightarrow R_1 \cdot 2 \Rightarrow (1, -\frac{6}{7}, \frac{4}{9} \Big| \frac{98}{90}) \quad \Rightarrow R_1 = (1, -\frac{6}{7}, \frac{4}{9} \Big| \frac{49}{45})
\]

We then update other rows to eliminate the first column for other rows:

1. \(R_2 - \frac{2}{3}R_1\) and 
2. \(R_3 - \frac{4}{5}R_1\)

This will produce an intermediate row echelon form, and further operations would yield a fully reduced row echelon form (RREF).

### Step 4: Continue and Check for Consistency

The system will either have a unique solution, infinite solutions, or no solution depending on the reduced matrix. I'll continue with this step.

---

To ensure precise computation, I'll implement these operations using Python. Let's perform these steps.The reduced row echelon form (RREF) of the given system is:

\[
\begin{pmatrix}
1 & 0 & 0 & \Big| & 0.2 \\
0 & 1 & 0 & \Big| & -0.8 \\
0 & 0 & 1 & \Big| & 0.5
\end{pmatrix}
\]

### Interpretation

This matrix corresponds to the following equations:

\[
\begin{aligned}
x_1 &= 0.2, \\
x_2 &= -0.8, \\
x_3 &= 0.5.
\end{aligned}
\]

Since each variable has a unique solution, the system is **consistent** and has a unique solution.

### Final Solution
The solution to the system of equations is:

\[
\boxed{x_1 = 0.2, \quad x_2 = -0.8, \quad x_3 = 0.5}
\]

Would you like more details on any part of the process, or do you have any other questions?

Here are 5 related questions to deepen your understanding:
1. How does the Gauss-Jordan elimination method differ from Gaussian elimination?
2. Why is it important to simplify fractions before applying row operations?
3. What are the conditions for a system of linear equations to have no solutions?
4. Can a system have a unique solution if the RREF matrix has a row of all zeros?
5. How can you verify the solution of a system of equations after solving it?

**Tip:** Always double-check the arithmetic when performing row operations manually to avoid simple mistakes that can lead to incorrect solutions.

Learn how to solve a system of linear equations using the Gauss-Jordan elimination method and determine its consistency. Step-by-step solution provided with detailed matrix operations and fractions simplification. Suitable for advanced high school and college students.

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