https://r2.erweima.ai/mathbot/image/17256539305447q7w8yf5.jpg

The image contains a system of linear equations, along with steps showing the matrix representation of the system in augmented form and the application of Gaussian elimination. It eventually leads to a conclusion that the solution is the "trivial" solution, implying the system has no unique or non-trivial solution.

Here is a breakdown of the key elements:

### System of Equations (b):
\[
\begin{aligned}
2x_1 + 3x_2 - x_3 + 4x_4 + 3x_5 &= 5, \\
3x_1 - 3x_2 - 2x_3 + 3x_4 - 5x_5 &= -10, \\
x_1 + 3x_2 + 4x_3 + 6x_4 + 10x_5 &= 15.
\end{aligned}
\]

### Matrix Augmentation and Gaussian Elimination:
The process of row reduction is used to convert the system into reduced row echelon form (RREF). The step-by-step operations are shown with several matrices where elementary row operations are applied. 

Key observations:
1. The system is expressed as an augmented matrix.
2. Several row operations (marked as \(R_i \to R_j\)) are performed to simplify the matrix.
3. At the final stage, the matrix has a row with zeros indicating a dependent variable or possibly indicating no non-trivial solutions.

### Conclusion:
The final statement, "por tanto la solución es la trivial," suggests that the only solution is the trivial one, meaning either:
- The system is inconsistent (no solution exists).
- Or the only solution is the zero vector (trivial solution) for all unknowns \(x_1 = x_2 = x_3 = x_4 = x_5 = 0\).

Would you like further details on Gaussian elimination or any specific part of this process?

### Related Questions:
1. What are the elementary row operations used in Gaussian elimination?
2. How can one identify if a system has no solution or infinitely many solutions?
3. What is the difference between a trivial and a non-trivial solution?
4. Can Gaussian elimination be applied to any system of linear equations?
5. How does the augmented matrix represent a system of equations?

**Tip**: When performing Gaussian elimination, always check for pivot elements in each column to ensure proper row operations and consistency in the system.

Solve the following system of linear equations using Gaussian elimination:
2x1 + 3x2 - x3 + 4x4 + 3x5 = 5,
3x1 - 3x2 - 2x3 + 3x4 - 5x5 = -10,
x1 + 3x2 + 4x3 + 6x4 + 10x5 = 15.

Learn how to solve a system of linear equations with five unknowns using Gaussian elimination. This detailed step-by-step solution illustrates the transformation of the system into matrix form and applies elementary row operations to reach the final solution.

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