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The system of linear equations provided is:

\[
\begin{aligned}
    x_1 - x_2 + 2x_3 &= -5, \\
    3x_1 - 4x_2 + 5x_3 &= -3, \\
    -2x_1 - 6x_3 &= 34.
\end{aligned}
\]

### Steps to solve the system:

1. We will express the system in augmented matrix form and use Gaussian elimination to find the solution.
2. Convert the system into a parametric vector form.

Let me perform these steps to find the solution.The solution to the system of equations is:

\[
\begin{aligned}
    x_1 &= 0.182 \\
    x_2 &= -6.273 \\
    x_3 &= -5.727
\end{aligned}
\]

This is the unique solution to the system, and no free parameters are present.

Would you like further clarification or a step-by-step breakdown of how Gaussian elimination works?

Here are 5 additional related questions you might find interesting:
1. How does Gaussian elimination work to solve systems of equations?
2. What are the conditions for a system to have no solutions or infinitely many solutions?
3. How can we solve systems of equations using matrix inversion?
4. What are the applications of systems of equations in real-world problems?
5. How do row operations affect the solution of a system of linear equations?

**Tip:** Systems of linear equations can also be solved graphically, but this method is only practical for small systems with two or three variables.

Solve the system of equations: x1 - x2 + 2x3 = -5, 3x1 - 4x2 + 5x3 = -3, -2x1 - 6x3 = 34.

Learn how to solve the system of linear equations involving x1, x2, and x3 using Gaussian elimination. This detailed guide breaks down the process of converting a linear system into matrix form and applying elimination steps. Suitable for linear algebra students.

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