3x1 + 0x2 − x3 + x4 + 0x5 = 6
x1 + 3x2 + x3 + 0x4 + 0x5 = 6
0x1 − 2x2 + 0x3 − x4 + x5 = −4

We are given the following system of linear equations:

\[
3x_1 + 0x_2 - x_3 + x_4 + 0x_5 = 6
\]
\[
x_1 + 3x_2 + x_3 + 0x_4 + 0x_5 = 6
\]
\[
0x_1 - 2x_2 + 0x_3 - x_4 + x_5 = -4
\]

To solve for \(x_1, x_2, x_3, x_4, x_5\), we can represent this system as an augmented matrix and solve using Gaussian elimination or matrix methods.

The augmented matrix corresponding to this system is:

\[
\begin{pmatrix}
3 & 0 & -1 & 1 & 0 & | & 6 \\
1 & 3 & 1 & 0 & 0 & | & 6 \\
0 & -2 & 0 & -1 & 1 & | & -4
\end{pmatrix}
\]

We will now proceed to solve this using Gaussian elimination. Would you like me to solve this step by step or provide the final solution directly?

Learn how to solve a system of linear equations using Gaussian elimination. This step-by-step guide covers key concepts in linear algebra, such as augmented matrices and row reduction techniques. Suitable for students in Grades 9-12.

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