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:

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?