Solved the problem using the echolan form. However, I am currently struggling to find the variables associated with this solution.

Question

The matrix is

$$\left( \begin{array}{cccc|c} 4 & -1 & 1 & -1 & 3 \\ 1 & 2 & -3 & 2 & 2 \\ 2 & -5 & 7 & -5 & -1 \\ 7 & -4 & 5 & -4 & 4 \\ \end{array} \right)$$ R1(-1/4)+R2 R1(-1/2)+R3 R1(-1/4)(7)+R4 (1/2)R3+R2 changed R3 whit R4

Solved the system of linear equations by transforming the matrix to gaussian form and I have reached the desired form and the last two rows of the matrix are zero.
I am currently struggling to find the values of $x_1$, $x_2$, $x_3$, and $x_4$. The question specifically asks for the value of S(Ω) and I am unsure how to proceed with calculating it.: $$\left( \begin{array}{cccc|c} 4 & -1 & 1 & -1 & 3 \\ 0 &\dfrac34 & -\dfrac{13}4 & \dfrac34 & 5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)$$

Answer 1

Considering the pivot method of Gauss https://www.youtube.com/watch?v=fD79MlN_ILc you will have:

$$\begin{pmatrix}4&-1&1&-1&3\\ 1&2&-3&2&2\\ 2&-5&7&-5&-1\\ 7&-4&5&-4&4\end{pmatrix}\to \begin{pmatrix}4&-1&1&-1&3\\ 0&\frac{9}{4}&-\frac{13}{4}&\frac{9}{4}&\frac{5}{4}\\ 0&0&0&0&0\\ 0&0&0&0&0\end{pmatrix}$$