Answer
$(\frac{1}{3}s-\frac{2}{3}t, \frac{1}{3}s+\frac{1}{3}t, s, t)$
Work Step by Step
Step 1. Establish the augmented matrix of the system and use the Gauss Eliminations method:
$\begin{vmatrix} 1 & -1 & 0 & 1 & 0 \\3 & 0 & -1 & 2 & 0\\1 & -4 & 1 & 2 & 0 \end{vmatrix} \begin{array}(\\R_2-3R_1\to R_2\\R_1-R_3\to R_3\\ \end{array}$
Step 2. Do the operations given on the right side of the matrix.
$\begin{vmatrix} 1 & -1 & 0 & 1 & 0 \\0 & 3 & -1 & -1 & 0\\0 & 3 & -1 & -1 & 0 \end{vmatrix} \begin{array}(\\.\\.\\ \end{array}$
Step 3. The last two rows are identical, and we need two parametric variables. Let $w=t, z=s$, from the second row, we have $3y-s-t=0$ which gives $y=\frac{1}{3}s+\frac{1}{3}t$. Use the first row, $x-y+t=0$, we get $x=\frac{1}{3}s-\frac{2}{3}t$. Thus the solutions are $(\frac{1}{3}s-\frac{2}{3}t, \frac{1}{3}s+\frac{1}{3}t, s, t)$
