I've tried solving this system of equations using Gauss-Jordan( I don't know how could I solve it any other way)
\begin{array}{c} x-y+2z-2t=-5 \\ 2x+3y-z+t=5 \\ 3x+y+2z-2t=-3 \\ 4x-y+5z-5t=-11 \end{array}
And using Gauss-Jordan, I've arrived to this matrix:
\begin{array}{rrrr|r} 1 & -1 & 2 & -2 & -5 \\ 2 & 3 & -1 & 1 & 5 \\ 1 & 0 & 1 & -1 & -2 \\ 0 & -1 & 1 & 1 & -3 \\ \end{array}
After this set of operations R3+R4, R3:7, R4-2R2, R4:7.