$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ 1 x +3 y + -2 z=3\\ 3 x +2 y + 1 z=2\\ 0 x +1 y + 3 z=5\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} x+3y-2z=3 \\ 3x+2y+z=2 \\ y+3z=5 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 3 & -2 & 3 \\ 3 & 2 & 1 & 2 \\ 0 & 1 & 3 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\cdot 1\text{-Zeile}1\cdot 3 \\z2s1=3\cdot 1-1\cdot 3 =0 \\ z2s2=2\cdot 1-3\cdot 3 =-7 \\ z2s3=1\cdot 1-(-2)\cdot 3 =7 \\ z2s4=2\cdot 1-3\cdot 3 =-7 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 3 & -2 & 3 \\ 0 & -7 & 7 & -7 \\ 0 & 1 & 3 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot (-7)\text{-Zeile}2\cdot 1 \\z3s2=1\cdot -7-(-7)\cdot 1 =0 \\ z3s3=3\cdot -7-7\cdot 1 =-28 \\ z3s4=5\cdot -7-(-7)\cdot 1 =-28 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 3 & -2 & 3 \\ 0 & -7 & 7 & -7 \\ 0 & 0 & -28 & -28 \\ \end{array} \\ \\ z=\frac{-28}{-28}=1\\y \cdot (-7)+7\cdot1=(-7)\\y= 2\\x\cdot 1+3\cdot2+(-2)\cdot1=3\\x= -1\\L=\{-1/2/1\} \end{array}$