$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ 1 x +1 + 0 z=1\\ 1 x +0 y + 1 z=6\\ 0 x +1 y + -1 z=5\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} x+y=1 \\ x+z=6 \\ y -z=5 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 6 \\ 0 & 1 & -1 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{1}{1}\\z2s1=1-1\cdot \frac{1}{1}=0 \\ z2s2=0-1\cdot \frac{1}{1}=-1 \\ z2s3=1-0\cdot \frac{1}{1}=1 \\ z2s4=6-1\cdot \frac{1}{1}=5 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 1 & 0 & 1 \\ 0 & -1 & 1 & 5 \\ 0 & 1 & -1 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{1}{-1}\\z1s2=1-(-1)\cdot \frac{1}{-1}=0 \\ z1s3=0-1\cdot \frac{1}{-1}=1 \\ z1s4=1-5\cdot \frac{1}{-1}=6 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 1 & 6 \\ 0 & -1 & 1 & 5 \\ 0 & 1 & -1 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{1}{-1}\\z3s2=1-(-1)\cdot \frac{1}{-1}=0 \\ z3s3=-1-1\cdot \frac{1}{-1}=0 \\ z3s4=5-5\cdot \frac{1}{-1}=10 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 1 & 6 \\ 0 & -1 & 1 & 5 \\ 0 & 0 & 0 & 10 \\ \end{array} \\ \\ \\ L=\{\} \end{array}$