$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ -2 x -8 + 0 z=1\\ 1 x +4 y + 0 z=-\frac{1}{2}\\ 8 x -2 y + -1 z=8\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} -2x-8y=1 \\ x+4y=-\frac{1}{2} \\ 8x-2y -z=8 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & -8 & 0 & 1 \\ 1 & 4 & 0 & -\frac{1}{2} \\ 8 & -2 & -1 & 8 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{1}{-2}\\z2s1=1-(-2)\cdot \frac{1}{-2}=0 \\ z2s2=4-(-8)\cdot \frac{1}{-2}=0 \\ z2s3=0-0\cdot \frac{1}{-2}=0 \\ z2s4=-\frac{1}{2}-1\cdot \frac{1}{-2}=0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & -8 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 8 & -2 & -1 & 8 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{8}{-2}\\z3s1=8-(-2)\cdot \frac{8}{-2}=0 \\ z3s2=-2-(-8)\cdot \frac{8}{-2}=-34 \\ z3s3=-1-0\cdot \frac{8}{-2}=-1 \\ z3s4=8-1\cdot \frac{8}{-2}=12 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & -8 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -34 & -1 & 12 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeilen vertauschen } \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & -8 & 0 & 1 \\ 0 & -34 & -1 & 12 \\ 0 & 0 & 0 & 0 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{-8}{-34}\\z1s2=-8-(-34)\cdot \frac{-8}{-34}=0 \\ z1s3=0-(-1)\cdot \frac{-8}{-34}=\frac{4}{17} \\ z1s4=1-12\cdot \frac{-8}{-34}=-1\frac{14}{17} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 0 & \frac{4}{17} & -1\frac{14}{17} \\ 0 & -34 & -1 & 12 \\ 0 & 0 & 0 & 0 \\ \end{array} \\ \\ \\ L= unendlich \end{array}$