$\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 +2 + 4 z=0\\ 4 x -\frac{1}{2} y + 2 z=5\\ 4 x -2 y + -1 z=8\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} -2x+2y+4z=0 \\ 4x-\frac{1}{2}y+2z=5 \\ 4x-2y -z=8 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 2 & 4 & 0 \\ 4 & -\frac{1}{2} & 2 & 5 \\ 4 & -2 & -1 & 8 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{4}{-2}\\z2s1=4-(-2)\cdot \frac{4}{-2}=0 \\ z2s2=-\frac{1}{2}-2\cdot \frac{4}{-2}=3\frac{1}{2} \\ z2s3=2-4\cdot \frac{4}{-2}=10 \\ z2s4=5-0\cdot \frac{4}{-2}=5 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 2 & 4 & 0 \\ 0 & 3\frac{1}{2} & 10 & 5 \\ 4 & -2 & -1 & 8 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{4}{-2}\\z3s1=4-(-2)\cdot \frac{4}{-2}=0 \\ z3s2=-2-2\cdot \frac{4}{-2}=2 \\ z3s3=-1-4\cdot \frac{4}{-2}=7 \\ z3s4=8-0\cdot \frac{4}{-2}=8 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 2 & 4 & 0 \\ 0 & 3\frac{1}{2} & 10 & 5 \\ 0 & 2 & 7 & 8 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{2}{3\frac{1}{2}}\\z1s2=2-3\frac{1}{2}\cdot \frac{2}{3\frac{1}{2}}=0 \\ z1s3=4-10\cdot \frac{2}{3\frac{1}{2}}=-1\frac{5}{7} \\ z1s4=0-5\cdot \frac{2}{3\frac{1}{2}}=-2\frac{6}{7} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 0 & -1\frac{5}{7} & -2\frac{6}{7} \\ 0 & 3\frac{1}{2} & 10 & 5 \\ 0 & 2 & 7 & 8 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{2}{3\frac{1}{2}}\\z3s2=2-3\frac{1}{2}\cdot \frac{2}{3\frac{1}{2}}=0 \\ z3s3=7-10\cdot \frac{2}{3\frac{1}{2}}=1\frac{2}{7} \\ z3s4=8-5\cdot \frac{2}{3\frac{1}{2}}=5\frac{1}{7} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 0 & -1\frac{5}{7} & -2\frac{6}{7} \\ 0 & 3\frac{1}{2} & 10 & 5 \\ 0 & 0 & 1\frac{2}{7} & 5\frac{1}{7} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-1\frac{5}{7}}{1\frac{2}{7}}\\z1s3=-1\frac{5}{7}-1\frac{2}{7}\cdot \frac{-1\frac{5}{7}}{1\frac{2}{7}}=0 \\ z1s4=-2\frac{6}{7}-5\frac{1}{7}\cdot \frac{-1\frac{5}{7}}{1\frac{2}{7}}=4 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 0 & 0 & 4 \\ 0 & 3\frac{1}{2} & 10 & 5 \\ 0 & 0 & 1\frac{2}{7} & 5\frac{1}{7} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{10}{1\frac{2}{7}}\\z2s3=10-1\frac{2}{7}\cdot \frac{10}{1\frac{2}{7}}=0 \\ z2s4=5-5\frac{1}{7}\cdot \frac{10}{1\frac{2}{7}}=-35 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline-2 & 0 & 0 & 4 \\ 0 & 3\frac{1}{2} & 0 & -35 \\ 0 & 0 & 1\frac{2}{7} & 5\frac{1}{7} \\ \end{array} \\ \\ x=\frac{4}{-2}=-2\\y=\frac{-35}{3\frac{1}{2}}=-10\\z=\frac{5\frac{1}{7}}{1\frac{2}{7}}=4\\L=\{-2/-10/4\} \end{array}$