Beispiel Nr: 05
$ \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 +2 + 3 z=4\\ 2 x +3 y + 2 z=6\\ 0 x +2 y + 6 z=0\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} x+2y+3z=4 \\ 2x+3y+2z=6 \\ 2y+6z=0 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 2 & 3 & 4 \\ 2 & 3 & 2 & 6 \\ 0 & 2 & 6 & 0 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{2}{1}\\z2s1=2-1\cdot \frac{2}{1}=0 \\ z2s2=3-2\cdot \frac{2}{1}=-1 \\ z2s3=2-3\cdot \frac{2}{1}=-4 \\ z2s4=6-4\cdot \frac{2}{1}=-2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 2 & 3 & 4 \\ 0 & -1 & -4 & -2 \\ 0 & 2 & 6 & 0 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{2}{-1}\\z1s2=2-(-1)\cdot \frac{2}{-1}=0 \\ z1s3=3-(-4)\cdot \frac{2}{-1}=-5 \\ z1s4=4-(-2)\cdot \frac{2}{-1}=0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & -5 & 0 \\ 0 & -1 & -4 & -2 \\ 0 & 2 & 6 & 0 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{2}{-1}\\z3s2=2-(-1)\cdot \frac{2}{-1}=0 \\ z3s3=6-(-4)\cdot \frac{2}{-1}=-2 \\ z3s4=0-(-2)\cdot \frac{2}{-1}=-4 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & -5 & 0 \\ 0 & -1 & -4 & -2 \\ 0 & 0 & -2 & -4 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-5}{-2}\\z1s3=-5-(-2)\cdot \frac{-5}{-2}=0 \\ z1s4=0-(-4)\cdot \frac{-5}{-2}=10 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 0 & 10 \\ 0 & -1 & -4 & -2 \\ 0 & 0 & -2 & -4 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-4}{-2}\\z2s3=-4-(-2)\cdot \frac{-4}{-2}=0 \\ z2s4=-2-(-4)\cdot \frac{-4}{-2}=6 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 0 & 10 \\ 0 & -1 & 0 & 6 \\ 0 & 0 & -2 & -4 \\ \end{array} \\ \\ x=\frac{10}{1}=10\\y=\frac{6}{-1}=-6\\z=\frac{-4}{-2}=2\\L=\{10/-6/2\}$