Beispiel Nr: 01
$ \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:} \\ 11 x +13 + 4 z=37\\ 12 x +14 y + 5 z=40\\ 9 x +3 y + 3 z=15\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 11x+13y+4z=37 \\ 12x+14y+5z=40 \\ 9x+3y+3z=15 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 13 & 4 & 37 \\ 12 & 14 & 5 & 40 \\ 9 & 3 & 3 & 15 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{12}{11}\\z2s1=12-11\cdot \frac{12}{11}=0 \\ z2s2=14-13\cdot \frac{12}{11}=-\frac{2}{11} \\ z2s3=5-4\cdot \frac{12}{11}=\frac{7}{11} \\ z2s4=40-37\cdot \frac{12}{11}=-\frac{4}{11} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 13 & 4 & 37 \\ 0 & -\frac{2}{11} & \frac{7}{11} & -\frac{4}{11} \\ 9 & 3 & 3 & 15 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{9}{11}\\z3s1=9-11\cdot \frac{9}{11}=0 \\ z3s2=3-13\cdot \frac{9}{11}=-7\frac{7}{11} \\ z3s3=3-4\cdot \frac{9}{11}=-\frac{3}{11} \\ z3s4=15-37\cdot \frac{9}{11}=-15\frac{3}{11} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 13 & 4 & 37 \\ 0 & -\frac{2}{11} & \frac{7}{11} & -\frac{4}{11} \\ 0 & -7\frac{7}{11} & -\frac{3}{11} & -15\frac{3}{11} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{13}{-\frac{2}{11}}\\z1s2=13-(-\frac{2}{11})\cdot \frac{13}{-\frac{2}{11}}=0 \\ z1s3=4-\frac{7}{11}\cdot \frac{13}{-\frac{2}{11}}=49\frac{1}{2} \\ z1s4=37-(-\frac{4}{11})\cdot \frac{13}{-\frac{2}{11}}=11 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 0 & 49\frac{1}{2} & 11 \\ 0 & -\frac{2}{11} & \frac{7}{11} & -\frac{4}{11} \\ 0 & -7\frac{7}{11} & -\frac{3}{11} & -15\frac{3}{11} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{-7\frac{7}{11}}{-\frac{2}{11}}\\z3s2=-7\frac{7}{11}-(-\frac{2}{11})\cdot \frac{-7\frac{7}{11}}{-\frac{2}{11}}=0 \\ z3s3=-\frac{3}{11}-\frac{7}{11}\cdot \frac{-7\frac{7}{11}}{-\frac{2}{11}}=-27 \\ z3s4=-15\frac{3}{11}-(-\frac{4}{11})\cdot \frac{-7\frac{7}{11}}{-\frac{2}{11}}=0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 0 & 49\frac{1}{2} & 11 \\ 0 & -\frac{2}{11} & \frac{7}{11} & -\frac{4}{11} \\ 0 & 0 & -27 & 0 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{49\frac{1}{2}}{-27}\\z1s3=49\frac{1}{2}-(-27)\cdot \frac{49\frac{1}{2}}{-27}=0 \\ z1s4=11-0\cdot \frac{49\frac{1}{2}}{-27}=11 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 0 & 0 & 11 \\ 0 & -\frac{2}{11} & \frac{7}{11} & -\frac{4}{11} \\ 0 & 0 & -27 & 0 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{\frac{7}{11}}{-27}\\z2s3=\frac{7}{11}-(-27)\cdot \frac{\frac{7}{11}}{-27}=0 \\ z2s4=-\frac{4}{11}-0\cdot \frac{\frac{7}{11}}{-27}=-\frac{4}{11} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline11 & 0 & 0 & 11 \\ 0 & -\frac{2}{11} & 0 & -\frac{4}{11} \\ 0 & 0 & -27 & 0 \\ \end{array} \\ \\ x=\frac{11}{11}=1\\y=\frac{-\frac{4}{11}}{-\frac{2}{11}}=2\\z=\frac{0}{-27}=0\\L=\{1/2/0\}$