Beispiel Nr: 13
$ \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:} \\ 6 x +4 + 9 z=32\\ 5 x +7 y + 10 z=17\\ 4 x +8 y + 5 z=100\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 6x+4y+9z=32 \\ 5x+7y+10z=17 \\ 4x+8y+5z=100 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 5 & 7 & 10 & 17 \\ 4 & 8 & 5 & 100 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{5}{6}\\z2s1=5-6\cdot \frac{5}{6}=0 \\ z2s2=7-4\cdot \frac{5}{6}=3\frac{2}{3} \\ z2s3=10-9\cdot \frac{5}{6}=2\frac{1}{2} \\ z2s4=17-32\cdot \frac{5}{6}=-9\frac{2}{3} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 0 & 3\frac{2}{3} & 2\frac{1}{2} & -9\frac{2}{3} \\ 4 & 8 & 5 & 100 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{4}{6}\\z3s1=4-6\cdot \frac{4}{6}=0 \\ z3s2=8-4\cdot \frac{4}{6}=5\frac{1}{3} \\ z3s3=5-9\cdot \frac{4}{6}=-1 \\ z3s4=100-32\cdot \frac{4}{6}=78\frac{2}{3} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 0 & 3\frac{2}{3} & 2\frac{1}{2} & -9\frac{2}{3} \\ 0 & 5\frac{1}{3} & -1 & 78\frac{2}{3} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{4}{3\frac{2}{3}}\\z1s2=4-3\frac{2}{3}\cdot \frac{4}{3\frac{2}{3}}=0 \\ z1s3=9-2\frac{1}{2}\cdot \frac{4}{3\frac{2}{3}}=6\frac{3}{11} \\ z1s4=32-(-9\frac{2}{3})\cdot \frac{4}{3\frac{2}{3}}=42\frac{6}{11} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 0 & 6\frac{3}{11} & 42\frac{6}{11} \\ 0 & 3\frac{2}{3} & 2\frac{1}{2} & -9\frac{2}{3} \\ 0 & 5\frac{1}{3} & -1 & 78\frac{2}{3} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{5\frac{1}{3}}{3\frac{2}{3}}\\z3s2=5\frac{1}{3}-3\frac{2}{3}\cdot \frac{5\frac{1}{3}}{3\frac{2}{3}}=0 \\ z3s3=-1-2\frac{1}{2}\cdot \frac{5\frac{1}{3}}{3\frac{2}{3}}=-4\frac{7}{11} \\ z3s4=78\frac{2}{3}-(-9\frac{2}{3})\cdot \frac{5\frac{1}{3}}{3\frac{2}{3}}=92\frac{8}{11} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 0 & 6\frac{3}{11} & 42\frac{6}{11} \\ 0 & 3\frac{2}{3} & 2\frac{1}{2} & -9\frac{2}{3} \\ 0 & 0 & -4\frac{7}{11} & 92\frac{8}{11} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{6\frac{3}{11}}{-4\frac{7}{11}}\\z1s3=6\frac{3}{11}-(-4\frac{7}{11})\cdot \frac{6\frac{3}{11}}{-4\frac{7}{11}}=0 \\ z1s4=42\frac{6}{11}-92\frac{8}{11}\cdot \frac{6\frac{3}{11}}{-4\frac{7}{11}}=168 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 0 & 0 & 168 \\ 0 & 3\frac{2}{3} & 2\frac{1}{2} & -9\frac{2}{3} \\ 0 & 0 & -4\frac{7}{11} & 92\frac{8}{11} \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{2\frac{1}{2}}{-4\frac{7}{11}}\\z2s3=2\frac{1}{2}-(-4\frac{7}{11})\cdot \frac{2\frac{1}{2}}{-4\frac{7}{11}}=0 \\ z2s4=-9\frac{2}{3}-92\frac{8}{11}\cdot \frac{2\frac{1}{2}}{-4\frac{7}{11}}=40\frac{1}{3} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 0 & 0 & 168 \\ 0 & 3\frac{2}{3} & 0 & 40\frac{1}{3} \\ 0 & 0 & -4\frac{7}{11} & 92\frac{8}{11} \\ \end{array} \\ \\ x=\frac{168}{6}=28\\y=\frac{40\frac{1}{3}}{3\frac{2}{3}}=11\\z=\frac{92\frac{8}{11}}{-4\frac{7}{11}}=-20\\L=\{28/11/-20\}$