Beispiel Nr: 04
$ \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 +3 + -4 z=16\\ 4 x +9 y + -1 z=58\\ 1 x +6 y + 2 z=34\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 2x+3y-4z=16 \\ 4x+9y -z=58 \\ x+6y+2z=34 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 4 & 9 & -1 & 58 \\ 1 & 6 & 2 & 34 \\ \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=9-3\cdot \frac{4}{2}=3 \\ z2s3=-1-(-4)\cdot \frac{4}{2}=7 \\ z2s4=58-16\cdot \frac{4}{2}=26 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 0 & 3 & 7 & 26 \\ 1 & 6 & 2 & 34 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{1}{2}\\z3s1=1-2\cdot \frac{1}{2}=0 \\ z3s2=6-3\cdot \frac{1}{2}=4\frac{1}{2} \\ z3s3=2-(-4)\cdot \frac{1}{2}=4 \\ z3s4=34-16\cdot \frac{1}{2}=26 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 0 & 3 & 7 & 26 \\ 0 & 4\frac{1}{2} & 4 & 26 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{3}{3}\\z1s2=3-3\cdot \frac{3}{3}=0 \\ z1s3=-4-7\cdot \frac{3}{3}=-11 \\ z1s4=16-26\cdot \frac{3}{3}=-10 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & -11 & -10 \\ 0 & 3 & 7 & 26 \\ 0 & 4\frac{1}{2} & 4 & 26 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{4\frac{1}{2}}{3}\\z3s2=4\frac{1}{2}-3\cdot \frac{4\frac{1}{2}}{3}=0 \\ z3s3=4-7\cdot \frac{4\frac{1}{2}}{3}=-6\frac{1}{2} \\ z3s4=26-26\cdot \frac{4\frac{1}{2}}{3}=-13 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & -11 & -10 \\ 0 & 3 & 7 & 26 \\ 0 & 0 & -6\frac{1}{2} & -13 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-11}{-6\frac{1}{2}}\\z1s3=-11-(-6\frac{1}{2})\cdot \frac{-11}{-6\frac{1}{2}}=0 \\ z1s4=-10-(-13)\cdot \frac{-11}{-6\frac{1}{2}}=12 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & 0 & 12 \\ 0 & 3 & 7 & 26 \\ 0 & 0 & -6\frac{1}{2} & -13 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{7}{-6\frac{1}{2}}\\z2s3=7-(-6\frac{1}{2})\cdot \frac{7}{-6\frac{1}{2}}=0 \\ z2s4=26-(-13)\cdot \frac{7}{-6\frac{1}{2}}=12 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & 0 & 12 \\ 0 & 3 & 0 & 12 \\ 0 & 0 & -6\frac{1}{2} & -13 \\ \end{array} \\ \\ x=\frac{12}{2}=6\\y=\frac{12}{3}=4\\z=\frac{-13}{-6\frac{1}{2}}=2\\L=\{6/4/2\}$