Beispiel Nr: 16
$ \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 +3 + -2 z=3\\ 3 x +2 y + 1 z=2\\ 0 x +1 y + 3 z=5\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} x+3y-2z=3 \\ 3x+2y+z=2 \\ y+3z=5 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 3 & -2 & 3 \\ 3 & 2 & 1 & 2 \\ 0 & 1 & 3 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{3}{1}\\z2s1=3-1\cdot \frac{3}{1}=0 \\ z2s2=2-3\cdot \frac{3}{1}=-7 \\ z2s3=1-(-2)\cdot \frac{3}{1}=7 \\ z2s4=2-3\cdot \frac{3}{1}=-7 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 3 & -2 & 3 \\ 0 & -7 & 7 & -7 \\ 0 & 1 & 3 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{3}{-7}\\z1s2=3-(-7)\cdot \frac{3}{-7}=0 \\ z1s3=-2-7\cdot \frac{3}{-7}=1 \\ z1s4=3-(-7)\cdot \frac{3}{-7}=0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 1 & 0 \\ 0 & -7 & 7 & -7 \\ 0 & 1 & 3 & 5 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{1}{-7}\\z3s2=1-(-7)\cdot \frac{1}{-7}=0 \\ z3s3=3-7\cdot \frac{1}{-7}=4 \\ z3s4=5-(-7)\cdot \frac{1}{-7}=4 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 1 & 0 \\ 0 & -7 & 7 & -7 \\ 0 & 0 & 4 & 4 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{1}{4}\\z1s3=1-4\cdot \frac{1}{4}=0 \\ z1s4=0-4\cdot \frac{1}{4}=-1 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 0 & -1 \\ 0 & -7 & 7 & -7 \\ 0 & 0 & 4 & 4 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{7}{4}\\z2s3=7-4\cdot \frac{7}{4}=0 \\ z2s4=-7-4\cdot \frac{7}{4}=-14 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & 0 & 0 & -1 \\ 0 & -7 & 0 & -14 \\ 0 & 0 & 4 & 4 \\ \end{array} \\ \\ x=\frac{-1}{1}=-1\\y=\frac{-14}{-7}=2\\z=\frac{4}{4}=1\\L=\{-1/2/1\}$