Beispiel Nr: 15
$ \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=9\\ 3 x +8 y + 9 z=5\\ 2 x +3 y + 6 z=7\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} x-2y+3z=9 \\ 3x+8y+9z=5 \\ 2x+3y+6z=7 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & -2 & 3 & 9 \\ 3 & 8 & 9 & 5 \\ 2 & 3 & 6 & 7 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\cdot 1\text{-Zeile}1\cdot 3 \\z2s1=3\cdot 1-1\cdot 3 =0 \\ z2s2=8\cdot 1-(-2)\cdot 3 =14 \\ z2s3=9\cdot 1-3\cdot 3 =0 \\ z2s4=5\cdot 1-9\cdot 3 =-22 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & -2 & 3 & 9 \\ 0 & 14 & 0 & -22 \\ 2 & 3 & 6 & 7 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 1\text{-Zeile}1\cdot 2 \\z3s1=2\cdot 1-1\cdot 2 =0 \\ z3s2=3\cdot 1-(-2)\cdot 2 =7 \\ z3s3=6\cdot 1-3\cdot 2 =0 \\ z3s4=7\cdot 1-9\cdot 2 =-11 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & -2 & 3 & 9 \\ 0 & 14 & 0 & -22 \\ 0 & 7 & 0 & -11 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 14\text{-Zeile}2\cdot 7 \\z3s2=7\cdot 14-14\cdot 7 =0 \\ z3s3=0\cdot 14-0\cdot 7 =0 \\ z3s4=(-11)\cdot 14-(-22)\cdot 7 =0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline1 & -2 & 3 & 9 \\ 0 & 14 & 0 & -22 \\ 0 & 0 & 0 & 0 \\ \end{array} \\ \\ \\ L= unendlich$