$\begin{array}{l} \\ \begin{array} \text{Gegeben:} \\ \text{Lineares Gleichungssytem} \\ a1 \cdot x_1 + b1\cdot x_2 + c1\cdot x_3 ....=d1 \\ a2\cdot x_1 + b2\cdot x_2 + c2\cdot x_3 .....=d2\\ a3\cdot x_1 + b3\cdot x_2 + c3\cdot x_3....=d3\\ ..... \\ \text{Gesucht: }x_1,x_2,x_3.... \\ \\ \end{array} \\ \textbf{Aufgabe:}\\ b\\ \textbf{Rechnung:}\\ \small \begin{array}{l} x_1+2x_2+3x_3=4 \\ 5x_1+6x_2+7x_3=8 \\ 9x_1+10x_2+11x_3=12 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{5}{1}\\z2s1=5-1\cdot \frac{5}{1}=0 \\ z2s2=6-2\cdot \frac{5}{1}=-4 \\ z2s3=7-3\cdot \frac{5}{1}=-8 \\ z2s4=8-4\cdot \frac{5}{1}=-12 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline1 & 2 & 3 & 4 \\ 0 & -4 & -8 & -12 \\ 9 & 10 & 11 & 12 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{9}{1}\\z3s1=9-1\cdot \frac{9}{1}=0 \\ z3s2=10-2\cdot \frac{9}{1}=-8 \\ z3s3=11-3\cdot \frac{9}{1}=-16 \\ z3s4=12-4\cdot \frac{9}{1}=-24 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline1 & 2 & 3 & 4 \\ 0 & -4 & -8 & -12 \\ 0 & -8 & -16 & -24 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{2}{-4}\\z1s2=2-(-4)\cdot \frac{2}{-4}=0 \\ z1s3=3-(-8)\cdot \frac{2}{-4}=-1 \\ z1s4=4-(-12)\cdot \frac{2}{-4}=-2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline1 & 0 & -1 & -2 \\ 0 & -4 & -8 & -12 \\ 0 & -8 & -16 & -24 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{-8}{-4}\\z3s2=-8-(-4)\cdot \frac{-8}{-4}=0 \\ z3s3=-16-(-8)\cdot \frac{-8}{-4}=0 \\ z3s4=-24-(-12)\cdot \frac{-8}{-4}=0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline1 & 0 & -1 & -2 \\ 0 & -4 & -8 & -12 \\ 0 & 0 & 0 & 0 \\ \end{array} \\ \\ \\ L= unendlich \end{array}$