$\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:}\\ h\\ \textbf{Rechnung:}\\ \small \begin{array}{l} 2x_1+4x_2+7x_3=9 \\ 3x_1+3x_2+3x_3=3 \\ x_1+3x_2+3x_3=3 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 4 & 7 & 9 \\ 3 & 3 & 3 & 3 \\ 1 & 3 & 3 & 3 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{3}{2}\\z2s1=3-2\cdot \frac{3}{2}=0 \\ z2s2=3-4\cdot \frac{3}{2}=-3 \\ z2s3=3-7\cdot \frac{3}{2}=-7\frac{1}{2} \\ z2s4=3-9\cdot \frac{3}{2}=-10\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 4 & 7 & 9 \\ 0 & -3 & -7\frac{1}{2} & -10\frac{1}{2} \\ 1 & 3 & 3 & 3 \\ \end{array} \\ \\ \small \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=3-4\cdot \frac{1}{2}=1 \\ z3s3=3-7\cdot \frac{1}{2}=-\frac{1}{2} \\ z3s4=3-9\cdot \frac{1}{2}=-1\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 4 & 7 & 9 \\ 0 & -3 & -7\frac{1}{2} & -10\frac{1}{2} \\ 0 & 1 & -\frac{1}{2} & -1\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{4}{-3}\\z1s2=4-(-3)\cdot \frac{4}{-3}=0 \\ z1s3=7-(-7\frac{1}{2})\cdot \frac{4}{-3}=-3 \\ z1s4=9-(-10\frac{1}{2})\cdot \frac{4}{-3}=-5 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & -3 & -5 \\ 0 & -3 & -7\frac{1}{2} & -10\frac{1}{2} \\ 0 & 1 & -\frac{1}{2} & -1\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{1}{-3}\\z3s2=1-(-3)\cdot \frac{1}{-3}=0 \\ z3s3=-\frac{1}{2}-(-7\frac{1}{2})\cdot \frac{1}{-3}=-3 \\ z3s4=-1\frac{1}{2}-(-10\frac{1}{2})\cdot \frac{1}{-3}=-5 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & -3 & -5 \\ 0 & -3 & -7\frac{1}{2} & -10\frac{1}{2} \\ 0 & 0 & -3 & -5 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-3}{-3}\\z1s3=-3-(-3)\cdot \frac{-3}{-3}=0 \\ z1s4=-5-(-5)\cdot \frac{-3}{-3}=0 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & 0 & 0 \\ 0 & -3 & -7\frac{1}{2} & -10\frac{1}{2} \\ 0 & 0 & -3 & -5 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-7\frac{1}{2}}{-3}\\z2s3=-7\frac{1}{2}-(-3)\cdot \frac{-7\frac{1}{2}}{-3}=0 \\ z2s4=-10\frac{1}{2}-(-5)\cdot \frac{-7\frac{1}{2}}{-3}=2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & 0 & 0 \\ 0 & -3 & 0 & 2 \\ 0 & 0 & -3 & -5 \\ \end{array} \\ \\ x_1=\frac{0}{2}=0\\x_2=\frac{2}{-3}=-\frac{2}{3}\\x_3=\frac{-5}{-3}=1\frac{2}{3}\\L=\{0/-\frac{2}{3}/1\frac{2}{3}\} \end{array}$