$\begin{array}{l} \\ \begin{array} \text{Gegeben:} \\ \text{Lineares Gleichungssytem} \\ a1 \cdot x_1 + b1\cdot x_2 + c1\cdot x_3 ....=d1 \\ a2\cdot x_1 + b2\cdot x_2 + c2\cdot x_3 .....=d2\\ a3\cdot x_1 + b3\cdot x_2 + c3\cdot x_3....=d3\\ ..... \\ \text{Gesucht: }x_1,x_2,x_3.... \\ \\ \end{array} \\ \textbf{Aufgabe:}\\ n\\ \textbf{Rechnung:}\\ \small \begin{array}{l} 2x_1+4x_2=9 \\ x_1+x_2=3 \\ \\ \end{array} \qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline2 & 4 & 9 \\ 1 & 1 & 3 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{1}{2}\\z2s1=1-2\cdot \frac{1}{2}=0 \\ z2s2=1-4\cdot \frac{1}{2}=-1 \\ z2s3=3-9\cdot \frac{1}{2}=-1\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline2 & 4 & 9 \\ 0 & -1 & -1\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{4}{-1}\\z1s2=4-(-1)\cdot \frac{4}{-1}=0 \\ z1s3=9-(-1\frac{1}{2})\cdot \frac{4}{-1}=3 \\ \end{array}\qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline2 & 0 & 3 \\ 0 & -1 & -1\frac{1}{2} \\ \end{array} \\ \\ x_1=\frac{3}{2}=1\frac{1}{2}\\x_2=\frac{-1\frac{1}{2}}{-1}=1\frac{1}{2}\\L=\{1\frac{1}{2}/1\frac{1}{2}\} \end{array}$