$\begin{array}{l} \\ \begin{array} \text{Gegeben:} \\ \text{Lineares Gleichungssytem} \\ a1 \cdot x_1 + b1\cdot x_2 + c1\cdot x_3 ....=d1 \\ a2\cdot x_1 + b2\cdot x_2 + c2\cdot x_3 .....=d2\\ a3\cdot x_1 + b3\cdot x_2 + c3\cdot x_3....=d3\\ ..... \\ \text{Gesucht: }x_1,x_2,x_3.... \\ \\ \end{array} \\ \textbf{Aufgabe:}\\ d\\ \textbf{Rechnung:}\\ \small \begin{array}{l} 3x_1+2x_2=1 \\ 2x_1-3x_2=5 \\ \\ \end{array} \qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline3 & 2 & 1 \\ 2 & -3 & 5 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{2}{3}\\z2s1=2-3\cdot \frac{2}{3}=0 \\ z2s2=-3-2\cdot \frac{2}{3}=-4\frac{1}{3} \\ z2s3=5-1\cdot \frac{2}{3}=4\frac{1}{3} \\ \end{array}\qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline3 & 2 & 1 \\ 0 & -4\frac{1}{3} & 4\frac{1}{3} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{2}{-4\frac{1}{3}}\\z1s2=2-(-4\frac{1}{3})\cdot \frac{2}{-4\frac{1}{3}}=0 \\ z1s3=1-4\frac{1}{3}\cdot \frac{2}{-4\frac{1}{3}}=3 \\ \end{array}\qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline3 & 0 & 3 \\ 0 & -4\frac{1}{3} & 4\frac{1}{3} \\ \end{array} \\ \\ x_1=\frac{3}{3}=1\\x_2=\frac{4\frac{1}{3}}{-4\frac{1}{3}}=-1\\L=\{1/-1\} \end{array}$