$\begin{array}{l} \\ \begin{array} \text{Gegeben:} \\ \text{Lineares Gleichungssytem} \\ a1 \cdot x_1 + b1\cdot x_2 + c1\cdot x_3 ....=d1 \\ a2\cdot x_1 + b2\cdot x_2 + c2\cdot x_3 .....=d2\\ a3\cdot x_1 + b3\cdot x_2 + c3\cdot x_3....=d3\\ ..... \\ \text{Gesucht: }x_1,x_2,x_3.... \\ \\ \end{array} \\ \textbf{Aufgabe:}\\ r\\ \textbf{Rechnung:}\\ \small \begin{array}{l} 4x_1+2x_2=22 \\ 5x_1+x_2=17 \\ \\ \end{array} \qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline4 & 2 & 22 \\ 5 & 1 & 17 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{5}{4}\\z2s1=5-4\cdot \frac{5}{4}=0 \\ z2s2=1-2\cdot \frac{5}{4}=-1\frac{1}{2} \\ z2s3=17-22\cdot \frac{5}{4}=-10\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline4 & 2 & 22 \\ 0 & -1\frac{1}{2} & -10\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{2}{-1\frac{1}{2}}\\z1s2=2-(-1\frac{1}{2})\cdot \frac{2}{-1\frac{1}{2}}=0 \\ z1s3=22-(-10\frac{1}{2})\cdot \frac{2}{-1\frac{1}{2}}=8 \\ \end{array}\qquad \small \begin{array}{cc|cc } x_1 & x_2 & & \\ \hline4 & 0 & 8 \\ 0 & -1\frac{1}{2} & -10\frac{1}{2} \\ \end{array} \\ \\ x_1=\frac{8}{4}=2\\x_2=\frac{-10\frac{1}{2}}{-1\frac{1}{2}}=7\\L=\{2/7\} \end{array}$