$\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:}\\ e\\ \textbf{Rechnung:}\\ \small \begin{array}{l} 3x_1+x_2 -x_3=0 \\ 15x_1+2x_2+4x_3=1 \\ 2x_1-2x_2+x_3=1 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 1 & -1 & 0 \\ 15 & 2 & 4 & 1 \\ 2 & -2 & 1 & 1 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{15}{3}\\z2s1=15-3\cdot \frac{15}{3}=0 \\ z2s2=2-1\cdot \frac{15}{3}=-3 \\ z2s3=4-(-1)\cdot \frac{15}{3}=9 \\ z2s4=1-0\cdot \frac{15}{3}=1 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 1 & -1 & 0 \\ 0 & -3 & 9 & 1 \\ 2 & -2 & 1 & 1 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{2}{3}\\z3s1=2-3\cdot \frac{2}{3}=0 \\ z3s2=-2-1\cdot \frac{2}{3}=-2\frac{2}{3} \\ z3s3=1-(-1)\cdot \frac{2}{3}=1\frac{2}{3} \\ z3s4=1-0\cdot \frac{2}{3}=1 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 1 & -1 & 0 \\ 0 & -3 & 9 & 1 \\ 0 & -2\frac{2}{3} & 1\frac{2}{3} & 1 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{1}{-3}\\z1s2=1-(-3)\cdot \frac{1}{-3}=0 \\ z1s3=-1-9\cdot \frac{1}{-3}=2 \\ z1s4=0-1\cdot \frac{1}{-3}=\frac{1}{3} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 0 & 2 & \frac{1}{3} \\ 0 & -3 & 9 & 1 \\ 0 & -2\frac{2}{3} & 1\frac{2}{3} & 1 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{-2\frac{2}{3}}{-3}\\z3s2=-2\frac{2}{3}-(-3)\cdot \frac{-2\frac{2}{3}}{-3}=0 \\ z3s3=1\frac{2}{3}-9\cdot \frac{-2\frac{2}{3}}{-3}=-6\frac{1}{3} \\ z3s4=1-1\cdot \frac{-2\frac{2}{3}}{-3}=\frac{1}{9} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 0 & 2 & \frac{1}{3} \\ 0 & -3 & 9 & 1 \\ 0 & 0 & -6\frac{1}{3} & \frac{1}{9} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{2}{-6\frac{1}{3}}\\z1s3=2-(-6\frac{1}{3})\cdot \frac{2}{-6\frac{1}{3}}=0 \\ z1s4=\frac{1}{3}-\frac{1}{9}\cdot \frac{2}{-6\frac{1}{3}}=\frac{7}{19} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 0 & 0 & \frac{7}{19} \\ 0 & -3 & 9 & 1 \\ 0 & 0 & -6\frac{1}{3} & \frac{1}{9} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{9}{-6\frac{1}{3}}\\z2s3=9-(-6\frac{1}{3})\cdot \frac{9}{-6\frac{1}{3}}=0 \\ z2s4=1-\frac{1}{9}\cdot \frac{9}{-6\frac{1}{3}}=1\frac{3}{19} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline3 & 0 & 0 & \frac{7}{19} \\ 0 & -3 & 0 & 1\frac{3}{19} \\ 0 & 0 & -6\frac{1}{3} & \frac{1}{9} \\ \end{array} \\ \\ x_1=\frac{\frac{7}{19}}{3}=\frac{7}{57}\\x_2=\frac{1\frac{3}{19}}{-3}=-\frac{22}{57}\\x_3=\frac{\frac{1}{9}}{-6\frac{1}{3}}=-\frac{1}{57}\\L=\{\frac{7}{57}/-\frac{22}{57}/-\frac{1}{57}\} \end{array}$