$\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:}\\ z \\ \textbf{Rechnung:}\\ \small \begin{array}{l} 2x_1+x_2+8x_3=2 \\ 8x_1 -x_2+x_3=1 \\ 3x_1-6x_2+2x_3=10 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 1 & 8 & 2 \\ 8 & -1 & 1 & 1 \\ 3 & -6 & 2 & 10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{8}{2}\\z2s1=8-2\cdot \frac{8}{2}=0 \\ z2s2=-1-1\cdot \frac{8}{2}=-5 \\ z2s3=1-8\cdot \frac{8}{2}=-31 \\ z2s4=1-2\cdot \frac{8}{2}=-7 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 1 & 8 & 2 \\ 0 & -5 & -31 & -7 \\ 3 & -6 & 2 & 10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{3}{2}\\z3s1=3-2\cdot \frac{3}{2}=0 \\ z3s2=-6-1\cdot \frac{3}{2}=-7\frac{1}{2} \\ z3s3=2-8\cdot \frac{3}{2}=-10 \\ z3s4=10-2\cdot \frac{3}{2}=7 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 1 & 8 & 2 \\ 0 & -5 & -31 & -7 \\ 0 & -7\frac{1}{2} & -10 & 7 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{1}{-5}\\z1s2=1-(-5)\cdot \frac{1}{-5}=0 \\ z1s3=8-(-31)\cdot \frac{1}{-5}=1\frac{4}{5} \\ z1s4=2-(-7)\cdot \frac{1}{-5}=\frac{3}{5} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & 1\frac{4}{5} & \frac{3}{5} \\ 0 & -5 & -31 & -7 \\ 0 & -7\frac{1}{2} & -10 & 7 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{-7\frac{1}{2}}{-5}\\z3s2=-7\frac{1}{2}-(-5)\cdot \frac{-7\frac{1}{2}}{-5}=0 \\ z3s3=-10-(-31)\cdot \frac{-7\frac{1}{2}}{-5}=36\frac{1}{2} \\ z3s4=7-(-7)\cdot \frac{-7\frac{1}{2}}{-5}=17\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & 1\frac{4}{5} & \frac{3}{5} \\ 0 & -5 & -31 & -7 \\ 0 & 0 & 36\frac{1}{2} & 17\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{1\frac{4}{5}}{36\frac{1}{2}}\\z1s3=1\frac{4}{5}-36\frac{1}{2}\cdot \frac{1\frac{4}{5}}{36\frac{1}{2}}=0 \\ z1s4=\frac{3}{5}-17\frac{1}{2}\cdot \frac{1\frac{4}{5}}{36\frac{1}{2}}=-0,263 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & 0 & -0,263 \\ 0 & -5 & -31 & -7 \\ 0 & 0 & 36\frac{1}{2} & 17\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-31}{36\frac{1}{2}}\\z2s3=-31-36\frac{1}{2}\cdot \frac{-31}{36\frac{1}{2}}=0 \\ z2s4=-7-17\frac{1}{2}\cdot \frac{-31}{36\frac{1}{2}}=7\frac{63}{73} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline2 & 0 & 0 & -0,263 \\ 0 & -5 & 0 & 7\frac{63}{73} \\ 0 & 0 & 36\frac{1}{2} & 17\frac{1}{2} \\ \end{array} \\ \\ x_1=\frac{-0,263}{2}=-0,132\\x_2=\frac{7\frac{63}{73}}{-5}=-1,57\\x_3=\frac{17\frac{1}{2}}{36\frac{1}{2}}=\frac{35}{73}\\L=\{-0,132/-1,57/\frac{35}{73}\} \end{array}$