$\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:}\\ u\\ \textbf{Rechnung:}\\ \small \begin{array}{l} 4x_1+2x_2+x_3=14 \\ 6x_1+x_2+x_3=8 \\ 8x_1+4x_2+x_3=18 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline4 & 2 & 1 & 14 \\ 6 & 1 & 1 & 8 \\ 8 & 4 & 1 & 18 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{6}{4}\\z2s1=6-4\cdot \frac{6}{4}=0 \\ z2s2=1-2\cdot \frac{6}{4}=-2 \\ z2s3=1-1\cdot \frac{6}{4}=-\frac{1}{2} \\ z2s4=8-14\cdot \frac{6}{4}=-13 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline4 & 2 & 1 & 14 \\ 0 & -2 & -\frac{1}{2} & -13 \\ 8 & 4 & 1 & 18 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{8}{4}\\z3s1=8-4\cdot \frac{8}{4}=0 \\ z3s2=4-2\cdot \frac{8}{4}=0 \\ z3s3=1-1\cdot \frac{8}{4}=-1 \\ z3s4=18-14\cdot \frac{8}{4}=-10 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline4 & 2 & 1 & 14 \\ 0 & -2 & -\frac{1}{2} & -13 \\ 0 & 0 & -1 & -10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{2}{-2}\\z1s2=2-(-2)\cdot \frac{2}{-2}=0 \\ z1s3=1-(-\frac{1}{2})\cdot \frac{2}{-2}=\frac{1}{2} \\ z1s4=14-(-13)\cdot \frac{2}{-2}=1 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline4 & 0 & \frac{1}{2} & 1 \\ 0 & -2 & -\frac{1}{2} & -13 \\ 0 & 0 & -1 & -10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{\frac{1}{2}}{-1}\\z1s3=\frac{1}{2}-(-1)\cdot \frac{\frac{1}{2}}{-1}=0 \\ z1s4=1-(-10)\cdot \frac{\frac{1}{2}}{-1}=-4 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline4 & 0 & 0 & -4 \\ 0 & -2 & -\frac{1}{2} & -13 \\ 0 & 0 & -1 & -10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-\frac{1}{2}}{-1}\\z2s3=-\frac{1}{2}-(-1)\cdot \frac{-\frac{1}{2}}{-1}=0 \\ z2s4=-13-(-10)\cdot \frac{-\frac{1}{2}}{-1}=-8 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x_1 & x_2 & x_3 & & \\ \hline4 & 0 & 0 & -4 \\ 0 & -2 & 0 & -8 \\ 0 & 0 & -1 & -10 \\ \end{array} \\ \\ x_1=\frac{-4}{4}=-1\\x_2=\frac{-8}{-2}=4\\x_3=\frac{-10}{-1}=10\\L=\{-1/4/10\} \end{array}$