$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ 2 x +3 + 4 z=175\\ 4 x +6 y + 5 z=287\\ 3 x +2 y + 8 z=257\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 2x+3y+4z=175 \\ 4x+6y+5z=287 \\ 3x+2y+8z=257 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & 4 & 175 \\ 4 & 6 & 5 & 287 \\ 3 & 2 & 8 & 257 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{4}{2}\\z2s1=4-2\cdot \frac{4}{2}=0 \\ z2s2=6-3\cdot \frac{4}{2}=0 \\ z2s3=5-4\cdot \frac{4}{2}=-3 \\ z2s4=287-175\cdot \frac{4}{2}=-63 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & 4 & 175 \\ 0 & 0 & -3 & -63 \\ 3 & 2 & 8 & 257 \\ \end{array} \\ \\ \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=2-3\cdot \frac{3}{2}=-2\frac{1}{2} \\ z3s3=8-4\cdot \frac{3}{2}=2 \\ z3s4=257-175\cdot \frac{3}{2}=-5\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & 4 & 175 \\ 0 & 0 & -3 & -63 \\ 0 & -2\frac{1}{2} & 2 & -5\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeilen vertauschen } \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & 4 & 175 \\ 0 & -2\frac{1}{2} & 2 & -5\frac{1}{2} \\ 0 & 0 & -3 & -63 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{3}{-2\frac{1}{2}}\\z1s2=3-(-2\frac{1}{2})\cdot \frac{3}{-2\frac{1}{2}}=0 \\ z1s3=4-2\cdot \frac{3}{-2\frac{1}{2}}=6\frac{2}{5} \\ z1s4=175-(-5\frac{1}{2})\cdot \frac{3}{-2\frac{1}{2}}=168\frac{2}{5} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & 6\frac{2}{5} & 168\frac{2}{5} \\ 0 & -2\frac{1}{2} & 2 & -5\frac{1}{2} \\ 0 & 0 & -3 & -63 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{6\frac{2}{5}}{-3}\\z1s3=6\frac{2}{5}-(-3)\cdot \frac{6\frac{2}{5}}{-3}=0 \\ z1s4=168\frac{2}{5}-(-63)\cdot \frac{6\frac{2}{5}}{-3}=34 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & 0 & 34 \\ 0 & -2\frac{1}{2} & 2 & -5\frac{1}{2} \\ 0 & 0 & -3 & -63 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{2}{-3}\\z2s3=2-(-3)\cdot \frac{2}{-3}=0 \\ z2s4=-5\frac{1}{2}-(-63)\cdot \frac{2}{-3}=-47\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 0 & 0 & 34 \\ 0 & -2\frac{1}{2} & 0 & -47\frac{1}{2} \\ 0 & 0 & -3 & -63 \\ \end{array} \\ \\ x=\frac{34}{2}=17\\y=\frac{-47\frac{1}{2}}{-2\frac{1}{2}}=19\\z=\frac{-63}{-3}=21\\L=\{17/19/21\} \end{array}$