$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ 6 x +4 y + 9 z=32\\ 5 x +7 y + 10 z=17\\ 4 x +8 y + 5 z=100\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 6x+4y+9z=32 \\ 5x+7y+10z=17 \\ 4x+8y+5z=100 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 5 & 7 & 10 & 17 \\ 4 & 8 & 5 & 100 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\cdot 6\text{-Zeile}1\cdot 5 \\z2s1=5\cdot 6-6\cdot 5 =0 \\ z2s2=7\cdot 6-4\cdot 5 =22 \\ z2s3=10\cdot 6-9\cdot 5 =15 \\ z2s4=17\cdot 6-32\cdot 5 =-58 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 0 & 22 & 15 & -58 \\ 4 & 8 & 5 & 100 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 6\text{-Zeile}1\cdot 4 \\z3s1=4\cdot 6-6\cdot 4 =0 \\ z3s2=8\cdot 6-4\cdot 4 =32 \\ z3s3=5\cdot 6-9\cdot 4 =-6 \\ z3s4=100\cdot 6-32\cdot 4 =472 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 0 & 22 & 15 & -58 \\ 0 & 32 & -6 & 472 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 22\text{-Zeile}2\cdot 32 \\z3s2=32\cdot 22-22\cdot 32 =0 \\ z3s3=(-6)\cdot 22-15\cdot 32 =-612 \\ z3s4=472\cdot 22-(-58)\cdot 32 =1,22\cdot 10^{4} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline6 & 4 & 9 & 32 \\ 0 & 22 & 15 & -58 \\ 0 & 0 & -612 & 1,22\cdot 10^{4} \\ \end{array} \\ \\ z=\frac{1,22\cdot 10^{4}}{-612}=-20\\y \cdot 22+15\cdot-20=(-58)\\y= 11\\x\cdot 6+4\cdot11+9\cdot(-20)=32\\x= 28\\L=\{28/11/-20\} \end{array}$