$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ 2 x +3 y + -4 z=16\\ 4 x +9 y + -1 z=58\\ 1 x +6 y + 2 z=34\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 2x+3y-4z=16 \\ 4x+9y -z=58 \\ x+6y+2z=34 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 4 & 9 & -1 & 58 \\ 1 & 6 & 2 & 34 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\cdot 2\text{-Zeile}1\cdot 4 \\z2s1=4\cdot 2-2\cdot 4 =0 \\ z2s2=9\cdot 2-3\cdot 4 =6 \\ z2s3=(-1)\cdot 2-(-4)\cdot 4 =14 \\ z2s4=58\cdot 2-16\cdot 4 =52 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 0 & 6 & 14 & 52 \\ 1 & 6 & 2 & 34 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 2\text{-Zeile}1\cdot 1 \\z3s1=1\cdot 2-2\cdot 1 =0 \\ z3s2=6\cdot 2-3\cdot 1 =9 \\ z3s3=2\cdot 2-(-4)\cdot 1 =8 \\ z3s4=34\cdot 2-16\cdot 1 =52 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 0 & 6 & 14 & 52 \\ 0 & 9 & 8 & 52 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 6\text{-Zeile}2\cdot 9 \\z3s2=9\cdot 6-6\cdot 9 =0 \\ z3s3=8\cdot 6-14\cdot 9 =-78 \\ z3s4=52\cdot 6-52\cdot 9 =-156 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline2 & 3 & -4 & 16 \\ 0 & 6 & 14 & 52 \\ 0 & 0 & -78 & -156 \\ \end{array} \\ \\ z=\frac{-156}{-78}=2\\y \cdot 6+14\cdot2=52\\y= 4\\x\cdot 2+3\cdot4+(-4)\cdot2=16\\x= 6\\L=\{6/4/2\} \end{array}$