$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x + b1\cdot y + c1\cdot z=d1\\ a2\cdot x + b2\cdot y + c2\cdot z=d2\\ a3\cdot x + b3\cdot y + c3\cdot z=d3\\ \\ \text{Gesucht:} \\\text{x,y,z} \\ \\ \textbf{Gegeben:} \\ 4 x -3 y + 2 z=10\\ 5 x +6 y + -7 z=4\\ 10 x +2 y + -3 z=7\\ \\ \\ \textbf{Rechnung:} \\\small \begin{array}{l} 4x-3y+2z=10 \\ 5x+6y-7z=4 \\ 10x+2y-3z=7 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline4 & -3 & 2 & 10 \\ 5 & 6 & -7 & 4 \\ 10 & 2 & -3 & 7 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}2=\text{Zeile}2\cdot 4\text{-Zeile}1\cdot 5 \\z2s1=5\cdot 4-4\cdot 5 =0 \\ z2s2=6\cdot 4-(-3)\cdot 5 =39 \\ z2s3=(-7)\cdot 4-2\cdot 5 =-38 \\ z2s4=4\cdot 4-10\cdot 5 =-34 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline4 & -3 & 2 & 10 \\ 0 & 39 & -38 & -34 \\ 10 & 2 & -3 & 7 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 4\text{-Zeile}1\cdot 10 \\z3s1=10\cdot 4-4\cdot 10 =0 \\ z3s2=2\cdot 4-(-3)\cdot 10 =38 \\ z3s3=(-3)\cdot 4-2\cdot 10 =-32 \\ z3s4=7\cdot 4-10\cdot 10 =-72 \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline4 & -3 & 2 & 10 \\ 0 & 39 & -38 & -34 \\ 0 & 38 & -32 & -72 \\ \end{array} \\ \\ \begin{array}{l}\text{Zeile}3=\text{Zeile}3\cdot 39\text{-Zeile}2\cdot 38 \\z3s2=38\cdot 39-39\cdot 38 =0 \\ z3s3=(-32)\cdot 39-(-38)\cdot 38 =196 \\ z3s4=(-72)\cdot 39-(-34)\cdot 38 =-1,52\cdot 10^{3} \\ \end{array}\qquad \small \begin{array}{ccc|cc } x & y & z & & \\ \hline4 & -3 & 2 & 10 \\ 0 & 39 & -38 & -34 \\ 0 & 0 & 196 & -1,52\cdot 10^{3} \\ \end{array} \\ \\ z=\frac{-1,52\cdot 10^{3}}{196}=-7\frac{36}{49}\\y \cdot 39+(-38)\cdot-7\frac{36}{49}=(-34)\\y= -8\frac{20}{49}\\x\cdot 4+(-3)\cdot-8\frac{20}{49}+2\cdot(-7\frac{36}{49})=10\\x= \frac{3}{49}\\L=\{\frac{3}{49}/-8\frac{20}{49}/-7\frac{36}{49}\} \end{array}$