Theo đề ra, ta có:
\(\hept{\begin{cases}\frac{10}{x-5}=\frac{6}{y-9}\\\frac{10}{x-5}=\frac{14}{z-21}\\xyz=6720\end{cases}}\Rightarrow\hept{\begin{cases}10y-90=6x-30\\10z-210=14x-70\\xyz=6720\end{cases}}\Rightarrow\hept{\begin{cases}10y=6x+60\\10z=14x+140\\xyz=6720\end{cases}}\Rightarrow\hept{\begin{cases}y=\frac{6x+60}{10}\\z=\frac{4x+140}{10}\\x.\frac{6x+60}{10}.\frac{4x+140}{10}=6720\left(1\right)\end{cases}}\)
Từ \(\left(1\right)\Rightarrow\frac{\left(6x^2+60x\right)\left(4x+140\right)}{100}=6720\)
\(\Rightarrow24x^3+840x^2+240x^2+8400=672000\)
\(\Rightarrow24x^3+840x^2+240x^2-663600=0\)
\(\Rightarrow x=21,94727494\)
\(\Rightarrow y=19,16836496\)
\(\Rightarrow z=22,77890998\)