\(\dfrac{3x-2y}{4}=\dfrac{4y-3z}{2}=\dfrac{2z-4x}{3}=\dfrac{12x-8y}{16}=\dfrac{8y-6z}{4}=\dfrac{6z-12x}{9}\)
\(=\dfrac{12x-8y+8y-6z+6z-12x}{16+4+9}=\dfrac{0}{29}=0\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{3x-2y}{4}=0\\\dfrac{4y-3z}{2}=0\\\dfrac{2z-4x}{3}=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}3x=2y\\4y=3z\\2z=4x\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\dfrac{x}{2}=\dfrac{y}{3}\\\dfrac{y}{3}=\dfrac{z}{4}\\\dfrac{x}{2}=\dfrac{z}{4}\end{matrix}\right.\)
\(\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}=\dfrac{x}{2}=\dfrac{-2y}{-6}=\dfrac{3z}{12}=\dfrac{x-2y+3z}{2-6+12}=\dfrac{8}{8}=1\)
\(\Rightarrow\left\{{}\begin{matrix}x=1.2=2\\y=1.3=3\\z=1.4=4\end{matrix}\right.\)