Theo đề, ta có: \(\left\{{}\begin{matrix}2x=3y\\4y=5z\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{x}{3}=\dfrac{y}{2}\\\dfrac{y}{5}=\dfrac{z}{4}\end{matrix}\right.\Leftrightarrow\dfrac{x}{15}=\dfrac{y}{10}=\dfrac{z}{8}=k\)
=>x=15k; y=10k; z=8k
Ta có: \(3x^2-y^2+z^2=1971\)
\(\Leftrightarrow675k^2-100k^2+64k^2=1971\)
\(\Leftrightarrow k^2=\dfrac{219}{71}\)
Trường hợp 1: \(k=\sqrt{\dfrac{219}{71}}\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=15\sqrt{\dfrac{219}{71}}\\y=10\sqrt{\dfrac{219}{71}}\\z=8\sqrt{\dfrac{219}{71}}\end{matrix}\right.\)
Trường hợp 2: \(k=-\sqrt{\dfrac{219}{71}}\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=-15\sqrt{\dfrac{219}{71}}\\y=-10\sqrt{\dfrac{219}{71}}\\z=-8\sqrt{\dfrac{219}{71}}\end{matrix}\right.\)