Đặt: \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}=k\)
\(x=2k;y=3k;z=5k\)
Mà: \(3x^2-2y^2+z^2=19\)
\(\Rightarrow3\cdot\left(2k\right)^2-2\cdot\left(3k\right)^2+\left(5k\right)^2=19\)
\(\Rightarrow12k^2-18k^2+25k^2=19\)
\(\Rightarrow19k^2=19\)
\(\Rightarrow k^2=1\)
\(\Rightarrow k=\pm1\)
Với k = 1 \(\Rightarrow\left\{{}\begin{matrix}x=2\\y=3\\z=5\end{matrix}\right.\)
Với \(k=-1\Rightarrow\left\{{}\begin{matrix}x=-2\\y=-3\\z=-5\end{matrix}\right.\)
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\(\dfrac{x}{2}=\dfrac{2y}{3}=\dfrac{3z}{4}\Rightarrow\dfrac{x}{2}=\dfrac{y}{\dfrac{3}{2}}=\dfrac{z}{\dfrac{4}{3}}\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{x}{2}=\dfrac{z}{\dfrac{4}{3}}=\dfrac{x-z}{2-\dfrac{4}{3}}=\dfrac{45}{2}\)
\(\Rightarrow\dfrac{x}{2}=\dfrac{45}{2}\Rightarrow x=45\)
\(\Rightarrow\dfrac{y}{\dfrac{3}{2}}=\dfrac{45}{2}\Rightarrow y=\dfrac{135}{4}\)
\(\Rightarrow\dfrac{z}{\dfrac{4}{3}}=\dfrac{45}{2}\Rightarrow z=30\)
