a. Đặt \(\dfrac{x}{3}=\dfrac{y}{4}=k\)
\(\Rightarrow x=3k;y=4k\)
\(\Rightarrow xy=3k\cdot4k=48\)\(\Rightarrow k^2=\dfrac{48}{12}=4\Rightarrow k=\pm2\)
\(\Rightarrow\left[{}\begin{matrix}k=2\Rightarrow\left\{{}\begin{matrix}\dfrac{x}{3}=2\Rightarrow x=6\\\dfrac{y}{4}=2\Rightarrow y=8\end{matrix}\right.\\k=-2\Rightarrow\left\{{}\begin{matrix}\dfrac{x}{3}=-2\Rightarrow x=-6\\\dfrac{y}{4}=-2\Rightarrow y=-8\end{matrix}\right.\end{matrix}\right.\)
Vậy \(\left[{}\begin{matrix}x=6;y=8\\x=-6;y=-8\end{matrix}\right.\)
a) \(\dfrac{x}{3}=\dfrac{y}{4}\Leftrightarrow4x=3y\)
ta có : \(x.y=48\Leftrightarrow3x.y=3.48=144\Leftrightarrow x.4x=144\Leftrightarrow4x^2=144\)
\(\Leftrightarrow x^2=\dfrac{144}{4}=36\Leftrightarrow\left[{}\begin{matrix}x=6\\x=-6\end{matrix}\right.\)
ta có : \(\left[{}\begin{matrix}\left\{{}\begin{matrix}x=6\\y=\dfrac{48}{6}=8\end{matrix}\right.\\\left\{{}\begin{matrix}x=-6\\y=\dfrac{48}{-6}=-8\end{matrix}\right.\end{matrix}\right.\) vậy ta có 2 tập nghiệm \(\left[{}\begin{matrix}\left\{{}\begin{matrix}x=6\\y=8\end{matrix}\right.\\\left\{{}\begin{matrix}x=-6\\y=-8\end{matrix}\right.\end{matrix}\right.\)
b) ta có : \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}\Leftrightarrow\dfrac{x^2}{4}=\dfrac{y^2}{9}=\dfrac{z^2}{16}\)
áp dụng tính chất dãy tỉ số bằng nhau
ta có : \(\dfrac{2x^2+3y^3-5z^2}{8+27-80}=\dfrac{-405}{-45}=9\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{x^2}{4}=9\\\dfrac{y^2}{9}=9\\\dfrac{z^2}{16}=9\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x^2=36\\y^2=81\\z^2=144\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=\pm6\\y=\pm9\\z=\pm12\end{matrix}\right.\)
vậy \(\left\{{}\begin{matrix}x=\pm6\\y=\pm9\\z=\pm12\end{matrix}\right.\)