\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
\(=\frac{12x-8y}{4^2}=\frac{6z-12x}{3^2}=\frac{8y-6z}{2^2}\)
\(=\frac{12x-8y+6z-12x+8y-6z}{4^2+3^3+2^2}\)
\(=\frac{\left(12x-12x\right)-\left(8y-8y\right)+\left(6z-6z\right)}{16+9+4}\)
\(=\frac{0-0-0}{16+9+4}=0\)
\(\Rightarrow\hept{\begin{cases}3x-2y=0\\2z-4x=0\\4y-3z=0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}\frac{x}{2}=\frac{y}{3}\\\frac{z}{4}=\frac{x}{2}\\\frac{y}{3}=\frac{z}{4}\end{cases}\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{4}}\)
Ta có :
\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)
\(\Rightarrow\frac{3xz-2y}{4z}=\frac{2yz-4xy}{3y}=\frac{4xy-3xz}{2x}\)
\(\Rightarrow\frac{\left(3xz-2y\right)+\left(2yz-4xy\right)+\left(4xy-3xz\right)}{4z+3y+2x}=0\)
\(\Rightarrow\hept{\begin{cases}3x-2y=0\\2z-4x=0\end{cases}\Rightarrow\hept{\begin{cases}3x=2y\\2z=4x\end{cases}}}\)
\(\Rightarrow\hept{\begin{cases}\frac{x}{2}=\frac{y}{3}\\\frac{x}{2}=\frac{z}{4}\end{cases}\Leftrightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\left(đpcm\right)}\)
