Question

Cho \(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}.\)CMR \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)

Answer 1

\(\frac{3x-2y}{4}=\frac{2z-4x}{3}=\frac{4y-3z}{2}\)

\(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}\)

\(\frac{12x-8y}{16}=\frac{6z-12x}{9}=\frac{8y-6z}{4}=\frac{\left(12x-8y\right)+\left(6z-12x\right)+\left(8y-6z\right)}{16+9+4}=\frac{0}{29}=0\)

\(\Rightarrow\frac{3x-2y}{4}=0\Rightarrow3x-2y=0\Rightarrow3x=2y\Rightarrow\frac{x}{2}=\frac{y}{3}\)( 1 )

\(\Rightarrow\frac{4y-3z}{2}=0\Rightarrow4y-3z=0\Rightarrow4y=3z\Rightarrow\frac{y}{3}=\frac{z}{4}\)( 2 )

Từ ( 1 ) và ( 2 ) \(\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)