Question

x^2 = yz ; y^2 = xz ; z^2 = xy

C/M: x = y = z

Answer 1

Giải:
Ta có: \(x^2=yz\Rightarrow\frac{x}{y}=\frac{z}{x}\)

\(y^2=xz\Rightarrow\frac{x}{y}=\frac{y}{z}\)

\(z^2=xy\Rightarrow\frac{z}{x}=\frac{y}{z}\)

\(\Rightarrow\frac{x}{y}=\frac{y}{z}=\frac{z}{x}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\frac{x}{y}=\frac{y}{z}=\frac{z}{x}=\frac{x+y+z}{x+y+z}=1\)

+) \(\frac{x}{y}=1\Rightarrow x=y\)

+) \(\frac{y}{z}=1\Rightarrow y=z\)

+) \(\frac{z}{x}=1\Rightarrow z=x\)

\(\Rightarrow x=y=z\left(đpcm\right)\)

Vậy \(x=y=z\)