$DK: x,y\geq 1$
chia hai vế pt cho $xy$
$\frac{\sqrt{x-1}}{x}+\frac{\sqrt{y-1}}{y}=1$(1)ta có $\sqrt{x-1}\leq \frac{x-1+1}{2}=\frac{x}{2}$( cauchy)
$\Rightarrow \frac{\sqrt{x-1}}{x}\leq \frac{1}{2}$
tượng tự $\sqrt{y-1}\leq \frac{1}{2}$
$\Rightarrow \frac{\sqrt{x-1}}{x}+\frac{\sqrt{y-1}}{y}\leq 1$
(1) xảy ra
$\Leftrightarrow (x;y)=(2;2)$