Positive $(x,y)$

$\begin{aligned}x\times (y+\dfrac{1}{y})&=35\\ \implies x&=\dfrac{35y}{y^2+1}\\ \text{Since }y \text{ and } y^2+1 &\text { are coprime for } y>1,\\ \implies y^2+1 \text{ divides } 35 &\text{ for x to be an integer.}\\ y=2 \text{ is the only } &\text{value satisfying the relation.}\\ \implies (x,y)=(14,2) &\text{ is the only solution to the above equation.}\end{aligned}$