2015/028) Solve in positive integers $3x + 5y = 2xy – 1$

Let is get y in terms of x
we have $3x + 1 = 2xy – 5y = y(2x-5)$
so $y =\dfrac{3x+1}{2x-5} = 1 + \dfrac{x+6}{2x-5}$

now  $\dfrac{x+6}{2x-5}$ should be positive

 

so $(x+6)(2x-5) \gt 0$ and $x \gt 0$

so we get $x \lt - 6$ or $x > 2.5$ but as $x \gt 0$ we have $x \gt 3.5$

and $(x + 6) \ge 2x – 5$ or $x \le 11$ else we shall have y non positive

checking for x between 3 and 11 we get solutions are (11,2) and (3,10)