How to obtain all the solution for this diophantine equation?

Question

How can I Find all solutions of the diophantine equation? :

$$xy=\frac{3x+y}{2}.$$

Answer 1

$$(x,y)=\left(x,\frac{3x}{2x-1}\right)$$

Hence :

$3x=k(2x-1)$

$3x=2kx-k$

$x=\frac{k}{2k-3}$ for some integer $k$

Answer 2

Given this answer that Bill gave you just one hour ago, this would be unfair to give you another similar one for this question as well.

Answer 3

Hint: (Inspired by a similar hint from Bill Dubuque)

Rewrite the equation as $$(2 x - 1) (2 y - 3) = 3$$ and equate $2x - 1$ with the factors of $3$.