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

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

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    @TheChaz: Oh, quite so (especially since the username on the other question was changed from the generic userxxxx to "Mily"). I was trying to say "That's the second time *this user* has posted..." not to imply there was a sudden wave of people with similar manners.2012-04-24

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

Hence :

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

$3x=2kx-k$

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

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    @pedja You are welcome. But I still think that what you have after the "Hence" is not the best way to proceed... sorry. Because it is not immediately clear for what $k$ the last expression is integer. Or, maybe, in this case with small numbers it is clear enough... That's OK. But I would also suggest looking at the Andre Nicolas's comment above.2012-04-24
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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.

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    Granted, I refer to this answer in *my* answer, but I still would have thought this better as a comment.2012-04-24
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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 $3$.

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    Sorry, I had no intent to criticize your answer or especially your pedagogy. And, as I said I even upvoted it. The problem at the link you gave in the comment is a completely different story. The problem at the link in the answer is (almost) exactly the same as this one. And can be solved using the exactly same method (with a little modification which you have shown in your answer). Moreover, it was asked by the same person.2012-04-24