solution of $y^2 - x = 15$ and $x^2 -xy = 2009$

Question

Find all the integer solutions to the equations:

\begin{eqnarray} y^2 - x &=& 15 \\ x^2 -xy &=& 2009 \end{eqnarray}

Not sure how to solve this :/, tried the usual algebra way (solving for something and substetuting) but didn't really work out

$x+15$ must be a square number, so $x$ has to be $1,10,21,34,49,\ldots$

Using $49$ it works and gives me a solution, but is that the only one? And if, how do I show it.

Answer 1

x² -xy = 2009

x(x-y)=1*2009 =2009 *1 =-2009 (-1)=(-1)(-2009)

x(x-y)=7*287 =... like that

x(x-y)=49*41=...like above

find x,y then compare with other equation