<s>[INST] Positive pairs of integral values satisfying $2xy − 4x^2 +12x − 5y = 11$ The number of positive pairs of integral values of $(x, y)$ that solves
$2xy − 4x^2 +12x − 5y = 11$ is?
I rearranged it to $(2x-5)(y+1-2x)=6$, which took quite a bit of time.
So it can be $2*3$ , $3*2$, $6*1$ or $1*6$ which gives us 2 possible positive integral pairs. Answer: 2.
Is there a faster way to do similar problems?
 [/INST] First note that:
$$y=\frac{11 - 12 x + 4 x^2}{2 x-5}$$
Applying division:
$$y=\frac{(2x-1)(2x-5)+6}{2x-5}=(2x-1)+\frac{6}
{2x-5}$$
To be an integer $2x-5$ must divide $6$ so $2x-5$ must be $-1,-3,1,2$ or $3$ or $6$. Check then by the positive integral solutions.
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