Find all integer solutions to hyperbola.

Question

The question is find all integer solutions to $6xy+4x-9y-7=0$

Now I did trial and error and found $(1,-1)$ to be an integer solution.

This question is in my summer math homework. We have not been taught about the behaviour of hyperbolas and only know what they look like. I do know some basic coordinate geometry, though. Is there a definitive way to approach this question? Any help, hint or a link to a similar question is highly appreciated. I couldn't find a similar question on MSE. Can someone please help me?

Answer 1

Hint: The equation is $(2x-3)(3y+2)=1$. There are a limited number of ways to factor $1$.

Answer 2

$$2x=\dfrac{9y+7}{3y+2}=3+\dfrac1{3y+2}$$

So, $3y+2$ must divide $1$

$\implies3y+2=\pm1$