a method/algorithm to solve a nonlinear Diophantine equation like this

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I have a Diophantine equation like this

10xy + 7y + 9x = 123456789..... (a 270 digit number!!)

I have tried an online calculator to solve it https://www.alpertron.com.ar/QUAD.HTM, but it is factoring that huge number, so it is taking a very very long time.

My question is : Is there any algorithm to solve Diophantine equations on this specific form

10xy + 7y + 9x = c   

where c is a huge ~300 digits number, without the need for factorization?

I need a solution that doesn't take forever to calculate,

the only constraint is that x and y are integers

thanks.