Is there any known process for solving nonlinear Diophantine equations such as the ones below?
$(8+3n)m = 11\;\;|\;\;n \in \{0,1\},\;m\in \Bbb Z^+$
$(5+(7+3x+2y)a+3z)b = 30\;\;|\;\; x,y,z \in \{0,1\},\;a,b\in \Bbb Z^+$
I'm aware these are easily solved by inspection, but for more complex examples I would imagine the answer won't be as obvious (if one even exists at all). All the variables are unique, there will never be any exponents.
At the very least I would like to know if there is any process that could be used to determine if there are any valid solutions for these equations. Ideally I would like to be able to generate all valid values for $n,m$ and $a,b,x,y,z$ respectively, but that might be a bit more complicated.
Any help at all would be greatly appreciated!
