So I get that you can determine the general solutions for a linear diophantine equation of the for $ax + by = c$ as:
$$x = x' + \frac{b}{\gcd(a,b)}\cdot t~~~~~\text{and}~~~~~~y = y' - \frac{a}{\gcd(a,b)}\cdot t$$
Where $t$ is an integer and $x'$ and $y'$ are the particular solutions.
My question is, how do you determine the particular solutions $x'$ and $y'$. I just sort of guess them, try to plug something in the simplified equation after I determine that $c$ is divisible by the $\gcd(a,b)$ so we have integer solutions.
Is there a method, or some kind of rule that is used that I'm not aware of ?