Is there any formula for solving such questions? or is it only possible to solve through trial and error?

There is no "formula" as such for solving these types of questions, however there is another approach which is slightly more easier to apply and remember.

Because we want the equation to have positive solutions only, the minimum value which $a, b$ can assume is 1.

If we plug the values of $a, b$ as 1 into the equation, we get $X$ as 14. The equation $5a+9b=14$ will only have 1 positive integral solution; that being; $a, b=1$.

Now, we want the equation to have six solutions, to increase the number of solutions, we can simply add the LCM of $5,9$ to $X$, that is, 45. Because we want 5 more solutions, we add $45\times5$ to 14, that is 239.

The logic behind adding 45 is simple, it will just increase the value of $a,b$ by 1.

Also, this question is probably more of a Algebra question than a Number System question.

Let me know if you need further explaination :D