This is how I would go about it:
Let be the number of balloons with 3 spots and be the number of balloons with 5 spots. We then have:
where in the total number of spots.
Next, I would let:
and and so there results:
Now, we want to find a specific instance that works for a particular .
In the case of , I would begin taking multiples of 5 away from 31 until I have a multiple of 3. In this case:
and so we find:
We know produces a valid solution , but does not, so going in the other direction we find:
But, produces an invalid solution, so we know we have found the only two solutions that work.
A similar process can be done for the other two values of .