Does other solutions exist for $29x+30y+31z = 366$? I was asked this trick question:
If $29x + 30y + 31z = 366$ then what is $x+y+z=?$ 
The answer is $12$ and it is said to be so because $29$ , $30$  and $31$ are respectively the number of days of months in a leap year. Therefore $x + y + z$ must be $12$, the total number of months.
How accurate is this? Is it possible to say so with just a single equation? Are there not other solutions to the equation? If yes, how can one proceed to find other solutions?

With $x,y,z$ being reals,  no, there's tons of answers.   With $x,y,z$ be natural numbers,  $12$ is the only answer.  Proof: $11$ is too small, because even if all $11$ was in the biggest number, $11\cdot 31=341<366$,  and $13$ is too big because $13\cdot 29=377>366$.   
The "middle" case if you allow negative integers but not rationals/reals, I'm not sure about
