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?

Not true. For instance, $x=y=0$ and $z=366/31$ is another solution, whose sum is not $12$.
