Are you considering the solutions to be ordered or unordered?
For example, if we wee trying to solve:
x + y + z + w = 12
Would the solution (x,y,z,w) = (9,1,1,1) = (1,9,1,1) = (1,1,9,1) = (1,1,1,9)
or would that be considered 4 separate solutions?
If they are considered the same, then this is a partitions problem, if they are different, then this is a combinations problem.