Is there a specific solution for diophantine equations with multiple variables that can be positive integer only?

Question

I have a problem with 6 variables, something like: $ai + bj + ck + dm + en + fo = g$ where $a-g$ are constants, positive integers (non-zero, not necessarily primes) and $i-o$ are variables, positive integers (including zero). I'd like to know if there is a more specific solution for this problem, than the Euclidean algorithm which can yield negative results?