Need help proving/disproving

I'm not sure how to best phrase this, so bear with me:
If you're given
abc = xyz
and
a+b+c = x+y+z
where a, b, c, x, y, z are all positive integers, is there a way to prove or disprove a = x, b = y, c = z?

I want to know if an integer shown in this way is unique.

The initial conditions are too vague to come up with any concise solution.  For example, if we say

a=x+y+z-b-c and substitute it into the first equation, we get

bc(x+y+z-b-c)=xyz, which is an equation of 5 variables and has no simple form to its solutions.

Further, from a1*a2*a3=b1*b2*b3 we note the gcd(ai,bi) needn't be greater than 1.  This fact implies simplification is not necessarily possible.