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.

Moar counterexamples:

1 7 15
4 17 1
 
1 7 18
2 21 3
 
1 7 18
3 21 2
 
1 8 12
2 16 3
 
1 8 12
3 16 2
 
1 9 10
2 15 3
 
1 9 10
3 15 2
 
1 10 20
2 25 4
 
1 10 20
4 25 2
 
1 11 16
2 22 4
 
1 11 16
4 22 2
 
1 12 14
2 21 4
 
1 12 14
4 21 2
 
1 13 18
3 26 3
 
1 14 20
2 28 5
 
1 14 20
5 28 2
 
1 15 18
2 27 5
 
1 15 18
5 27 2