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.

Here's a generalization:

What if all of the sums of products of k-combinations with 0 < k <= n match up? So if

a + b + c = x + y + z
ab + ac + bc = xy + xz + yz
abc = xyz

then up to permutation, a = x;b=y;c=z.