How can z=0????....Another GMAT Prep

Hi...
Heres the problem

If xy+z = x(y+z), which of the following must be true?
1) x=0 & z=0
2) x=1 & y=1
3) y=1 & z=0
4) x=1 & y=0
5) x=1 & z=0

My answer was # 4)...
This is how i approached this problem
xy+z = x(y+z)
=> xy+z = xy+xz
Eliminating xy from both sides...
=> z=xz
=> x=1 ( PROVIDED Z does not = 0)

But the OA is 5)...
Request someone to explain...

Thanks