Problem Solving

Hi,
Firstly, I have done a search and I couldn't find this question. (I searched for both "xy+z" and "x(y+z)" )
Secondly, I apologise if it is a silly question BUT ...
I ended up getting each of the options other than d as a “must be true”, so obviously I have missed out something basic in my Quant studies. Can someone please tell me what’s wrong with my calculations?

The question is:

If xy + z = x(y+z), which must be true?---

1) x=0 and z=0

substituting the values of x and z in the stem’s equation,
0*y + 0 = 0( y + 0)
and so 0=0 ….. True

2) x=1 and y = 1

substituting the values of x and y in the equation, we get
1*1 +z =1(1+z) and from there,
1+z =1+z …. True

3)y=1 and z =0
substituting the values of y and z in the equation, we get
x*1 +0 =x(1+0)
and so x = x….. True

4)x=0 or y=0
substituting the value of x, we get
0*y +z= 0(y+z)
and so, z=0, which may or may not be true…… so discarded.

5)x=1 or z=0 (official answer)


Thanks.
Shilo