Stuck in Boolean Algebra equation

Question

I have this equation in Boolean Algebra:

$x*y*z+x'*y*z+x*y'*z+x*y*z' = y*z+x*z+x*y$

I got this:

$= yz(x+x')+xy'z+xyz'$

$= yz+xy'z+xyz'$

And from here I tried multiple things but it goes wrong like:

$yz+x(y'z+yz')$

$yz + x$

Since y'*y and z'*z =0 they disappear so this seemed to be the only answer for me what am I doing wrong?

Answer 1

Hint: You could add in a couple of $xyz$ terms into the left side of the original equation.

Answer 2

So using the hint of @paw88789 I got this:

$xyz+x'yz+xy'z+xyz'$

$=yz(x+x')+xz(y+y')+xy(z+z')$

$=yz + xz + xy$

Is this right?

If it is, can I have a little more explanation on why I can use xyz a couple times and if I can do this with all the other terms too?