Simplification of a boolean polynomial

Question

I have to simplify the following Boolean polynomial using $x\land y$ = $xy$ and $x\lor y$ = x+y :

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

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

=$xy'+xy'+xz'+z$

=$xy'+xz'+z$

My book gives the following answer: $xy'+x+z$

How do i get from the last step that I did to this answer ?

Answer 1

Note that $$z'=1-z$$ Hence: $$ xz'+z=x(1-z)+z=x-xz+z\underbrace{=}_{\text{mod 2}}x+xz+z=x+z+xz $$

Now note that $x+z+xz$ is false if and only if both of $x,z$ are false, thus this expression is equivalent to $x+z$.

Substituting $xz'+z$ with $x+z$ gives the answer written in the book