Boolean Algebra x+y=0 proof

Question

So I am having a problem solving this proof of Boolean algebra. I am trying to prove that if x + y = 0 then x = 0

This is what I have tried

x + y
x + (y.1)
(x + y).(x + 1)
(x + y).1
(x + y)

But as you can see I am just going in circles, can anyone offer any advice?

Answer 1

If $x+y=0$ then we have $$\eqalign{x &=x+0\cr &=x+yy'\cr &=(x+y)(x+y')\cr &=0(x+y')\cr &=0\ .\cr}$$