#1
Let x, y be given. We show that x + y = y + x.
Observe 0 = (x + y) + (x + y). Add (y + x) from the left on both sides to get y + x = (y + x) + (x + y) + (x + y)
= (y + (x + x) + y) + (x + y) = (y + y) + (x + y) =  x + y. Q.E.D.
#2
Let x, y be given such that 0 = x + y. Add x from the left on both sides to get x = (x + x) + y = y. Q.E.D.