prove -x-y=-y-x, for all integers

Answer 1

I do not know what grade you're in, but questions of this sort could go quite deep into algebra.

Since x,y are integers, which belong to a ring in which additions are commutative, so if we write the subtractions as inverses of addition, the above would be invoking commutativity in the ring-addition:

-x-y
=(-x)+(-y) inverse of addition
=(-y)+(-x) commutativity of addition
=-y-x inverse of addition

For high-school algebra, then it would be the commutative property of addition and subtraction.