Flipping Signs Using Arithmetic Axioms

Question

This may be a dumb question, but I'm in a computer science class called Applied Logic, where we have to develop formal proofs, and I'm very inexperienced with them.

So my question is:

Is there a way to prove that (1*(-x)) = ((-1)*x) using basic arithmetic axioms. These axioms specifically:

x + 0 = x {+ identity}

(x) + x = 0 {+ complement}

x 1 = x { identity}

x 0 = 0 { null}

x + y = y + x {+ commutative}

x y = y x { commutative}

x + (y + z) = (x + y) + z {+ associative}

x (y z) = (x y) z { associative}

x (y + z) = (x y) + (x z) {distributive law}

Answer 1

$1x+(-1)x=(1-1)x=0x=0$, thus by uniqueness of inverses, $-x=(-1)x$