Prove that for any integers x and y and any integer k that if $x \equiv_k y$, then $y \equiv_k x$.

Question

If anyone has a better method please feel free to comment it below.

My proof:

This is a direct proof. Assume $x \equiv_k y$ is true. From $x \equiv_k y$ we know that:

x = y + kq, where q is q ∈ Z $\space\space\space\space\space\space$ (1)

Now if we take (1) and solve for y we get:

y = x - kq $\space\space\space\space\space\space$ (2)

Equation (2) tells us that $y \equiv_k x$ is possbile provided that the value for $kq$ used in (1) is negated. $\blacksquare$

Also is my proof right?