Solution verification: showing $xRy$ on $\Bbb Z$ iff $3x-7y$ is even is an equivalence relation

Question

A relation $R$ is defined on the set $\mathbb{Z}$ of integers by $xRy$ if $3x − 7y$ is even. Prove that $R$ is an equivalence relation.

Proving the given relation is reflexive is pretty straightforward. I am unsure as to whether my proofs for the given relation being symmetric and transitive are acceptable

• Symmetric: $3x -7y = 2k$ To Prove: $3y - 7x $ is even Sum of both: $3y - 7x + 3x -7y = 2(-2x - 2y)$. Hence, for the sum to be even, $3y-7x$ is even as $3x-7y$ is even.
• Transitive: $3x-7y = 2k$, $3y - 7z = 2r$ To prove: $3x - 7z$ is even Sum of both: $3x - 7z -4y = 2(k+r) = 3x-7z = 2(k+r+2)$.

Is this an acceptable proof to prove the given relation is an equivalent relation?

Answer 1

Your answer is correct.

Symmetry: Let $3x-7y$ be even. Since $4x+4y$ is even, $3x-7y+4x+4y=7x-3y$ is even.

Transitivity: Let $3x-7y$ and $3y-7z$ be even. Then their sum is $3x-4y-7z$, which is even and hence $3x-4y-7z+4y =3x-7z$ is also even.