How to solve congruence with two variables x and y

Question

I'm having trouble solving the following congruence:

$$2x + 2y ≡ 0 \pmod{7}.$$

What I've tried so far is writing the congruence as follows: since the remainder is $0,$ we know $7$ is divisible by $2x + 2y,$ so $2x + 2y = 0 + 7k,$ and I got pretty much stuck on this step, so any further help is greatly appreciated.

Answer 1

Multiplying both sides by $4$, $$8x+8y\equiv x+y\equiv0\mod{7}$$ Hence $$y\equiv -x\mod{7}$$ $$\therefore y=-x+7k$$ where $k\in\mathbb{Z}$.

Answer 2

$7 | 2x+2y =2(x+y) \implies 7|2 $ or $7|x+y$. But $7\not|2$, so this means $7|x+y$.

Conversely, $7|x+y\implies7|2x+2y$.