Find all solutions for $3x+y \equiv 2 \pmod{9}$

Question

I want to find the integer solutions of

$$3x+y \equiv 2 \pmod{9}$$

We know that $y \equiv2\pmod{3}$, but I don't know how this helps to find the solution which is $ \mathbb{L} = (0, 2) + (1,−3) \cdot \mathbb{Z}$.

I would appreciate any hints.

Answer 1

$$y\equiv2-3x\pmod9$$

As $x\equiv0,1,2\pmod3,3x\equiv0,3,6\pmod9$

$$\implies y\equiv2,2-3,2-6\pmod9$$