Find all solutions of the following linear congruence $2x+3y≡1\pmod 7$ I've never done one with two variables, I understand that $(2, 3, 7) = 1$ and $1|1$ so there are $1*7$ solutions, and $x=0,1,2,3,4,5,6$. I have no idea how to proceed from here, and any help would be greatly appreciated!

\begin{align}2x + 3y - 1 = 7k &\Rightarrow  3y = -2x + 1 - 7k = -3(x+2k) + x + 1 - k\\ &\Rightarrow y = -x - 2k + \dfrac{1+x-k}{3}
\end{align}
Since
\begin{align}
y \in \mathbb{Z} &\Rightarrow  3\mid (1+x-k)\\ &\Rightarrow 1+x-k = 3n \\&\Rightarrow x = k-1 +3n\\ &\Rightarrow y = -(k-1+3n) - 2k + n = -3k + 4n + 1$, 
\end{align}
with $n,k \in \mathbb{Z}$.
