Prove that 2m + 3n = 0 (mod 5) is an equivalence relation on the integers

Question

The relation on $\mathbb{Z}$, given by m ~ n iff 2m+3n $\equiv$ 0 (mod 5). How do you prove that this is an equivalence relation?

I got that it is reflexive so far, but I am stuck on if it is symmetric.

Answer 1

To show symmetry ($n\sim m \iff m\sim n$):

$2n+3m\equiv0\pmod5$

$\iff4(2n+3m)\equiv0\pmod5$

$\iff8n+12m\equiv0\pmod5$

$\iff 3n+2m=2m+3n\equiv0\pmod5$