Proving transitivity in equivalence relation a ~ b iff 2a+3b is div by 5

Homework Statement

Relation on set of integers.

a~b if and only if 2a+3b is divisible by 5
show that ~ is an equivalence relation

The Attempt at a Solution

I have already proved that the relation is reflexive and symmetric, but I'm unsure of my approach at proving transitivity.

if the relation is transitive, then: m~n and n~q implies m~q

by the relation: 5 divides 2m + 3n, so let 2m+3n=5r
5 divides 2n + 3q, so let 2n + 3q= 5t (r,t are integers)

==> 2m=5r-3n and 3q=5t-2n

if m~q, then 2m+3q must be divisible by 5.
2m + 3q= (5r-3n)+(5t-2n)= 5(r+t)-3n-2n
= 5(r+t)-5n

sum of numbers divisible by "a" = a number divisible by "a", so 5(r+t)-5n is divisible by 5, implying that 2m+3q is divisible by 5 which implies that m~q, which proves transitivity of the relation.

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