Showing final part of equivalence relation

Let S be the set of all ordered pairs (m,n) of positive integers. For (a1,a2) in S and (b1,b2) in S, define (a1,a2)~(b1,b2) if a1+b2=a2+b1.

I need to show we have an equivalence relation.

The first two parts I have already determined to be true, so i will omit my work on that.

For the third part I need to show if for all a,b,c in S if (a,b) is in ~ and (b,c) is in ~ then (a,c) is in ~.

We have:

a1+b2=a2+b1

b1+c2=b2+c1

We want to show:

a1+c2=a2+c1

I'm unsure of how to get there.

I tried using b1=a1-a2+b2 and b2=a2-a1+b1

a1-a2+b2+c2=a2-a1+b1+c1

That didn't really get me anywhere, though.