An alternate way to arrive at the equation 4s/r₁ - 4s/r₂ = 4s/r is as follows: 

As you know, pressure inside soap bubble =  P_ext + 4s/r

Assume P₁ = pressure inside smaller bubble S₁ (same at all points inside smaller bubble).

P₂ = pressure inside bigger bubble S₂ (same at all points inside bigger bubble). 

Now you can find P₁ with a trick also.

Instead of atmospheric pressure as external pressure, use P₂ as external pressure to calculate P₁  (only applicable at common surface). (easier to think if you consider a point close to common surface in smaller bubble)

So, P₁ = P₂  + 4s/r, where r is the radius of common surface. (1) 

You can calculate pressure in smaller and bigger bubble using external atmospheric also. 

So, P₁ = P_atm + 4s/r₁ (2)

P₂ = P_atm + 4s/r₂ (3)

Now putting eqn (2) and (3) in (1), you get: 

P_atm + 4s/r₁ = P_atm + 4s/r₂ + 4s/r

=> 4s/r₁ = 4s/r₂ + 4s/r

=> 4s/r₁ - 4s/r₂ = 4s/r