First we find their cistern-filling rates in cisterns per minuter.

Let the time required for the faster pipe to fill the cistern be x minutes
So the faster pipe's filling rate is 1 cistern per x minutes or
 or  cisterns per minute

Then the time required for the slower pipe to fill the cistern is x+72
minutes.  So the slower pipe's filling rate is 1 cistern per x+72 minutes 
or 
 or  cisterns per minute.

Since the faster pipe's rate is 7 times faster than the slower pipe's rate,
we have:

        
        
Cross-multiply:
      x + 72 = 7x
          72 = 6x
          12 = x

So the faster pipe's filling rate is  =  cistern per minute.
And the slower pipe's filling rate is  =  cistern per minute.

Now we have their respective filling rates in cisterns per minute.

Let t = the number of minutes required to fill the pipe when both pipes
are open.  So their combined filling rate is 1 cistern per t minutes 
or  or  cisterns per minute.

The equation comes from:

   

   

Multiply through by 84t

    7t + t = 84
        8t = 84
         t =  = 10.5 minutes.

Edwin