Suppose A begins with £a and B begins with £b 

If A gives B £2, then A will have £(a-2) and B will have £(b+2).

In that case their amounts would be equal, so we set them equal:

                 £(a-2) = £(b+2)
                    a-2 = b+2
                      a = b+4

If B gives A £2, then B will have £(b-2) and A will have £(a+2).

In that case A's amount would be twice B's amount, so we have:

                 £(a+2) = 2·£(b-2)
                    a+2 = 2(b-2)
                    a+2 = 2b-4
                      a = 2b-6

So we have this system of 2 equations in two unknowns:

                     

                     b+4 = 2b-6
                      10 = b

                       a = 2b-6 = 2(10)-6 = 20+6 = 14

So A has £14 and B has £10

Checking:  If A gives B £2, A will have only £12 and B will have £12. 
That checks because they are the same.  If B gives A £2, B will have only £8
and A will have £16. That checks because £16 is twice as much as £8.

Edwin