Let M be the amount of money each person had initially.

Let A = amount Betty gave to Alan

Let C = amount Betty gave to Cathy.


Then Betty has  (M-A-C).
     Alan  has  (M+A).
     Cathy has  (M+C).


And the condition says


M - A - C = M + A - 5     (1)    ( . . . so that Alan has $5 more than Betty)
M + C     = M + A - 1     (2)    ( . . . Cathy has $1 more than Alan)


Simplify by canceling M:


-A -C = A - 5             (1')
C     = A - 1             (2')


Write in the standard form

2A + C = 5                (1'')
 A - C = 1                (2'')


Now add eq(1'')  and eq(2'').  You will get

3A = 6  ====>  A = 2.


Then from equation (2'')  C = A - 1 = 1.


Answer.   A = 2, C = 1.  In other words,  B gives 2 to A and B gives 1 to C.


Check.    Do the check on your own.