From the condition, you have these two equations


(1)   J - 20 = P + 20         ("If Julian gave Patrick 20 marbles, both of them would have an equal number of marbles")

(2)   3*(P - 20) = J + 20     ("If Patrick gave Julian 20 marbles, Julian would have three times as many marbles as Patrick")


From eq(1), express  J = P+40  and substitute it into eq(2).  You will get


3*(P-20) = (P+40) + 20,    or


3P - 60 = P + 60,


3P - P = 60 + 60


2P = 120  ====>  P = 120/2 = 60.


Answer.  Patrick had initially 60 marbles;  Julian had P+40 = 60+40 = 100 marbles.


Check on your own that the equations (1)  and (2)  are hold in this case.