We have two moments of history here: "now" and "then" (at particular moment in the past). Let's mark as the following:
T - Tom's age now
E - Elizabeth's age now
To - Tom's age then
Eo - Elizabeth's age then

So we have:
T - E =  To - Eo (their difference is constant)   (1)
T + E = 49                                                         (2)
T = 2Eo                                                            (3)
To = E                                                               (4)

Using (4) into (1): 
T - E =  E - Eo  
Then using (3) here:
2Eo - E =  E - Eo  
2Eo + Eo =  E + E
3Eo =  2E
E = 3/2 Eo                                                         (5)

Using (5) and (3) into (2):
T + E = 49
2Eo + 3/2Eo = 49  
3 1/2Eo = 49
7/2Eo = 49
Eo = 49 : 7/2
Eo = 14

So, Elizabeth was 14. It means now Tom is 2*14=28. Elizabeth now is 49 - 28 = 21

Answer: Elizabeth is 21, Tom is 28