We will let
x = first number
y = second number
1/x = reciprocal of the first number which is larger
1/y = reciprocal of the second number
(1) 1/x + 1/y = 7
1/y = 7 - 1/x
(2) 1/x = 1/y + 7/3
1/x = (7 - 1/x) + 7/3
1/x = 7 + 7/3 - 1/x
1/x + 1/x = (21+7)/3 - 1/x
2/x = 28/3
28x = 6
x = 6/28
x = 3/14
(1) 1/x + 1/y = 7
1/(3/14) + 1/y = 7
14/3 + 1/y = 7
1/y = 7 - 14/3
1/y = (21-14)/3
1/y = 7/3
7y = 3
y = 3/7
Therefore the two numbers are 3/14 and 3/7.
