I will assume that "Ann" and "Sue" are supposed to be the same person....
Joy starts swimming at Ann's 8th session.
Ann starts doing 10 lengths and increases the number by 2 each session. So in her 8th session, the number of lengths she swims is 10+7(2) = 24.
In Joy's first session, she swims 15 lengths, and she increases that number by 5 each session.
Let x be the number of Joy's sessions after her first (i.e., after Ann's 8th).
The number of lengths Ann swims x sessions after Joy's first is 24+2x; the number Joy swims that same session is 15+5x.
The number of lengths they swim will be the same when 24+2x = 15+5x.
24+2x = 15+5x
3x = 9
x = 3
It will be 3 sessions after her first (i.e., her 4th session) when Joy swims the same number of lengths as Ann.
Ann's sessions:
session number 1 2 3 4 5 6 7 8 9 10 11 12...
lengths 10 12 14 16 18 20 22 24 26 28 30 32...
Joy's sessions:
session number 1 2 3 4 5...
lengths 15 20 25 30 35...
Of course, you could solve the problem without formal algebra, simply by writing out a table like the one shown.