This problem is going to reduce to a quadratic equation as follows.
Let R = rate driving in rain and D = rate driving when no rain.
Let Tr = time driving in the rain and Td=time driving when not raining.
R*Tr = 294 and D*Td = 260
D = R + 3 and Tr + Td = 11
R*Tr = 294
(R + 3) (11-Tr) = 260
Tr = 294/R
(R+3)[11-(294/R)] = 260
This last equation will give you a quadratic with nasty coefficients:
11R2 - 521R-882=0
I will leave evaluation of the quadratic formula to you.