Suppose that a cyclist began a 627mi ride across a state at the western edge of the state, at the same time that a car traveling toward it leaves the eastern end of the state. if the bicycle and car met after 9.5hr and the car traveled 31.6mph faster than the bicycle, find the average rate of each.

This is the concept of relative speed; We are required to get the speed of the cyclist and the speed of the driver;
Let the speed of the cyclist be x;
The speed of the driver=x+31.6
The distance between them at start=627 miles
Time taken for them to meet=9.5 hr
Distance is given by:
Distance=speed*time
Relative speed=x+(x+31.6)=2x+31.6
Time=9.5 hr
hence;
627=9.5(2x+31.6)
627=19x+300.2
putting like term together we get:
627-300.2=19x
19x=326.8
x=326.8/19
x=17.2 mph
thus the speed of the cyclist=17.2 mph
The speed of the driver=x+31.6=17.2+31.6=48.8 mph