Two cyclists start at the same point and travel in opposite directions. one cyclist travels 10 km per hour faster than the other. if the two cyclists are 174 km apart after 3 hours, what is the rate of each cyclist?

Distance = rate times time.

Let the first cyclist's rate be r1 and the 2nd cyclist's rate be r2.

Let r2 = r1 + 10   (km per hour).

The total distance between them after 3 hours is 174 km; using the distance= rate times time formula, we also get:

r1(3 hrs) + r2(3 hrs) = 174 km
Then
r1(3 hrs) + (r1+10)(3hrs) = 174 km

Then 3r1 + 3r1 + 30 = 174 

6r1 = 144, and r1 = 24 km/hr.  Then r2 = (24 + 10) km/hr = 34 km/hr.