let

d1 = 234 mi the distance the car travels

d2 = 204 mi the distance the motorcycle travels

v1 = the speed of the car

v2 = the speed of the motorcycle

the cars speed is 10 miles per hour more than the motorcycle's:

v1 = 10 + v2

since the both travel the same time and we know that time = distance/speed we have:

d1/v1 = d2/v2

234/v1 = 204/v2 cross-multiply

204v1 = 234v2

by solving the system of equations:

v1 = 10 + v2

204v1 = 234v2

we find:

v1 = 78 mph

v2 = 68 mph

click here to see the step by step solution of the system of equations:

the speed of the car is 78 mph.

the speed of the motorcycle is 68 mph.