Acar and a motorcycle leave at noon from the same location, heading in the same direction. the average speed of the caris 30 mph slower than twice the speed of the motorcyclein two hours, the car is 20 miles ahead of the motorcycle. find thespeed of both the car and the motorcycle, in miles per hour.

Speeds are s_c for car, s_m for motorcycle.

time is 2

distance of car is:
d_c = d_m + 20, d_m is distance of motorcycle.

speed is defined as:
s = d/t, distance over time.

hence:
d_c/2 = s_c = d_m/2 + 10
s_c = s_m + 10

from problem statement we know:
s_c = 2s_m - 30

so we have 2 simultaneous equations:
s_c = s_m + 10
s_c = 2s_m - 30

multiply second by -1 and sum them both:
 s_c =  s_m + 10
-s_c = -2s_m + 30

0 = -s_m + 40
s_m = 40
that is the speed of the motorcycle

 s_c =  s_m + 10
 s_c =  40 + 10
s_c = 50

that is the speed of car, both speeds in miles per hour