From this information-
a + c_2_ ---> ac + c Slow
ac + a ---> a_2_c Fast
2a + c_2_ ---> a_2_c + c Overall
-you know that it is 1st order in a, 1st in c2, and 2nd overall.
how? what about a reaction being fast or slow tells you if it's in first or second order?
The slow step is the rate determining step; therefore, the slow step is the one in which we get the order of the reaction.
Since the coefficient for a in the slow reaction is 1, the reaction is first order with respect to a. The coefficient for C2 is 1, also; therefore, the reaction is first order with respect to C2 also. Thus rate = k(a)^x(c2)^y and we know x = 1 and y =1 because they have been experimentally determined in the slow step.