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You want to get the equations to match the given rate. So if you write the rate law for the second step, you'll notice that there is an intermediate in the rate (in this case, N2O2), and you do not want an intermediate in the final rate law. As a result, you look at the first step and write the rate law of the reverse reaction is equal to the forward rate, and divide to get [N2O2] by itself. You then plug that into the first rate law you wrote for step 2 and you'll notice that it matches the given rate law if k= k2 x (k1/k'1). Because this matches the given, you know that the second step has to be the slow step, since the slow step matches the given rate law.
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