In the analysis of the quicksort algorithm a recurrence is presented in 2.12. I feel I understand the simplification steps down to the point where we have
$$ nC_n - (n - 1)C_{n-1} = 2n + 2C_{n-1},\ \ \ \ \ \text{for}\ n > 2. $$
Then the book says
It turns out that this relation also holds when $n = 1$, because $C_1 = 2$. Therefore the original recurrence for $C_n$ reduces to a much simpler one: $$ \begin{align} C_0 &= 0 \\ nC_n &= (n + 1)C_{n-1} + 2n,\ \ \ \ \ \ \text{for}\ n > 0 \end{align} $$
I don't understand how the fact that the relation holding for $n = 1$ allows for simplification from the first equation (in this question, not the book section) to the second simpler one. How is this transformation done?