## An equilibrium supply and demand question via implicit differentiation

I don't entirely remember how these concepts go; the information is muddled by a few other topics. The question is listed below, word for word.

The demand and supply for a good are given by the linear functions

$\displaystyle D=30-2p-y$, $\displaystyle S=p$,

where $\displaystyle D$ is the quantity demanded, $\displaystyle S$ is the quantity supplied, $\displaystyle p$ is the price, and $\displaystyle y$ is the aggregate consumers' income, which is exogenous. Denote the equilibrium quantity by $\displaystyle Q$ and the equilibrium price by $\displaystyle P$.

(i) Find the effect of a change in income on equilibrium price and quantity, that is, $\displaystyle \frac{dQ}{dy}$ and $\displaystyle \frac{dP}{dy}$, by the method of implicit differentiation.

(ii) Check the result by first solving for the equilibrium price and quantity explicitly and then finding $\displaystyle \frac{dQ}{dy}$ and $\displaystyle \frac{dP}{dy}$.

(iii) Repeat (i) and (ii) for the following demand and supply functions

$\displaystyle D=a-bp+cy$, $\displaystyle S=\alpha+\beta p$,

with $\displaystyle a>0, b>0, c>0, \alpha>0, \beta>0, \alpha<a+cy$.

I'm doing my best to try and remember how this all works, but some help on the subject would be appreciated.