Why do keys need both X and Y coordinates, if X can be solved for Y using the curve equation? Reference:
https://en.bitcoin.it/wiki/File:PubKeyToAddr.png
 and 
https://en.bitcoin.it/wiki/Secp256k1

Why do we need both X and Y to make a private key?
A private key is just a number modulo the order of the curve.

A public key is the (X,Y) coordinate pair corresponding to that number (the private key) multiplied by the base point (which is a property of the curve used).

If you're talking about public keys: you're almost right. The Y coordinate can indeed be computed from the X coordinate, if you know the sign (given the formula y^2 = x^3 + 7, there are two solutions for Y for every X).

In fact, if you're using a recent version of several wallet clients (bitcoind/bitcoin-qt since 0.6.0 for example), this trick is used. It's called compressed public keys, and it means that when spending a transaction output, the public key stored in the spending script (and thus the block chain) only contains the X coordinate and a marker byte to denote which of both Y coordinates is used. This is slightly slower to validate, but saves space.

In practice, public keys are encoded in the following legal ways:


0x02 + [32-byte X coordinate] (if the Y coordinate is even)
0x03 + [32-byte X coordinate] (if the Y coordinate is odd)
0x04 + [32-byte X coordinate] + [32-byte Y coordinate]


(the two solutions for Y always have different oddness, but as we're talking about a coordinate in a finite field rather than a real number, it does not actually have a 'sign')