Why in this 21st century, solving Ax=b fast is still of importance?

The reasons are simple. Firstly I hope you know that Ax=b is not just popping out from the usually `boring' linear algebra lectures. It comes mainly from solving other hard equations such as differential or integral equations. Secondly high resolutions demanded by computer solution of practical problems often require more than N0=1,000,000 variables (say in oceanography or even in digital image restoration, N=10*N0 is quite normal). Do you know that even with the fastest computer getting the solution of a linear system of such a large size may take at least 10 years (depending if the matrix has any convenient zeros that can be avoided in calculations)?

Thus the practical answer to the question is simply: how many 10 years does an average researcher have in his or her life time?

Do we wait for existing mathematical methods to get the answer for us or get our hands dirty in inventing fast methods?

Computational mathematicians are doing a lot of the latter.