Gaussian elimination, the classic algorithm for solving systems of n linear equations in n unknowns, requires about in multiplications, which is the algorithm's basic operation.

a$.$ How much longer should you expect Gaussian elimination to work on a system of $1000$ equations versus a system of $500$ equations?

b$.$ You are considering buying a computer that is $1000$ times faster than the one you currently have. By what factor will the faster computer increase the sizes of systems solvable in the same amount of time as on the old computer?