If you have a set ofnpoints, you can find a polynomial of degreenor less that interpolates them using systems of equations. But Lagrange interpolation can get the job done much faster.

My question is this: Is there a way to do the first part in reverse? I.e., can systems of equations be transformed into corresponding ordered pairs in a relatively hassle-free manner, interpolated, and then the resulting coefficients become the solutions?

Example: Let's say I take the following systems of equations: a + 2b + 3c = 6, 2a - 2b + 4c = 4, and 3a - 5b + c = -1. Could I put them into a 3X4 matrix, turn it into ordered pairs that all fit the function y = ax^2 + bx + c (this is the critical part), interpolate to get that function, and then read out what a, b, and c are? Or would that take too many steps?