Minimizing c2

Suppose we have data (xi, yi,si) and we want to fit the data to a line:  y = A + Bx.  First we calculate c2:

Then we minimize it with respect to the parameters of the theory, A and B. To find the best values of both A and B, we need to carry out this minimization simultaneously. So we take the partial derivative of c2 w.r.t. A and B, and set them both to 0 simultaneously.

We can rearrange these two equations a bit and see that there are only 2 unknowns, A and B. All other information can be determined directly from our data (xi, yi,si).