Suppose that you have a collection of data points $(x_i,y_i)$ and you are trying to optimize the parameters of a line $y=ax+b$ to get the best fit, where we're defining best fit to mean least squares. Then we want to minimize
$$\sum_i (ax_i+b-y_i)^2 $$
This will happen when the partial derivatives with respect to both $a$ and $b$ are zero. If we take the partial derivative with respect to $a$, we obtain
$$2\sum_i (ax_i+b-y_i)x_i=0$$
and if we take the partial with respect to $b$ we obtain
$$2\sum_i (ax_i+b-y_i)=0$$.
The equations on the board are just these rewritten.