**New**: The GRAIL mission is a mission taking place around the moon that’s comprised of two separate satellites. The mission’s goal is to map out the gravitational contour of the moon by leveraging a KA (part of the microwave region of the EM spectrum) -band emitter whose sensitivity is so great it can be used to determine relative displacement of the two satellites down to approximately one micrometer (the size of a human red blood cell). As the satellites orbit the moon, displacement is monitored and the relative gravitational effects are noted; lending to an understanding of the internal makeup of the moon.

**Extended**: An ordinary least squares approach to solving the best-fit of a 2D line on a standard Cartesian coordinate system can be viewed in two discrete ways. The first, which I’ve always known, is to treat the task as an optimization problem whereby each individual point is plugged into a 1 degree polynomial as such:

In this particular system, and using the least-squares approach, we’re trying to solve for an x and b which minimize the squared error of each point’s polynomial. This means

, or to find the values m and b which minimize the residual.

This is done by computing the sum (as above), and taking the partial derivative of the sum of the polynomials with respect to each variable, so where

We compute

and

Where the optimal equation is expressed by

However, another way to view the problem is as a covariance of the x and y variables over the variance of x.