b. General case. Do exactly what you did above but instead of the three specific points, use k points with unknown values: ( x₀, y₀ ), ( x₁, y₁ ), . . . , ( x_k-1, y_k-1 )

i. Again, the function D(m , b) represents the sum of the squares of the residuals. Write a formula for D(m, b). It is easier now, and will be much easier in the next part, if you work with these quantities using sigma notation.

ii. Now optimize D(m, b). Take the partial derivatives with respect to each of the two variables and set the results equal to zero. This, again, will lead to two linear equations in two unknowns. Note that it is very important that we think of the ( x, y) points as constants, even though we do not know their values.

iii. Solve the two equations to the extent that they are each written in the following form: b =a fraction that involves a m,x_i, y_i ,k and preferably sigma signs. Note that all symbols may not be needed to present the equations in their required form.

iv. Use your equations from iii to find the equation of the best-fit line to the following data: (1,2), (1,3), (2,4), (2,2), (4,8), (3,5), (4,5). When you plug in the data, you should end up with two linear equations in two unknowns.

v. Manipulate your equations from iii to end up with one of the standard equations for linear regression. Take your two equations of the form b =something:E and set the two somethings equal to each other. Cross multiply and manipulate.