Steps to solve the following question

59. Suppose that a scientist has reason to believe that two quantities * and y are related linearly, that is, y = mx + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (21 , 31) , (x2, 12), ... (In, Wn ), and then plots these points. The points don't lie exactly on a straight line, so the scientist wants to find constants m and b so that the line y = max + b "fits" the points as well as possible (see the figure). Letdi = yi - (max; + b) be the vertical deviation of the point (x,, y:) from the line. The method of least squares determines m and b so as to minimize >d?, the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when m ) & + bn = i=1 i=1 and m ) *+6) * = i=1 i=1 i=1 Thus the line is found by solving these two equations in the two unknowns m and b. (See Section 1.2 for a further discussion and applications of the method of least squares.)