The following table lists what we know and what we want to determine:
Since the demand curve is linear, then it has the form
and because this linear curve will be true for all q and p, then it will, in particular, hold for the point (q', p'):
where m is the slope and b is the y-intercept of the line. Also observe that
Since the supply curve S(q) is known for all q, then in particular we have
The consumer's surplus (33,000) is the area of the triangle shown shaded in the graph below
and since the area of a triangle is (base x height)/2, then the equation which results is
The producer's surplus of 132,000 is the area shown shaded in the graph below
which results in the equation
Note: Slide your mouse over the terms in equation (4) to see what each term represents in the graph above.
Now you are ready to solve Oinkle Sam's problem. Equations (1) through (4) give you 4 equations in the 4 unknowns m, b, q', and p'. After you have solved this system of equations, the equation of the linear demand curve is specified by the m and b, and the equilibrium point is specified by the q' and p'.