We are asked to maximize the objective function P=4x+5y subject to the following constraints:
The last constraint says that we are dealing in the first quadrant.
The corner method is a simple example of Dantzig's simplex method. We evaluate the objective function at the vertices (corners) of the convex polygon formed by the constraints. If the constraints do not form a polygon there may or may not be optimal points. But if the feasible region is closed then there will be both a maximum and a minimum.
We make use of the fact that any maxima or minima will occur at the vertices.
First we graph the feasible region. Each of the inequalities is graphed as a line which divides the plane into two half-planes; every solution to the inequality lies in a half-plane. The solution set of all of the inequalities is the feasible region. Every point in the feasible region satisfies all of the inequalities.
In this case we get a triangle. (See attachment.)
We find the vertices by finding where each pair of lines cross. This can be accomplished using linear combinations, substitution, Cramer's method, inverse matrices, Gaussian elimination, and others.
x+y=10 and 3x+y=12 implies x=1,y=9.
x+y=10 and -2x+3y=9 implies x=29/5 and y=21/5
3x+y=12 and -2x+3y=9 implies x=27/11,y=51/11.
We evaluate the objective function at the vertices.
Thus the maximum occurs at the point (1,9) with P=49.