We are asked to maximize the objective function P=4x+5y subject to the following constraints:

`x+y<=10`
`3x+y>=12`
`-2x+3y>=9`
`x,y>=0`

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.

P(1,9)=4(1)+5(9)=49
P(27/11,51/11)=33
P(21/5,29/5)=45.8

Thus the maximum occurs at the point (1,9) with P=49.

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