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Linear programming involving two variables

Solving a Linear Programming Problem
Many applications in business and economics involve a process called optimization, which is used to find such quantities as minimum cost, maximum profit, and minimum use of resources. In this section, you will study an optimization strategy called linear programming.
A two-dimensional linear programming problem consists of a linear objective function and a system of linear inequalities called constraints. The objective function gives the quantity that is to be maximized (or minimized), and the constraints determine the set of feasible solutions [1].
A linear programming problem may be defined as the problem of maximizing or minimizing a linear function subject to system of linear constraints. The constraints.
may be equalities or inequalities. The linear function is called the objective function, of the form . The solution set of the system of inequalities is the set of possible or feasible solution, which are of the form .
If a linear programming problem can be optimized, an optimal value will occur at one of the vertices of the region representing the set of feasible solutions.

Figure 2.1.
For example,

T, the maximum or minimum value of:

f (x, y) = ax +by + cf (x, y)= ax +by +c , over
the set of feasible solutions graphed occurs at point: A, B, C, D, E or F.

For the function

⇒ Objective function
subject to the set of constraints that determines the region in Figure 2.1., every point in the region satisfies each constraint. So, it is not clear how to go about finding the point that yields a maximum or minimum value of z. Fortunately, it can be shown that when there is an optimal solution, it must occur at one of the vertices of the region. This means that you can find the optimal value by testing z at each of the vertices