Find the feasible region of the constraints' equations. The constraints are linear, so for convenience, this means solving systems of linear equations to find the limits and then taking everything inside as the feasible region. If the equations are simple, which they usually are, being linear, a graph can help you see where the constraints intersect. The feasible region is where all the constraints intersect. For example, if you are given the constraints "x is greater than 0," "y is greater than zero," "4x + 2y is less than 12" and "x + 2y is less than 4" then after graphing you will find the feasible region is a quadrilateral.
Find the set of possible solutions. This set is the extreme values on the feasible region's boundaries. You can find this by looking at the feasible region and finding all the vertices, the sharp points. Using the previous example, these extreme values are (0, 0), (2, 0), (3, 0) and (8/3, 4/3).
Determine the solution. The solution is the point that maximizes, or minimizes depending on the goal of the equation, the objective function. Plug in each possible solution and compare the resulting values to make this determination. For the given example, assuming the objective function is "maximize x + y," plugging in the points (0, 0), (2, 0), (3, 0) and (8/3, 4/3) yields the values 0, 2, 3 and 4, respectively. Since 4 is the largest value and it corresponds to the point (8/3, 4/3), the solution to this linear programming problem is (8/3, 4/3).