Problem 27 - College Algebra Essentials

Question

In your own words, describe how to solve a linear programming problem.

Step-by-Step Solution

Answer

To solve a linear programming problem, understand the problem, formulate it into an objective function and constraints, graph the feasible region, identify the optimum solution, and interpret the results.

Step 1: Understanding the problem

The first step is to understand the problem thoroughly. The variables, the objective function (the function that needs to be optimized), and the constraints (the inequalities or equalities that the variables must satisfy) should be identified.

Step 2: Formulating the problem

The next step is to formulate the problem in terms of the objective function and the constraints. The objective can either be maximized or minimized. Linearity involves multiplying the variables by constants and summing the results. The constraints usually take the form of inequalities.

Step 3: Graphing the feasible region

The feasible region (the set of all points that satisfy all the constraints) should be graphed on a coordinate plane. It is essential to denote the feasible region either by shading it or by highlighting it in some way.

Step 4: Identifying the optimal solution

The optimal solution is the point (or points) in the feasible region at which the objective function is maximized (or minimized). It always happens at a corner point (the points at which the boundary lines intersect) of the feasible region. Consider each corner point and find the one for which the objective function has the maximum (or minimum) value.

Step 5: Interpreting the solution

The last step involves interpreting the results in the context of the original problem.