Describe the graphical method to solve a simple linear programming problem with two variables.

Describe the graphical method to solve a simple linear programming problem with two variables.

• A. Plot constraints, identify the feasible region, and evaluate the objective function at corner points
• B. Write the algebraic equations only and solve for the optimum
• C. Use calculus to find the maximum of the objective function
• D. Randomly sample points in the plane to estimate the optimum

Answer: (A)

The key idea is that with two variables you can see the whole problem on a plane. Plot each constraint as a line by turning the inequality into an equal line, then shade the side that satisfies the constraint. The intersection of all those shaded half-planes forms the feasible region, which is typically a polygon. Because the objective is a linear function, its highest value over this region occurs at one of the corner points (vertices) of that polygon. So you evaluate the objective at each vertex and pick the largest value. If the best value happens along an entire edge, any point on that edge gives the same optimum. This graphical approach gives a direct, visual way to identify where the optimum lies and why corner points matter. The other methods described—solving algebraically without considering the whole feasible region, using calculus, or randomly sampling points—either aren’t tailored to the geometry of the problem, are unnecessary for linear objectives, or don’t reliably find the true optimum.

The key idea is that with two variables you can see the whole problem on a plane. Plot each constraint as a line by turning the inequality into an equal line, then shade the side that satisfies the constraint. The intersection of all those shaded half-planes forms the feasible region, which is typically a polygon. Because the objective is a linear function, its highest value over this region occurs at one of the corner points (vertices) of that polygon. So you evaluate the objective at each vertex and pick the largest value. If the best value happens along an entire edge, any point on that edge gives the same optimum.

This graphical approach gives a direct, visual way to identify where the optimum lies and why corner points matter. The other methods described—solving algebraically without considering the whole feasible region, using calculus, or randomly sampling points—either aren’t tailored to the geometry of the problem, are unnecessary for linear objectives, or don’t reliably find the true optimum.