Linear Programming Graph Guide
What the Graph Shows
Linear programming helps you choose the best result under limits. A graph makes the process easier to see. Each constraint becomes a line on the coordinate plane. The allowed side of each line creates a half plane. The overlap is called the feasible region. Any point inside it respects every entered limit.
Inputs and Limits
This calculator uses two decision variables, named x and y. It accepts maximize and minimize problems. You can enter several inequalities or equalities. You can also keep non negative variables on. That option is common in production, diet, shipping, and budget models. Negative units rarely make sense in those cases.
Corner Point Method
The strongest part of graphical solving is vertex testing. A linear objective changes at a steady rate. Because of this, the best finite answer occurs at a corner point. The tool finds intersections between boundary lines. Then it checks which points satisfy all constraints. It evaluates the objective at every feasible vertex. The best value is reported with its x and y values.
Interpreting the Result
Use the graph as a visual check. Parallel lines can remove possible solutions. Equalities can force the answer onto one line. A very small feasible region may still contain a valid optimum. A missing upper limit may make a maximizing problem unbounded. The message area warns when the entered model appears open in an improving direction.
Reports and Planning
The example table shows a typical resource allocation case. You can replace it with your own values. Use the CSV export for spreadsheets. Use the PDF export for reports, homework, or client notes. Always review units before trusting the answer. Coefficients should describe the same time period, product batch, or planning horizon. Clean data gives a clean graph. Clear constraints give a useful decision.
Advanced Review
Advanced users can study sensitivity by changing one number at a time. Increase a resource limit and solve again. Watch which vertex becomes best. Change an objective coefficient and compare the slope of the profit line. These small trials show which limits control the model. They also show whether extra capacity is valuable. For teaching, the graph connects algebra with geometry. For planning, it turns many limits into one visible choice. This makes the final recommendation easier to explain and audit in real work today.