Add Math Linear Programming Grapher

• Enter up to three constraints (x ≥ 0 and y ≥ 0 are assumed), using ≤ or ≥.
• Enter your objective function P = px + qy.
• See the shaded region, the corner points, and the optimum at a vertex.

Linear programming rewards a clean graph and testing the vertices. Seeing the region and corners built correctly makes the method click.

Linear programming is one of the more visual chapters, and it rewards two things: drawing the feasible region accurately, and remembering that the optimum of a linear objective always sits at a corner of that region. This grapher does both so you can see how they fit together, it shades the region satisfying all your constraints (including x ≥ 0 and y ≥ 0), marks every corner point, and evaluates your objective at each to report the maximum and minimum.

The most common exam mistakes are shading the wrong side of a line and forgetting to test all the vertices; seeing the correct region and corners here makes both habits stick. Note that many real problems need whole-number answers, if an optimum vertex has fractional coordinates, the exam may want the best integer point inside the region, which is a step to reason through by hand.

Use the grapher to check your own graph and vertices, then present your own labelled sketch and vertex table in the exam, which is what earns the method marks.

A quick tool is genuinely useful, but it is worth being clear about what it can and cannot do. It can check a result, illustrate an idea, or save time on planning, but the SPM exam rewards method, not answers, and marks come from working you produce yourself under exam conditions.

So use a tool the way you would use the answer at the back of a book: attempt the problem first, then use the tool to confirm your result or to see where your working went wrong. A student who leans on a tool to avoid the working tends to feel confident right up until the exam, where no tool is allowed and the method has to come from memory and practice.

Used the other way, to check and to build intuition, the same tool speeds up real learning. If you find you can get an answer from the tool but cannot yet reproduce the steps yourself, that gap is exactly what a lesson is for.