Last updated October 10, 2026 · Report an error
The corner point method for linear programming relies on a key fact: the optimal solution to a linear program always occurs at a vertex (corner point) of the feasible region. This tool finds those vertices by intersecting every pair of constraint boundary lines and keeping only the ones that satisfy every constraint.
How it works
Every pair of constraint boundary lines is solved together as a 2x2 system to find a candidate intersection point. Each candidate is then checked against every constraint (including x≥0 and y≥0 if included) — only points satisfying all of them are true corner points of the feasible region.
- Enter number of constraints.
- Enter include x ≥ 0 and y ≥ 0?.
- Enter constraint 1: a.
- Enter constraint 1: b.
- Enter constraint 1: relation.
- Enter constraint 1: right side.
- Enter constraint 2: a.
- Enter constraint 2: b.
- Enter constraint 2: relation.
- Enter constraint 2: right side.
- Enter constraint 3: a (optional).
- Enter constraint 3: b (optional).
- Enter constraint 3: relation.
- Enter constraint 3: right side (optional).
- Enter constraint 4: a (optional).
- Enter constraint 4: b (optional).
- Enter constraint 4: relation.
- Enter constraint 4: right side (optional).
- Enter constraint 5: a (optional).
- Enter constraint 5: b (optional).
- Enter constraint 5: relation.
- Enter constraint 5: right side (optional).
- Click Calculate to see your results.
Examples
A simple feasible region
With x+y≤10, 2x+y≤15, x≥0, and y≥0, the feasible region is a quadrilateral with corner points at (0,0), (7.5,0), (5,5), and (0,10).
Who should use it
- Operations research and linear programming coursework.
- Setting up the feasible region before manually testing an objective function.
Industry applications
- Operations research
- Business and economics (resource allocation problems)
Advantages
- Automatically tests every pairwise intersection against every constraint.
- Optional built-in x≥0, y≥0 non-negativity constraints.
Limitations
- Limited to 2 variables and 5 constraints (plus non-negativity).
- Does not evaluate an objective function or solve the full optimization.
Common mistakes to avoid
- Forgetting to include the non-negativity constraints (x≥0, y≥0) that are implicit in most real-world linear programming problems.
Best practices
- Sketch the constraints roughly on paper first so you have an intuition for which corner points to expect.
Tips
- Once you have the corner points, evaluating a linear objective function at each one (and comparing) always finds the optimal solution — that's the rest of the corner point method.