Find corner points of a feasible region from a system of linear inequalities, evaluate an objective function at each vertex, and identify the optimal solution for linear programming problems.
The corner-point method (also called the vertex method) is the foundational technique for solving two-variable linear programming problems. The key insight behind it is the Fundamental Theorem of Linear Programming: if a linear objective function has an optimal value on a bounded feasible region, that optimum occurs at one of the region's corner points — the vertices where constraint boundary lines intersect.
This calculator automates the entire process. You enter a system of linear inequalities (one per line, using familiar notation like 2x+y<=20 or x>=0), and the tool parses each constraint into a boundary line. It then computes every pairwise intersection of those lines, checks which intersections satisfy all constraints (i.e., lie inside the feasible region), and lists the feasible corner points in a detailed evaluation table.
Next, you specify a linear objective function such as 5x+4y and choose whether to maximize or minimize. The calculator evaluates the objective at every corner point and highlights the optimal vertex. A polygon visualization draws the feasible region with labeled corners, and a bar chart compares the objective values side by side.
Whether you are a student working through homework on LP, a teacher preparing examples, or an analyst quickly prototyping a resource-allocation model, this tool saves you from tedious manual intersection and substitution. Five built-in presets — a simple triangle, a classic four-constraint LP, a minimization problem, a square, and a diamond — let you explore immediately.
Solving a linear-programming problem by hand means computing every pairwise intersection of constraint lines, checking each against all constraints, and evaluating the objective at every feasible vertex — a combinatorial workload that scales quickly with more constraints. This calculator automates the entire vertex-enumeration process, draws the feasible region, and highlights the optimum in seconds. Students verifying LP homework, teachers demonstrating the graphical method, and analysts prototyping allocation models all save significant time.
For lines a₁x + b₁y = c₁ and a₂x + b₂y = c₂: x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁) y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁) A point is feasible if aᵢx + bᵢy ≤ cᵢ for all constraints. Objective: f(x,y) = cx·x + cy·y evaluated at each feasible vertex.
Result: Maximum of 52 at (8, 3)
Corner points: (0,0), (10,0), (8, ≈5.33), (0,8). Evaluating 5x+4y: 0, 50, 52, 32. Maximum is 52 at intersection of 2x+y=20 and x+3y=24.
The theorem states that if a linear objective function has an optimal value on a bounded convex feasible region, that optimum is attained at one (or more) vertices. The proof relies on the convexity of the feasible region: any interior point can be written as a convex combination of vertices, and because the objective is linear, its value at that interior point is a weighted average of its values at the vertices — which can never exceed the maximum vertex value. This is why checking only the corners is sufficient: you are guaranteed not to miss a better solution hiding in the interior.
For two-variable problems, the graphical method is both intuitive and rigorous. First, graph each constraint as a boundary line and shade the halfplane that satisfies the inequality. The feasible region is the intersection of all shaded halfplanes. Next, identify every corner point by solving pairs of boundary-line equations simultaneously. Discard any intersection that violates another constraint. Finally, evaluate the objective function at each remaining vertex and pick the best value. This calculator automates all three steps, but understanding the geometry helps you catch errors and build intuition for larger problems.
Real-world LPs often involve hundreds or thousands of variables and constraints, far beyond what graphical methods can handle. The Simplex method (developed by George Dantzig in 1947) generalises the corner-point idea to higher dimensions: it moves along edges of the feasible polytope from vertex to adjacent vertex, always improving the objective, until it reaches the optimum. Interior-point methods offer an alternative by traversing the interior of the polytope. Both approaches rely on the same theorem — the optimum is at a vertex — making the two-variable graphical method a perfect conceptual entry point for understanding industrial-strength optimisation.
A corner point (vertex) is a point where two or more constraint boundary lines intersect and the point satisfies all constraints — it lies inside or on the boundary of the feasible region. Use this as a practical reminder before finalizing the result.
Because a linear function on a convex polygon achieves its extreme values at the vertices. This is the Fundamental Theorem of Linear Programming.
For maximization, an unbounded region in the direction of increasing objective means the problem has no finite maximum. The calculator will still list all finite corner points found.
Yes — enter as many constraints as you like, one per line. The calculator checks all pairwise intersections.
Use the form ax+by<=c or ax+by>=c. Examples: x>=0, y>=0, 2x+y<=20. Coefficients of 1 can be omitted (x instead of 1x).
No — corner points may have fractional coordinates. For integer LP, you would need branch-and-bound or cutting planes on top of this.