What is linear programming?

It's optimizing (maximizing or minimizing) a linear objective function subject to a set of linear constraints — inequalities or equations that restrict the feasible values of the variables.

Why does the optimum always occur at a corner point?

The feasible region of a linear program is a convex polygon (in 2D) or polytope (in higher dimensions), and a linear objective function's value changes monotonically across any straight edge — so its extreme value can never occur strictly inside an edge, only at a vertex.

What format do constraints use?

Each constraint is entered as a,b,op,c representing a·x + b·y {op} c, where op is =, or = — for example, 2,3,<=,12 means 2x + 3y ≤ 12.

Why is this solver limited to 2 variables?

The corner-point method used here relies on intersecting pairs of 2D lines directly; solving linear programs with 3 or more variables exactly requires a general-dimension algorithm like the simplex method, which is out of scope for this calculator.

What does 'infeasible' mean?

It means no point satisfies every constraint (including x ≥ 0 and y ≥ 0) simultaneously — the feasible region is empty, so there is no valid solution to optimize.

What does 'unbounded' mean?

It means the feasible region extends infinitely in a direction that keeps improving the objective, so no finite optimum exists — this calculator detects likely unboundedness with a heuristic check of far-away feasible points, not a formal simplex proof.

How many constraints can I enter?

This calculator supports up to 10 constraints, which keeps the corner-point search (checking every pair of boundary lines) comfortably fast while covering typical textbook-style problems.

Where is linear programming used in real life?

It's used in production planning, resource allocation, diet and blending optimization, transportation and logistics routing, and financial portfolio optimization.

Linear Programming Solver — Maximize or Minimize a 2-Variable Objective

Q: Why is this solver limited to 2 variables?

The corner-point method used here relies on intersecting pairs of 2D lines directly; solving linear programs with 3 or more variables exactly requires a general-dimension algorithm like the simplex method, which is out of scope for this calculator.

Q: What does 'infeasible' mean?

It means no point satisfies every constraint (including x ≥ 0 and y ≥ 0) simultaneously — the feasible region is empty, so there is no valid solution to optimize.

Q: What does 'unbounded' mean?

It means the feasible region extends infinitely in a direction that keeps improving the objective, so no finite optimum exists — this calculator detects likely unboundedness with a heuristic check of far-away feasible points, not a formal simplex proof.

Q: How many constraints can I enter?

This calculator supports up to 10 constraints, which keeps the corner-point search (checking every pair of boundary lines) comfortably fast while covering typical textbook-style problems.

Q: Where is linear programming used in real life?

It's used in production planning, resource allocation, diet and blending optimization, transportation and logistics routing, and financial portfolio optimization.

Our Linear Programming Solver maximizes or minimizes a linear objective c₁x + c₂y subject to linear constraints and x, y ≥ 0, using the exact corner-point method: intersect every pair of constraint boundary lines, keep the intersections that satisfy every constraint (the vertices of the feasible region), and evaluate the objective at each to find the optimum.

Quick Answer

Enter an objective and linear constraints to instantly find the optimal (x, y) solution using the exact corner-point method.

Objective

Constraints (one per line, a,b,op,c)

Each line means a·x + b·y {op} c, where op is <=, >=, or =, e.g. 1,0,<=,4.

How to Use the Linear Programming Solver — 2-Variable Corner-Point Method Online

1
Choose maximize or minimize, and enter the objective coefficients c1 (for x) and c2 (for y).
2
Enter constraints one per line as a,b,op,c (meaning a·x + b·y {op} c), e.g. 1,0,<=,4.
3
Click 'Calculate' to find every feasible corner point and the optimal solution.

Why Use Linear Programming Solver — 2-Variable Corner-Point Method Online?

In a 2-variable linear program, the feasible region (the set of points satisfying every constraint plus x ≥ 0, y ≥ 0) is always a polygon, and a fundamental theorem of linear programming guarantees the optimal solution always occurs at one of that polygon's corners (vertices) — never strictly inside an edge or the interior. This calculator exploits that directly: it intersects every pair of boundary lines (each constraint plus the x = 0 and y = 0 axes), discards any intersection that violates another constraint, and evaluates the objective at each surviving vertex to find the maximum or minimum. This corner-point method is exact and simple for two variables, avoiding the need to implement a general-dimension simplex method.

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