Corner Point Calculator - LP Optimum at Every Vertex

A corner point calculator is a linear programming tool that finds every corner point of a 2-variable feasible region and evaluates the objective at each one. You enter the objective P = p_x x + p_y y, choose maximize or minimize, and add up to 3 linear constraints. The page reports the optimal P, the (x, y) that achieves it, the corner point count, and a feasibility status.

• • Linear programming homework: Confirm a problem that asks you to maximize profit or minimize cost subject to 2 or 3 constraints.
• • Operations research classwork: Solve a small production-mix or diet-style problem by entering the coefficients and constraint rows.
• • Quick what-if analysis: Change one coefficient at a time and watch the optimal corner move as the feasible region changes.
• • Teaching the corner point method: Demonstrate why the optimum of a bounded LP lands at a corner point, and what an infeasible system looks like.

The corner point method is one of the two classic ways to solve a linear program, alongside the simplex method, and the easier of the two to teach and check by hand for 2-variable problems. Use this page to confirm a homework answer, sanity-check a production-mix problem, or pre-validate a corner point before charting it.

If you only need to solve a single 2x2 system - say, the intersection of two constraint lines - the System of Equations Calculator page handles that algebraic step on its own without the optimization layer.

The page turns each inequality into a line, solves every pair of lines as a 2x2 system, keeps only the intersections that satisfy the original constraints, then evaluates the objective at each survivor.

• p_x, p_y: Coefficients of the objective function P = p_x x + p_y y.
• a_i, b_i, c_i: Coefficients and right-hand side of constraint i in a_i x + b_i y ≤ c_i.
• sign: Inequality or equality sign: ≤, ≥, or =.
• D = a_i b_j - a_j b_i: Determinant of a pair of constraints; zero means the lines are parallel and the pair produces no corner point.
• (x, y): A candidate corner point, kept only if it satisfies every original constraint.

Each pair is solved with the 2x2 determinant formula. A zero determinant means the lines are parallel and that pair is skipped. Every surviving candidate is then checked against all constraints, including the implicit x ≥ 0 and y ≥ 0 non-negativity. If no corner point survives, the LP is reported as infeasible; otherwise the page evaluates P = p_x x + p_y y at each and picks the largest value (max) or the smallest value (min).

According to Khan Academy, a linear programming problem asks for the best value of a linear objective function over a feasible region defined by linear constraints, and the optimum always occurs at a corner point (extreme point) of that region.

If you would rather see the same 2x2 intersection solved by substitution than by the determinant, the Substitution Method Calculator page walks the algebra through both equations explicitly.

Four ideas show up in every linear programming problem and shape how the page reads the answer.

A reported corner point and a 'feasible' status mean the LP has a finite optimum at that vertex. 'Infeasible' means the constraints contradict each other. An unbounded LP is a separate outcome: the feasible region extends to infinity in the direction of improvement, so the true optimum is infinite.

If you need a refresher on evaluating a single linear expression or shading a single half-plane, the Linear Inequality Calculator page handles that single step in isolation.

Five short steps cover the standard workflow from a textbook LPP to a verified answer.

1. 1 Enter the objective function: Type p_x and p_y in the two top fields. The defaults 30 and 40 match the textbook example P = 30x + 40y.
2. 2 Choose maximize or minimize: Pick the direction that matches the problem statement. The page reports the largest P for max and the smallest P for min.
3. 3 Enter the first two constraints: For each, fill in a_i, b_i, c_i, and the inequality sign. Use the sign selector to set ≤, ≥, or = as needed.
4. 4 Add a third constraint or leave it as a placeholder: For an LP with only two real constraints, set a_3 = b_3 = 0 to make the third row a no-op. The page also adds implicit x ≥ 0 and y ≥ 0.
5. 5 Read the optimal P, x, and y: The primary output is the optimal value of P, with the (x, y) that achieves it and the corner-point count. The status line says 'feasible', 'infeasible', or 'error'.

Try the textbook LPP: p_x = 30, p_y = 40, maximize, with (2, 3, 18, ≤), (1, 1, 9, ≤), (1, 2, 16, ≤). The page reports optimal P = 270, x = 9, y = 0, with 3 corner points and a 'feasible' status.

When you want to see the feasible region plotted on the x-y plane rather than read it from a list of corner points, the Graphing Inequalities (1D) Calculator page gives you the boundary and the shaded side.

These benefits matter most when the textbook answer is a number and you need a quick, auditable way to confirm it.

• • Skip the intersection arithmetic: Solving a 2x2 system by hand is easy to get wrong on a sign flip. The page does the determinant and cross-multiplication for every pair of constraints.
• • Catch infeasible problems early: If the constraints contradict each other, the page reports 'infeasible' and zero corner points, faster than drawing a shaded region that never closes up.
• • Switch between max and min in one click: Many LPPs come in matching pairs - max profit and min cost. The same form handles both without re-entering the constraint rows.
• • See the corner point count and status at a glance: The results panel reports the optimal P, the optimal (x, y), the corner-point count, and a status, so you can spot a single-corner degenerate LP as easily as a textbook one.
• • Implicit non-negativity is always on: The page adds x ≥ 0 and y ≥ 0 to the constraint list, matching the textbook convention. For LPs that need x or y free, switch to a full LP solver.

The page is most useful as a check, not a replacement for understanding the method. Use it to confirm a homework answer or pre-validate a corner point before charting it.

If a future problem in your course moves to a curved boundary, the Graphing Quadratic Inequalities Calculator page shows how the same shading logic extends to quadratic inequalities.

The algorithm is the same in every case, but a few factors change how the result should be read.

• • The page covers the 2-variable, up-to-3-constraint case. For 3+ decision variables or 4+ user constraints, the simplex method on a full LP solver is the right next step.
• • Integers and integrality constraints are not supported. If x and y must be whole numbers, the corner point returned here is a real-number optimum and may need to be rounded.
• • Numerical tolerance is 1e-9 for the determinant and 1e-6 for corner-point deduplication.

An unbounded LP is the one outcome the page cannot fully resolve on its own. The reported 'feasible' status is correct, but the optimal P it returns is only a finite bound, not the true optimum.

According to Wolfram MathWorld, a linear program optimizes a linear objective function subject to linear equality and inequality constraints, and the fundamental theorem of linear programming states a finite optimum is attained at a vertex of the feasible polytope.

When you would rather solve the same pair of constraint lines by row reduction than by the 2x2 determinant, the Elimination Method Calculator page walks through the algebraic step on the same kind of linear system.