Corner Point Calculator
Our free linear algebra calculator solves corner point problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Solve pairs of constraint equations: a1x + b1y = c1 and a2x + b2y = c2 using Cramers rule
Each corner point is found by solving two constraint equations simultaneously. The intersection point (x, y) is computed as x = (c1*b2 - c2*b1)/(a1*b2 - a2*b1) and y = (a1*c2 - a2*c1)/(a1*b2 - a2*b1). Only feasible points satisfying all constraints and non-negativity are kept.
Worked Examples
Example 1: Maximize Profit with Resource Constraints
Problem:Maximize Z = 3x + 5y subject to: x + y <= 10, 2x + y <= 14, x + 3y <= 18, x >= 0, y >= 0
Solution:Find all constraint intersections:\n(1) and (2): x + y = 10, 2x + y = 14 gives x = 4, y = 6 -> Z = 42\n(1) and (3): x + y = 10, x + 3y = 18 gives x = 6, y = 4 -> Z = 38\n(2) and (3): 2x + y = 14, x + 3y = 18 gives x = 4.8, y = 4.4 -> Z = 36.4\nAxes intersections: (0,0)->Z=0, (7,0)->Z=21, (0,6)->Z=30, (10,0)->infeasible\nCheck all for feasibility
Result:Maximum Z = 42 at corner point (4, 6)
Example 2: Minimize Cost Problem
Problem:Minimize Z = 2x + 3y subject to: x + y <= 10, 2x + y <= 14, x + 3y <= 18, x >= 0, y >= 0
Solution:Use same corner points as above:\n(0, 0): Z = 0\n(4, 6): Z = 8 + 18 = 26\n(6, 4): Z = 12 + 12 = 24\n(7, 0): Z = 14\n(0, 6): Z = 18\nMinimum is at the origin.
Result:Minimum Z = 0 at corner point (0, 0)
Frequently Asked Questions
What is a corner point in linear programming?
A corner point (also called a vertex or extreme point) is a point where two or more constraint boundaries intersect within the feasible region of a linear programming problem. The feasible region is formed by the intersection of all constraint inequalities, and it forms a convex polygon in two dimensions. The Fundamental Theorem of Linear Programming states that if an optimal solution exists, it must occur at a corner point. This dramatically simplifies optimization because instead of checking infinitely many feasible points, you only need to evaluate the objective function at the finite set of corner points.
How do you find corner points of a feasible region?
To find corner points, you solve every pair of constraint equations simultaneously as a system of two linear equations. For each pair, compute the intersection point using methods like substitution, elimination, or Cramers rule. Then check whether each intersection point satisfies all other constraints (feasibility check) and non-negativity requirements. Points that fail any constraint are discarded. The remaining feasible intersection points are the corner points of the feasible region. For n constraints in two variables (including non-negativity), there are at most n-choose-2 possible intersection points to check.
Why does the optimal solution occur at a corner point?
The optimal solution occurs at a corner point because the objective function is linear and the feasible region is a convex polygon. A linear function over a convex set achieves its maximum and minimum values at extreme points (vertices). Geometrically, imagine sliding a straight line (level curve of the objective function) across the feasible region. The last point the line touches as it exits the region must be a vertex. If the objective function line is parallel to a constraint boundary, the optimum may occur along an entire edge, but it still includes the corner points at both ends of that edge.
How does the objective function affect which corner point is optimal?
The objective function determines which corner point is optimal by assigning a value to each corner point. Different objective functions typically select different corner points as optimal. For maximization, the corner point with the highest objective value is chosen. For minimization, the lowest value is selected. Changing the objective function coefficients changes the slope of the objective function line, which may cause a different corner point to become optimal. Sensitivity analysis examines how much the coefficients can change before the optimal corner point switches, which is important for understanding the robustness of the solution.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy