Corner Point Calculator
Our free linear algebra calculator solves corner point problems. Get worked examples, visual aids, and downloadable results.
Calculator
Adjust values & calculateAll Corner Points (4)
Formula
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.
Last reviewed: December 2025
Worked Examples
Example 1: Maximize Profit with Resource Constraints
Example 2: Minimize Cost Problem
Background & Theory
The Corner Point Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Corner Point Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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.
What is the difference between a corner point and a basic feasible solution?
In theory, corner points and basic feasible solutions are equivalent concepts viewed from different perspectives. A corner point is a geometric concept describing a vertex of the feasible region polygon. A basic feasible solution is an algebraic concept from the simplex method, where exactly as many constraints are active (satisfied as equalities) as there are variables. Every corner point corresponds to a basic feasible solution and vice versa. The simplex method moves from one basic feasible solution to an adjacent one, which geometrically corresponds to moving along an edge from one corner point to the next. This equivalence is fundamental to why the simplex method works.
How many corner points can a feasible region have?
For a two-dimensional linear programming problem with n constraints (including non-negativity), the maximum number of corner points is n-choose-2, which equals n(n-1)/2. However, many intersection points will be infeasible, so the actual number of corner points is usually much smaller. For three user-defined constraints plus two non-negativity constraints (5 total), there are at most 10 possible intersection points but typically 4-6 feasible corner points. In higher dimensions, the number of potential vertices grows combinatorially, which is one reason the simplex method is preferred over exhaustive enumeration for large problems.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy