Corner Point Calculator

Find corner points (vertices) of a feasible region from linear inequalities by computing intersection points.

Enter Linear Equations (ax + by = c)

Equation 1
Equation 2
Equation 3 (optional)
Set all coefficients to 0 to use only 2 equations. Format: ax + by = c

Result

Corner Points (Vertices)
(4, 6), (2, 8), (6, 2)
3 intersection points found

Step-by-Step Solution

Solve pairs of equations simultaneously

Understanding Corner Points

Corner points (also called vertices) are the intersection points of the boundary lines of a feasible region in linear programming. These points are critical because, according to the fundamental theorem of linear programming, if an optimal solution exists, it occurs at one of the corner points of the feasible region.

A feasible region is defined by a system of linear inequalities. The corner points are found by solving pairs of boundary equations (the equality forms of the inequalities) simultaneously.

Key Concepts

2x2 System of Equations

Solve two linear equations to find their intersection point.

a1*x + b1*y = c1, a2*x + b2*y = c2

Cramer's Rule

Use determinants to solve systems of linear equations.

x = (c1*b2 - c2*b1) / D, D = a1*b2 - a2*b1

Feasible Region

The set of all points satisfying all constraints simultaneously.

Region bounded by all inequalities

Parallel Lines

When determinant D = 0, lines are parallel (no intersection).

D = a1*b2 - a2*b1 = 0

Objective Function

Evaluated at each corner point to find the optimum.

Z = cx + dy (maximize or minimize)

Number of Vertices

For n lines, at most C(n,2) intersection points exist.

Max vertices = n! / (2!(n-2)!)

How to Find Corner Points

  1. Write boundary equations: Convert each inequality to an equation (replace the inequality sign with =).
  2. Pair the equations: Take every possible pair of equations from your system.
  3. Solve each pair: Use substitution, elimination, or Cramer's rule to find the intersection point.
  4. Check feasibility: Verify each intersection point satisfies all original inequalities.
  5. List corner points: The feasible intersection points are the corner points of the feasible region.

Example: x + y = 10, 2x + y = 14

Subtract equation 1 from equation 2: (2x + y) - (x + y) = 14 - 10, which gives x = 4. Substituting back: 4 + y = 10, so y = 6. The intersection point is (4, 6).

Applications of Corner Points

  • Linear programming: Finding optimal solutions for resource allocation, production planning, and scheduling.
  • Operations research: Solving transportation, assignment, and network flow problems.
  • Economics: Maximizing profit or minimizing cost subject to constraints.
  • Engineering: Design optimization within physical constraints.
  • Portfolio optimization: Allocating investments to maximize returns within risk constraints.

Tips

  • Always include non-negativity constraints (x >= 0, y >= 0) when applicable.
  • If two lines are parallel (same slope, different intercept), they have no intersection point.
  • An unbounded feasible region may have infinitely many solutions or no optimal solution.
  • Check that corner points satisfy all original inequalities to ensure they are in the feasible region.