Corner Point Calculator — Linear Programming Intersection
Enter the coefficients of two linear constraint equations (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) to find their corner point — the vertex where they intersect — and see both lines plotted.
x-coordinate of the intersection (corner) point
- 1
Determinant
D = 2 × 3 − 1 × 1 = 5D ≠ 0 confirms the lines intersect at a unique point. - 2
x numerator (Dₓ)
10 × 3 − 12 × 1 = 18 - 3
Corner point x
Dₓ ÷ D = 18 ÷ 5 = 3.6000 - 4
y numerator (Dᵧ)
2 × 12 − 1 × 10 = 14 - 5
Corner point y
Dᵧ ÷ D = 14 ÷ 5 = 2.8000
How does this calculator work?
To find the corner point where two linear constraints a₁x + b₁y = c₁ and a₂x + b₂y = c₂ meet, compute the determinant D = a₁b₂ − a₂b₁, then x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D. If D = 0 the lines are parallel — no unique intersection. Evaluate the objective function at each corner point to find the LP optimum.
Formula
How this is calculated
In linear programming, the optimal solution to a two-variable objective function always occurs at a corner point (vertex) of the feasible region — the polygonal area bounded by the constraint inequalities. To locate these vertices you convert each pair of boundary constraint lines to equations and solve the system. This calculator handles one pair at a time.
The solution uses Cramer's rule. Given two equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the determinant of the coefficient matrix is D = a₁b₂ − a₂b₁. If D ≠ 0 the system has a unique solution: x = (c₁b₂ − c₂b₁) / D and y = (a₁c₂ − a₂c₁) / D. This (x, y) pair is the corner point.
If D = 0 the lines are parallel (no intersection) or coincident (infinitely many solutions) — neither yields a unique corner point. After finding all corner points of the feasible region, substitute each into the objective function and compare to identify the optimum. The built-in plot shows the two constraint lines and marks their intersection so you can visually confirm the result.
Frequently asked questions
A corner point (vertex) is a point where two or more constraint boundary lines intersect at the edge of the feasible region. The fundamental theorem of linear programming guarantees that if an optimal solution exists it occurs at a corner point, so you only need to evaluate the objective function at the vertices rather than at every feasible point.
D = 0 means the coefficient matrix is singular — the two lines have the same slope (are parallel). Either they never intersect (no solution) or they are the same line (infinitely many solutions). In both cases there is no unique corner point for that pair of constraints.
Repeat this calculation for every pair of constraints that share a boundary. With n constraints there are n(n−1)/2 pairs to check. Also include the axis intercepts (set x=0 or y=0 in each constraint). Discard any point that violates any constraint — only the points satisfying all constraints simultaneously are valid corner points.
Also known as
TG we-Calculate Editorial Team. (2026). Corner Point Calculator — Linear Programming Intersection [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/corner-point-calculator
TG we-Calculate Editorial Team. "Corner Point Calculator — Linear Programming Intersection." TG we-Calculate. 2026. https://we-calculate.com/calculator/corner-point-calculator.
TG we-Calculate Editorial Team, "Corner Point Calculator — Linear Programming Intersection," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/corner-point-calculator
@misc{wecalculate_corner_point_calculator, title = {Corner Point Calculator — Linear Programming Intersection}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/corner-point-calculator}}, year = {2026}, note = {TG we-Calculate} }
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