About the Corner Point Calculator
A corner point — also called a vertex — is the point where two straight boundary lines cross. The term comes from linear programming, where a set of linear inequalities (constraints) carves out a feasible region shaped like a polygon, and each corner of that polygon is a corner point. This calculator finds that intersection point exactly, given the two lines in standard form.
The formula: solving two lines with Cramer's rule
Write each boundary line in standard (general) form:
- Line 1: a1x + b1y = c1
- Line 2: a2x + b2y = c2
The corner point (x, y) where the two lines meet is solved with Cramer's rule for a 2×2 system:
- Determinant: D = a1b2 − a2b1
- x = (c1b2 − c2b1) / D
- y = (a1c2 − a2c1) / D
If D ≠ 0, the two lines cross at exactly one point — the corner point. If D = 0, the lines are parallel: they either never meet (no solution) or lie exactly on top of each other, sharing infinitely many points (coincident lines) if c1b2 also equals c2b1.
Why corner points matter in linear programming
The corner-point (fundamental) theorem of linear programming states that if a linear objective function has a maximum or minimum value over a feasible region, that optimum occurs at a corner point of the region. This is why the graphical method for solving small linear programs works by finding every corner point of the feasible region — using exactly the two-line intersection this calculator performs — then plugging each corner point into the objective function to see which one scores best.