Identify the vertices of a feasible region for linear programming problems.
Last updated: 2026-05-24T22:58:31.624Z | By ForgeCalc Engineering
* Assumes x ≥ 0 and y ≥ 0
Enter numeric coefficients for constraints in the form ax + by ≤ c. Values must parse as numbers to participate in the computation.
In linear programming, corner points (or vertices) are the points where the boundary lines of the feasible region intersect. The feasible region is the set of all possible points that satisfy a system of linear inequalities.
The Fundamental Theorem of Linear Programming states that if an optimal solution exists, it must occur at one of these corner points. This makes identifying them the most critical step in solving optimization problems.
Constraints: x + y ≤ 10, 2x + y ≤ 15, x ≥ 0, y ≥ 0
Step 1: Intersections with axes (0,0), (10,0), (0,10), (7.5,0), (0,15)
Step 2: Intersection of lines x + y = 10 and 2x + y = 15. Subtracting gives x = 5, then y = 5. Point: (5,5)
Step 3: Test points (0,0) is valid. (7.5,0) is valid. (5,5) is valid. (0,10) is valid. (10,0) is invalid (2*10+0 > 15).
Final Answer: Corner points are (0,0), (7.5,0), (5,5), (0,10)
They are the only candidates for the maximum or minimum value of a linear objective function. This is known as the Corner Point Theorem.
If the region is unbounded, a maximum value might not exist, but a minimum value (if it exists) will still occur at a corner point.
Yes. If the objective function line is parallel to one of the constraint lines, every point on that segment (including two corner points) is an optimal solution.
A region defined by $n$ constraints can have at most $n(n-1)/2$ intersection points, though usually far fewer are actually corner points of the feasible region.
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