Enter your objective function and up to three constraints, and instantly get every corner point of the feasible region with the optimal one highlighted.
Linear Programming Solver
Assumes x ≥ 0 and y ≥ 0. Leave a constraint’s a and b as 0 to skip it.
Objective: Z = a·x + b·y
Constraints (a·x + b·y ≤ or ≥ value)
How It’s Calculated
The graphical method says the optimal value of a linear objective always occurs at a corner (vertex) of the feasible region — never strictly inside it. This tool finds every corner by intersecting each pair of boundary lines (the axes plus your constraints), keeps only the points that satisfy every constraint simultaneously, evaluates Z at each one, and picks whichever gives the maximum or minimum you asked for.
Worked Example
Maximize Z = 3x + 5y subject to x ≤ 4, y ≤ 6, 3x + 2y ≤ 18, x,y ≥ 0. The feasible corners are (0,0), (4,0), (4,3), (2,6) and (0,6). Evaluating Z at each: the maximum is at (2,6), giving Z = 3(2)+5(6) = 36.
Frequently Asked Questions
Q: Why does the optimal solution always sit at a corner?
Because the objective function is linear, its value changes at a constant rate across the feasible region — it can only be pushed to its extreme by moving as far as the boundary allows, which happens at a vertex where two boundaries meet.
Q: What if there’s no feasible region?
That means your constraints contradict each other — no (x,y) with x,y ≥ 0 can satisfy all of them at once. The problem has no solution as stated.
This is the full graphical method covered in 2nd PUC Linear Programming. See how Daniel Classes teaches 1st and 2nd PUC Mathematics →