Graphical Method Solver
Visualize 2-variable linear programs: plot constraints, shade the feasible region, evaluate corners, and compare with Simplex.
Related solvers
Plot
Green fill = feasible region · Black = corners · Red = optimum
When to use the graphical method
The graphical method solves linear programs with exactly two decision variables by drawing constraint half-planes, finding the feasible polygon, and evaluating Z at every corner.
Corner-point theorem
Because both the objective and the constraints are linear, an optimum (if one exists) always occurs at a vertex of the feasible region. Evaluate Z = c₁x₁ + c₂x₂ at each corner and pick the best.
Special cases
Infeasible — empty intersection. Unbounded — Z improves forever along a ray. Multiple optima — an iso-profit line lies along an edge.
Link to Simplex
Each corner corresponds to a basic feasible solution. Simplex walks those corners algebraically for any number of variables; this page also runs Simplex so you can compare tableaus with the picture.
Max example → (8, 4), Z = 32
Max 3x₁ + 2x₂ s.t. 2x₁ + x₂ ≤ 20, x₁ + 2x₂ ≤ 16, x ≥ 0.
Min / ≥ example → (2, 2), Z = 10
Min 3x₁ + 2x₂ s.t. x₁ + x₂ ≥ 4, 2x₁ + x₂ ≥ 6.
Frequently Asked Questions
1. What is the main limitation of the Graphical Method?
2. Where does the optimal solution occur in a linear program?
3. What represents the feasible region?
4. How do we find the optimal point graphically?