Calculator Inputs
Enter up to six linear conditions in the form ax + by relation c. Blank rows are ignored automatically.
Example Data Table
| Constraint | a | b | Relation | c | Meaning |
|---|---|---|---|---|---|
| 1 | 1 | 1 | ≤ | 12 | x + y ≤ 12 |
| 2 | 1 | 0 | ≥ | 2 | x ≥ 2 |
| 3 | 0 | 1 | ≥ | 1 | y ≥ 1 |
| 4 | 2 | 1 | ≤ | 18 | 2x + y ≤ 18 |
With objective Z = 3x + 2y, the calculator inspects corner points, calculates polygon measures, and identifies the best feasible vertex.
Formula Used
General constraint: ax + by ∘ c, where ∘ can be ≤, ≥, or =.
Boundary line: ax + by = c.
Intersection of two boundaries: solve the linear system
a₁x + b₁y = c₁ and a₂x + b₂y = c₂.
Determinant: D = a₁b₂ − a₂b₁.
Coordinates: x = (c₁b₂ − c₂b₁) / D and y = (a₁c₂ − a₂c₁) / D, when D ≠ 0.
Objective function: Z = px + qy, evaluated at each feasible vertex.
Polygon area: the shoelace formula is used after sorting feasible vertices counterclockwise.
This approach is ideal for two-variable linear inequality systems, classroom graphing checks, and small linear-programming style region studies.
How to Use This Calculator
- Enter each condition using coefficients for x and y, then choose ≤, ≥, or =.
- Set the plotting box so the visible graph covers the region you want to inspect.
- Optionally enter an objective function to test maximum or minimum corner values.
- Press Calculate Region to display the result above the form.
- Review the plot, the feasible vertices, area, perimeter, centroid, and optimal point.
- Use the download buttons to export the summary as CSV or PDF.
Frequently Asked Questions
1. What does this calculator actually find?
It plots boundary lines, tests feasible points, builds the visible feasible polygon, lists vertices, measures area and perimeter, and checks an objective function.
2. Can it solve inequalities with equalities too?
Yes. You can mix ≤, ≥, and = conditions. Equality rows act as exact boundary lines and still participate in vertex detection.
3. Why does the plotting box matter?
The graph and polygon are clipped to the chosen x and y limits. Wider limits reveal more of large or partly unbounded regions.
4. Does it work for more than two variables?
No. This page is designed for two-variable systems only, because feasible regions are graphed in the x-y plane.
5. How is the optimum point selected?
The calculator evaluates the objective function at every feasible vertex and returns the highest or lowest value, depending on your chosen mode.
6. What if no region appears?
Your conditions may be inconsistent, or the feasible area may lie outside the chosen plotting box. Try wider bounds or review the inequalities.
7. Why are some blank rows allowed?
Blank rows let you use fewer than six constraints without editing the page structure. Empty rows are ignored during calculation.
8. Can I use this for linear programming practice?
Yes. It is useful for classroom examples, corner-point method practice, and quick checks of bounded feasible sets and objective values.