Calculator Input
Example Data Table
| Constraint | Input | Boundary Line | Meaning |
|---|---|---|---|
| C1 | x + y ≤ 10 | x + y = 10 | Total combined units cannot exceed 10. |
| C2 | x ≤ 8 | x = 8 | The x variable has an upper limit. |
| C3 | y ≤ 6 | y = 6 | The y variable has an upper limit. |
| C4 | x ≥ 0, y ≥ 0 | Coordinate axes | Negative values are not allowed. |
Formula Used
Each inequality is treated as a boundary equation:
ax + by = c
Two boundary lines are intersected by solving:
a1x + b1y = c1
a2x + b2y = c2
The determinant method is used:
D = a1b2 - a2b1
x = (c1b2 - c2b1) / D
y = (a1c2 - a2c1) / D
Every intersection is tested in all inequalities. A point is feasible only when every active constraint is satisfied. If an objective function is entered, the calculator evaluates:
Z = px + qy
For a bounded linear region, the best objective value occurs at a feasible vertex.
How to Use This Calculator
- Enter each inequality in the form
ax + by ≤ c,ax + by ≥ c, orax + by = c. - Turn on only the constraints you want to use.
- Keep
x ≥ 0andy ≥ 0checked when variables cannot be negative. - Enter an optional objective function for maximization or minimization.
- Adjust graph limits when your region is outside the default view.
- Press the submit button to see the graph, vertices, and objective values.
- Use the CSV or PDF button to save the result.
Understanding Feasible Region Graphs
Why Feasible Regions Matter
A graph feasible region calculator helps students solve linear inequality systems with more confidence. It turns each restriction into a boundary line. Then it tests the half plane allowed by that line. The overlap becomes the feasible region. This page also lists corner points. Those points are important in linear programming.
Cleaner Workflows
Manual graphing is useful, but it can be slow. Small arithmetic mistakes can change the final shape. This tool reduces that risk. You can enter coefficients for x and y, choose the inequality sign, and set the right side value. You can also add non-negative restrictions. These are common in business, production, diet, transport, and resource planning questions.
Corner Point Method
The calculator finds intersections from every pair of boundary lines. Each intersection is checked against all active restrictions. Passing points are feasible vertices. If an objective function is entered, the tool evaluates it at those vertices. The best listed point is then shown. When the region is unbounded, the page warns you. This matters because some maximum or minimum values may not exist.
Visual and Exported Output
The graph gives a visual check. Boundary lines are drawn on the selected viewing window. Bounded polygon regions are shaded when enough vertices are available. Equalities, less-than restrictions, and greater-than restrictions can be mixed. The table gives exact numerical output for further work. You can export the result as a CSV file. You can also create a simple PDF report.
Better Learning Habits
Use this calculator as a learning aid, not as a replacement for understanding. Always read each inequality carefully. Check whether units match. Make sure signs face the correct direction. For exams, write the boundary equations first. Then shade the correct side. Finally, test the corner points. This process builds a strong link between algebra and geometry.
Real Planning Uses
Feasible regions also support real decisions. A factory may limit labor, material, and machine time. A school project may limit budget and hours. A diet model may limit calories and nutrients. In each case, the feasible region shows every allowed combination. The best choice usually sits on a vertex, so accurate vertex detection is very valuable.
Clear visuals make constraint errors easier to spot before final answers are submitted during homework checks.
FAQs
1. What is a feasible region?
A feasible region is the set of all points satisfying every active inequality. It is the overlap of accepted half planes from all boundary lines.
2. Can this calculator handle greater-than inequalities?
Yes. You can choose less-than, greater-than, or equality signs. The calculator tests each intersection against the selected direction.
3. Why are vertices important?
Vertices are corner points of the feasible region. In linear programming, maximum or minimum objective values usually occur at these corners.
4. What does unbounded mean?
Unbounded means the region continues forever in at least one direction. A maximum or minimum may not exist for some objective functions.
5. Should I keep x and y non-negative?
Keep them checked when negative values make no sense. Examples include products, people, hours, money, ingredients, and distance.
6. Why is my graph blank?
Your graph limits may not include the important lines or vertices. Expand the x and y ranges, then submit the form again.
7. Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report of the current result.
8. Is this useful for linear programming?
Yes. It supports constraint graphing, vertex detection, and objective testing. These are key steps in two-variable linear programming problems.