2026-09-29 9 min

Linear Programming Calculator

Free linear programming calculator: maximize px + qy subject to ax + by ≤ c and dx + ey ≤ f (x, y ≥ 0) with corner-point enumeration and feasible-region graph.

Calculator

Objective and Constraints

Profit per x (p)

Profit per y (q)

Constraint 1: a

Constraint 1: b

Constraint 1: limit c

Constraint 2: d

Constraint 2: e

Constraint 2: limit f

Results update live as you type

Status:max 3x + 5yOPT (16, 42) = 258

Solution Summary

Objectivemax 3x + 5y

Constraints1x+2y≤100; 3x+1y≤90

Optimum(16, 42) = 258

Results

Optimal

Maximum profit

258

At x = 16, y = 42

Optimal mix

(16, 42)

Produce this much x and y

Feasible corners

(0,0)=0 | (0,50)=250 | (30,0)=90 | (16,42)=258

Every vertex evaluated

Line intersection

(16, 42)

Where both constraints bind

Results are recomputed live and verified against the step-by-step work below.

Visualizations

Corner values (feasible)

0C1

250C2

90C3

258C4

Formula

Linear objectives grow steadily, so the best feasible point is always pushed into a corner of the polygon.

max px + qy s.t. ax + by ≤ c, dx + ey ≤ f, x,y ≥ 0

Alternative Form

Optimum sits at a corner: enumerate every feasible vertex and keep the best

Variables

Symbol	Name	Unit	Description
p, q	Objective coeffs	numbers	Profit per unit of x and y
a..c	Constraint 1	numbers	First resource limit
d..f	Constraint 2	numbers	Second resource limit
corner	Optimum corner	point	Feasible vertex with top value

Explanation

List every candidate corner (axis intercepts plus the line intersection), drop infeasible ones, and evaluate the objective at the rest.

Derivation

Feasible sets of linear inequalities are convex polygons; a linear function over a polygon attains its extremes at vertices.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the objective and constraints.

max px + qy

obj:3x + 5y

Intermediate: 1x + 2y ≤ 100; 3x + 1y ≤ 90

Result: Model stated

Step 2

Collect candidate corners: origin, axis intercepts, line crossing.

Corners

n:6

Result: 6 candidates

Step 3

Drop candidates violating any constraint or negativity.

Feasibility

feas:4

Result: Feasible set isolated

Step 4

Evaluate px + qy at each feasible corner.

Scores

best:258

Result: Best = 258

Step 5

Report the winning mix and profit.

Optimum

mix:(16, 42)

Result: Profit 258 at (16, 42)

Final Answer

258

Worked Examples

See how the calculator works with different scenarios.

Scenario

Two resource caps.

Inputs

p:3

q:5

a:1

b:2

cc:100

d:3

e:1

f:90

Calculation

Corners include (0,0), (30,0), (0,50), crossing (16,42)

Result

OPT (16, 42) = 258

Interpretation

The crossing binds both resources and wins.

Understanding Your Result

What This Means

The headline result is the exact answer for the linear programming calculator inputs you entered, with supporting values shown beside it.

Why It Matters

Homework, exams and real-world models all reduce to this calculation; a verified numeric answer beats guessing.

Professional Interpretation

Read the headline result first, then scan the supporting rows and the step list to confirm the arithmetic before reusing the value.

Recommendations

•Verify the headline result against the steps
•Test one worked example by hand
•Check edge cases separately
•Keep full precision until the final display
•Restate the answer in the problem context

Educational Content

Introduction

This linear programming calculator evaluates your inputs with closed-form arithmetic and shows every intermediate value.

Concept Explanation

List every candidate corner (axis intercepts plus the line intersection), drop infeasible ones, and evaluate the objective at the rest.

Applications

▸Homework and exams

▸Engineering checks

▸Finance models

▸Science labs

▸Programming logic

▸Everyday estimation

Benefits

•Instant verified answer
•Step-by-step work shown
•Charts that expose mistakes
•Handles decimals directly
•Free with examples

Limitations

•Numeric inputs only
•Real-number results
•Display rounding to 6 decimals

Best Practices

•Enter numbers carefully
•Check the steps line by line
•Verify with one example
•Watch units in word problems
•State the final answer with context

Common Misconceptions

Myth: One input is enough.

Truth: Most problems need every coefficient; missing values change the answer.

Myth: Rounding early is fine.

Truth: Keep full precision until the final display to avoid drift.

Myth: The chart is decoration.

Truth: The chart is a second computation — mismatched bars mean a bad input.

Myth: Any rearrangement works.

Truth: Only equivalent operations preserve equality; follow the steps.

Practical Tips

•Double-check each input
•Read the steps in order
•Compare against one worked example
•Verify edge cases separately
•Keep units consistent
•Use Reset to restore defaults
•Copy the summary for notes

Common Mistakes

Typing a value in the wrong field

Match each label to the formula variables before calculating.

Dividing by zero

Check denominators and leading coefficients first.

Ignoring the steps

The step list shows exactly where a sign or arithmetic slip hides.

Rounding mid-calculation

Round only the displayed answer.

Forgetting to verify

Substitute the answer back into the original statement.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers: Inputs and outputs are real numbers.

Exact inputs: Coefficients are taken as given.

Standard notation: Conventional symbols and operator precedence.

Display rounding: Six decimals in outputs.

Limitations

Be aware of these limitations when using this calculator:

Numeric scope: Symbolic parameters need the advanced pages.

Single scenario: One input set per calculation.

No units conversion: Convert units before entering values.

Frequently Asked Questions

Answers to common questions about this calculator.

Why check only corners?

A linear objective over a convex polygon always peaks at a vertex, so interior points can never beat the best corner.

What are the candidate corners?

The origin, each axis intercept of both lines, and the crossing of the two lines — then infeasible ones are dropped.

What if the lines are parallel?

Then there is no crossing; the optimum sits at one of the intercepts.

How are ties handled?

The first best corner wins; every tied corner shares the same optimal value along that edge.

What does infeasible mean?

No (x, y) satisfies all constraints at once — the model needs looser caps.

How do I verify?

Plug the reported mix into both constraints and confirm each holds with room to spare or exact binding.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Showing your steps

Write one operation per line.

Checking answers

Substitute back every time.

Reading interval notation

Brackets include, parentheses exclude.

Word-problem setup

Name variables before computing.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Solution set: All values satisfying the given statement.
Interval notation: Compact bracket notation for ranges of the number line.
Critical value: A point where the expression is zero or undefined, splitting sign regions.
Verification: Substituting the answer back to confirm the original statement.
Closed form: An exact formula evaluated directly, not an approximation loop.
Edge case: Inputs where the usual rule breaks, such as zero denominators.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Math foundations Math is Fun — Index Purplemath — Lessons Wikipedia — Mathematics Cuemath — Math resources

Applicable Standards

Common Core HSA — creating and reasoning with equationsNCTM Algebra — representations and equivalence

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.