Linear Programming
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Formulating Linear Programs
Part of: Formulation and Feasible Regions
A shop chooses quantities x and y to maximise profit Z = ax + by under limited resources written as linear inequalities, plus non-negative restrictions. Decision variables are the choices, constraints are the limits, and the objective prices each plan.
Why it matters. Factories, farms and shops plan production through these formulations.
Formulas
- Objective Z = ax + by prices each plan
- Resources become ≤ inequalities
- Negative production is banned
Key words
- Objective function
- : The linear score Z = ax + by to optimise.
- Decision variables
- : The quantities x and y being chosen.
- Constraints
- : Linear inequalities limiting the choices.
- Non-negative restrictions
- : The rules x ≥ 0 and y ≥ 0.
Try it in two minutes
- Name the two choice quantities x and y.
- Price each unit into the objective.
- Write each limited resource as an inequality.
Try at home. Formulate max profit for two products with machine and labour limits.
Feasible Region and Corners
Part of: Formulation and Feasible Regions
All constraints overlap in the feasible region; inside and boundary points are feasible choices. Corners where two boundary lines cross host every optimum, and regions fittable inside a circle are bounded.
Why it matters. Corners shrink infinite choices into a short test list.
Formulas
- Overlap all shades for the region
- Corners host every optimum
- Bounded fits inside a circle
Key words
- Feasible region
- : The common overlap of all constraints.
- Feasible solution
- : Any point inside or on the region.
- Corner point
- : Two boundary lines crossing inside the region.
- Bounded region
- : A region fittable inside a circle.
Try it in two minutes
- Shade each inequality half plane.
- Keep the overlap of all shades.
- Dot the overlap corners.
Try at home. Test (5, 5) against x + 2y ≤ 20 and 2x + y ≤ 20.
Corner Point Method
Part of: Corner Method and Unbounded Cases
On bounded regions the largest corner value M is the maximum and the smallest m is the minimum. The method lists corners, evaluates Z at each, and crowns the extremes.
Why it matters. Three evaluations replace infinite guessing.
Formulas
- List corners, then score each
- Bounded M is max; bounded m is min
- Compare all scores before crowning
Key words
- Corner Point Method
- : Evaluating Z at corners to find optima.
- Maximum value
- : The largest corner score M.
- Minimum value
- : The smallest corner score m.
- Optimal solution
- : A corner crowning an extreme score.
Try it in two minutes
- List every corner point.
- Evaluate Z at each corner.
- Crown the largest and smallest.
Try at home. Maximise Z = 2x + 5y over the triangle (0,0), (6,0), (0,4).
Unbounded Regions and Special Cases
Part of: Corner Method and Unbounded Cases
Unbounded regions spill endlessly, so corner extremes M and m win only when the beyond half plane shares nothing with the region. Tied corners crown whole segments, and infeasible points stay outside every fence.
Why it matters. Real plans often run unbounded, and half-plane checks rescue the verdict.
Formulas
- Unbounded M needs empty beyond-M sharing
- Tied corners crown whole segments
- Outside points never win
Key words
- Unbounded region
- : A region spilling past all circles.
- Half-plane test
- : Checking beyond-M territory for sharing.
- Multiple optima
- : A tied segment of winning points.
- Infeasible solution
- : A point outside the region.
Try it in two minutes
- Spot the endless direction first.
- Score corners for M and m.
- Test the beyond half plane for sharing.
Try at home. Minimise Z = x + y over x ≥ 0, y ≥ 0, x + 2y ≥ 8.
What your child will practise
- Corner Point Method: Space Quick Fire (Quick Fire)
- Feasible Region and Corners: Candy Puzzle (Clue Puzzle)
- Feasible Region and Corners: Desert Quest (Flip and Match)
- Formulating Linear Programs: Jungle Quick Fire (Quick Fire)
- Unbounded Regions and Special Cases: Space Puzzle (Clue Puzzle)
- Quiz: Corner Point Method (8 questions)
- Quiz: Feasible Region and Corners (8 questions)
- Quiz: Formulating Linear Programs (8 questions)
- Quiz: Unbounded Regions and Special Cases (8 questions)