Linear Programming Class 12

Linear Programming applies linear inequalities to optimisation under constraints. Students formulate decision variables, objective functions and restrictions, graph feasible regions and find maximum or minimum values at corner points. Teachoo provides NCERT solutions, examples and concept-wise practice for graphical methods and practical resource-allocation problems.

Formulating a linear programming problem

An LPP contains decision variables, a linear objective function and linear constraints. If x and y represent quantities, non-negativity conditions x≥0 and y≥0 are usually required. Words such as “at most,” “at least,” “available,” “required,” “profit” and “cost” determine inequality directions and the objective.

Each constraint should come from one independent resource or condition. Units must match before coefficients are written. The objective may be maximised, such as profit, or minimised, such as cost.

Graphical feasible region

Replace each inequality by its boundary line, draw the line and test a point to select the correct half-plane. The common intersection is the feasible region. It can be bounded, unbounded or empty. Corner points are found by solving boundary equations, not by estimating from the drawing.

The corner-point method evaluates the objective at every feasible vertex. If a unique greatest or least value occurs, that vertex gives the optimum. If two adjacent corner points have the same optimum, every feasible point on the connecting segment is also optimal. An unbounded region does not automatically mean the objective is unbounded; the objective direction must be analysed.

Topics and resources on Teachoo

• NCERT exercises and worked examples;

• decision variables and objective functions;

• translation of verbal constraints;

• non-negativity conditions;

• graphical solution of inequalities;

• feasible, infeasible, bounded and unbounded regions;

• corner-point optimisation;

• multiple optimal solutions;

• production, diet and allocation applications;

• board and case-based questions.

Learning outcomes

Students should be able to formulate an LPP, graph all constraints accurately and identify the feasible region and vertices. They should evaluate the objective, interpret the optimum and diagnose infeasible, unbounded or multiple-solution cases.

Board and competency preparation

Define variables with units. Build a small table translating each resource into a constraint. Label every boundary and shade lightly so the common region stays visible. Calculate exact intersections algebraically and present objective values in a table before stating the contextual conclusion.

Common mistakes to avoid

Do not reverse “at least” and “at most.” Include non-negativity unless the problem states otherwise. A corner outside one constraint is not feasible. Do not read approximate coordinates from the graph when exact equations are available. An unbounded feasible region may still have a finite optimum.

Deeper reasoning and concept connections

Study Linear Programming through comparison and justification. Place two related examples side by side, identify the decisive difference and explain why one method works in each case. Then create a new example and a deliberate non-example. This forces the definition to do real work and exposes gaps that passive reading hides.

Students should also practise reversing questions. After solving for an answer, ask what question could have produced it, whether more than one answer is possible and which extra condition would make the result unique. Reverse reasoning develops flexibility and is especially useful for missing-value, assertion–reason and error-analysis questions. The goal is to understand the network of ideas, not merely the order of a textbook solution.

How to solve unfamiliar and competency-based questions

Begin by separating facts from conclusions. Facts are given by the question or a known property; conclusions must be derived. Draw or rewrite the problem so each fact has a visible place. If several methods are possible, prefer the one with fewer assumptions and an easy final check. Record intermediate results rather than doing everything mentally.

Competency questions often change context without changing mathematics. Replace names and story details with variables, shapes, sets or data values. After solving, restore the context and check feasibility: counts should be whole where required, lengths and areas should have suitable units, probabilities should lie between 0 and 1, and constructed figures should satisfy every stated condition.

What complete mastery looks like

For Linear Programming, a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.

Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.

Additional frequently asked questions

What should a student know before starting Linear Programming?

Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.

How can a student check an answer in Linear Programming?

Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.

How many questions are enough for strong preparation?

There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.

How should Teachoo solutions be used without becoming dependent on them?

Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.

Frequently asked questions

What is a feasible region?

It is the set of points satisfying every constraint simultaneously.

Why are corner points tested?

For a linear objective over a polygonal feasible region, an attainable optimum occurs at a corner point, possibly along an entire boundary segment.

Can a linear programming problem have no solution?

Yes. If the constraints have no common point, the problem is infeasible.