Optimisation and the linearity that makes this class tractable, and how a stated situation becomes decision variables, constraints and an objective

What to assume they know

Linear inequalities in two unknowns, solved graphically, from Class XI
Turning a stated condition into an inequality, with the sign pointing the way the words point
Forming and simplifying a linear expression in two unknowns
Dividing an inequality through by a positive number without disturbing it
A straight line from its equation, and the two points where it meets the axes
Systems of linear equations, and solving two of them together
Reading a point as an ordered pair, and the four quadrants
Reading a rate — rupees per item, spaces per item — as a multiplier
The idea of a function of two variables, at the level of substituting a pair and getting a number
Arithmetic with rupees, and percentages of nothing more than simple profit

What they should be able to do

Name the larger family a linear programming problem belongs to, and say what membership of that family requires
State the two extra demands that cut the family down to the class this chapter can solve
Identify, in a stated situation, which quantity is being pushed up or down and which quantities the chooser actually controls
Introduce a letter for each unknown quantity and state, in a sentence, exactly what that letter counts
Turn each stated limit into one inequality, with the correct direction of the inequality sign
Simplify a constraint by dividing through by a common positive factor, and say why the purchases it allows are unchanged
Check that the two sides of a constraint carry the same unit before believing it
Assemble the objective from the rates that pay, not from the rates that cost, and say which letters in it are fixed by the problem and which the solver sets
Distinguish a constraint from the objective, and explain why the two are never interchangeable
Explain why the non-negative conditions are constraints and not a separate kind of thing, and why the chapter still gives them a name of their own
Set the finished problem out in the chapter's own layout — an instruction to maximise or minimise, then the linking phrase, then the list
Read the finished formulation back against the original paragraph and account for every number in it
Read the chapter's formal definition clause by clause and match each clause to a part of the running example
Say what the word programming meant when the subject was named, and what it does not mean
Place the subject's origin in the nineteen-forties and name the method this chapter does not teach
State the scope of what follows — two variables, a graph, and two results that arrive unproved — and recognise that the chapter supplies exactly one worked formulation and no practice items at all

Where it usually goes wrong

"Programming means writing a computer program." It does not, and the chapter says so in the same sentence in which it glosses linear. The word means drawing up a plan of action, and it predates the machines. Left unaddressed this misreading survives the whole chapter.
"Linear just means the answer comes out neat." Linear is a condition on the problem, checked before anything is drawn: every constraint a straight line, the objective a straight-line expression, no squares, no products of the two variables, no rate that changes with quantity. If it fails, nothing in this chapter applies.
"Every optimisation problem is a linear programming problem." The chapter is explicit that this is one class among many; it says outright that the class is special. Any problem asking for a largest or smallest is an optimisation problem; only the ones that pass the linearity check are in this chapter.
"A letter is a letter; I do not need to say what it counts." Every later check depends on the meaning. Without it the coefficient of one in the shelf constraint cannot be justified, the non-negative conditions cannot be motivated, and the final pair of numbers cannot be read back as a purchase.
"Dividing the constraint by five hundred changes the problem." It does not, because five hundred is positive. Say this in one line, because a student who distrusts the step will keep the large numbers and mis-plot the line's intercepts.
"Both limits are about money." One is money and one is space. They happen to bind at different places, which is exactly why the answer is a mixture rather than all of one item. If a student treats them as the same kind of limit, the shelf constraint looks redundant.
"The inequality signs are obvious from the words." The words of this one paragraph give caps, so every limit in it points the same way; at least would point the other. Verified by reading all ten printed items: five of the ten items of the chapter's own exercise carry at least one limit pointing the other way — three of the five state every one of their limits that way and two mix the directions in a single item. A student who has watched only this formulation will assume caps are the only kind there is.
"The objective function is one of the constraints." It is the one expression that is not constrained — it has no inequality sign attached and it is not required to be anything in particular. Students who list it among the inequalities shade a half plane for it and get a smaller region than the problem has.
"The constants in the objective are the costs." They are built from whatever the question asks to be made large or small, which here is profit. In the running example they are the two profit figures, and the costs live in a constraint instead. A student who swaps them gets a legitimate-looking graph and a wrong answer, and nothing on the page catches it. This is the most common formulation error in the whole chapter and it is invisible once the graph is drawn.
"Non-negativity is a technicality you can leave out." It is a pair of constraints like any other, it does real work on the drawing — it is what confines the region to one quadrant — and the chapter gives it a name of its own. Drop it and half the corner points disappear.
"The non-negative conditions come from the dealer." They come from what the letters count. Nothing in the paragraph forbids a negative table; arithmetic does.
"Guessing three plans was a waste of time." It is what establishes that the answer is not at an extreme — neither all tables nor all chairs wins — and it is why the rest of the chapter is needed.
"Since the chapter's examples are already in symbols, formulating is the easy part." It is the part the chapter does not drill at all. Every item on the page starts after this step is finished. A student sitting an examination that asks for a formulation has practised it exactly once, by watching.
"There is only one method, since only one is taught." The chapter names the simplex method in its Historical Note and describes it as the efficient one, and it says in its introduction that other methods exist. What is on offer here is the drawing, and the drawing works only in two variables.
"The Historical Note's diet and shipping problems are exercises later in the chapter." They are not. They appear once, as history, and this edition contains no worked problem of either kind. See Notes.

Questions to check understanding

Given a stated situation, say whether it is an optimisation problem and, if so, whether it is a linear programming problem, and justify each answer separately
Pick out the objective and the constraints from a paragraph, without solving anything
Name a change to the running example that would break linearity, and say which clause of the definition it breaks
Given a stated situation with two items and two limits, introduce two letters and write a sentence saying what each counts
Turn a stated limit into an inequality and justify the direction of the sign
Simplify a constraint by dividing through, and state why the simplification is legitimate
Write the objective for a stated situation, say which of the given rates you used and which you deliberately did not, and say which numbers in it are fixed by the problem and which are chosen
Set a finished formulation out in the chapter's layout, with the linking phrase in place
Given a formulation and the paragraph it came from, find the one number that was mis-assigned
Check a given constraint by units and say whether it can possibly be right
Explain why the two non-negative conditions are constraints, and say what the drawing would look like without them
State the four clauses of the definition and give the running example's match for each
Say what the word programming refers to in the name of the subject
Place the naming of the simplex method in its decade and say why this chapter does not teach it
Given a formulation only, write a plausible situation it could have come from — the inverse exercise, which nothing in the chapter asks but which tests the same skill

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

The stated paragraph (§12.1, Part II p. 394). Six numbers and one question. A dealer stocks two items only. Capital of fifty thousand rupees; shelf room for sixty pieces in total; a table costs two thousand five hundred rupees and a chair five hundred; the profit is two hundred and fifty rupees on a table and seventy-five on a chair. The question put to the reader is how many of each he should buy, and the paragraph closes by assuming everything bought is sold. That closing assumption is doing real work — without it the profit figures would not be a function of what he buys — and the chapter never returns to it. Every abstraction.
The three trial plans (§12.2, Part II p. 395). Verified: tables only — fifty thousand over two thousand five hundred is twenty tables, and twenty times two hundred and fifty is five thousand rupees of profit. Chairs only — the money would buy a hundred chairs, since fifty thousand over five hundred is a hundred, but the shelf caps him at sixty, and sixty times seventy-five is four thousand five hundred. Mixed — ten tables and fifty chairs is exactly sixty pieces, and ten times two hundred and fifty plus fifty times seventy-five is six thousand two hundred and fifty. The chapter runs all three. Narrator's warning: that third figure is the answer to the whole chapter, arrived at by guessing, twelve pages before it is confirmed. Present the three as trials that show the answer is not obvious, and do not announce that the third is the winner.
The genus and the species (§12.1, Part II p. 394). The chapter introduces optimisation problems first and linear programming problems second, explicitly as a subclass. Keep that order: a student who meets the special case first has nothing to contrast it against, and the word linear then sounds decorative.
What is linear here, item by item (not in the book). Verified against the chapter's own five numbered lines: the investment condition is linear because a table costs the same whether it is the first bought or the twentieth; the storage condition is linear because a table and a chair each take one space; the objective is linear because profit per item does not fall with volume. Each of those is an assumption about the world, not a fact, and the chapter never says so. This is the single best thirty seconds in the explanation and it is entirely added here to write.
A non-linear near-miss to contrast (not in the book). A bulk discount, a storage cost that rises with the square of the stock, or a profit that falls as the market floods — any one of these breaks linearity and puts the problem outside this chapter. The chapter offers no such contrast anywhere; without one the word linear is a label rather than a condition.
The two letters, and what they count (§12.2.1, Part II p. 395). The chapter sets one letter to the number of tables and the other to the number of chairs. Insist on the second half of that sentence. A letter without a stated meaning produces constraints that cannot be checked and an objective nobody can read back.
The two non-negative lines (§12.2.1, Part II p. 395). The chapter writes them first, numbers them as its first two conditions, and labels the pair. They come before the interesting constraints, not after. Verified reading: the reason is that they follow from what the letters count — you cannot buy a negative table — and not from anything the dealer said.
The investment constraint, before and after (§12.2.1, Part II p. 395). The chapter first writes the money condition with the true prices, two thousand five hundred a table and five hundred a chair, against the fifty thousand of capital; then it divides through and prints the simpler form, with five and one as the coefficients against a hundred. Verified: the common factor is five hundred, it is positive, and dividing an inequality by a positive number leaves the set of solutions untouched — so the two lines allow exactly the same purchases. The chapter shows the second line without saying why it is allowed to. Say why.
The storage constraint (§12.2.1, Part II p. 395). Both coefficients are one, because a table takes one shelf space and so does a chair. Worth pausing on: students who have just seen coefficients of two thousand five hundred and five hundred expect every constraint to have big numbers in it, and the coefficient of one looks to them like a missing number.
The objective, in the particular and in the general (§12.2, Part II p. 396). In the running example it is two hundred and fifty times the number of tables plus seventy-five times the number of chairs. Verified against the paragraph: those two constants are the two profit figures. The two cost figures, two thousand five hundred and five hundred, are already spent in the investment constraint and must not reappear here — this is the single mistake most worth pre-empting in the whole video. The chapter then writes the general form as a sum of the two variables, each multiplied by a constant, fixing two letters as the constants and two as the variables; it is worth ten seconds that the constants come from the problem and the variables come from the solver.
The finished programme (§12.2, Part II p. 396). The chapter lays it out in three parts: an instruction naming the objective and saying it is to be maximised; the linking phrase; and then the constraints, the two interesting ones first and the non-negative pair last, on one line together. Note that the printed layout reverses the numbering of §12.2.1, where the non-negative pair came first. Both orders are correct.
Reading it back (not in the book). Verified: every one of the six numbers in the paragraph appears exactly once in the finished programme — fifty thousand and the two prices inside the money constraint (as a hundred, five and one after the division), sixty in the shelf constraint, and the two profits in the objective. Nothing in the paragraph is unused and nothing in the programme is unaccounted for. Run this check; it is one of only two self-checks a student has before drawing.
The units check (not in the book). Verified: the money constraint carries rupees on both sides before the division, and after it both sides are counted in lots of five hundred rupees — the divisor rescales the two sides alike, which is exactly why the verdict on any purchase is unchanged. The shelf constraint carries pieces on both sides; the objective is in rupees. A constraint that puts rupees on one side and pieces on the other is wrong before it is drawn. The chapter performs no such check anywhere, and this is the cheapest error-catcher available to a student.
The three bold-lead definitions (§12.2, Part II p. 396). The chapter sets out, in this order and each with its name in bold at the head of the paragraph: the objective function; constraints, with the non-negative restrictions named inside that same paragraph; and the optimisation problem. Read off the printed page. Note that the chapter defines the general term third, after the two parts — that inversion is worth flagging, because the explanation will want to go outside in.
The formal definition (§12.2, Part II p. 396). Four clauses, all of which have to hold at once: an optimal value is wanted; the thing optimised is a linear function; the variables are held at or above zero; and a set of linear inequalities has to be satisfied. Build it one clause at a time and strike each against the running example.
The two words of the name (§12.2, Part II p. 396). The chapter glosses both in a single sentence — linear pointing at the form of every relation, and programming pointing at the drawing up of a plan. Say plainly that programming here has nothing to do with a computer; students who have written code read the title wrongly and never recover.
The Historical Note (Part II pp. 404–405). The narrative in order: the Second World War and the planning of operations; nineteen forty-one, the first formulation, by a Russian mathematician and an American economist working separately — the transportation problem; nineteen forty-five, an English economist and the question of a cheapest adequate diet; nineteen forty-seven, an American economist and the simplex method, described as an iterative procedure that finishes in finitely many steps; nineteen seventy-five, the Nobel prize in economics shared for this work. Two names carry printed slips — see Notes — and neither should be shown as printed.
The chapter frontispiece (Part II p. 394). The opening page carries a QR code marked with the Part II catalogue number and the chapter number, an epigraph attributed to G. Polya about a student solving a problem they invented themselves, and a photographic portrait captioned with the surname of the Russian mathematician. Unlike other chapters of this book the caption carries no dates. Caption the portrait on a card; do not reproduce the page or the epigraph's wording.
The Summary (Part II p. 404). Verified on the printed page: it is a single bullet. It restates the definition of the problem and names the objective function, the linear constraints and the decision variables. It says nothing whatever about how to solve one. That is a fact about the printed Summary, not an oversight in this brief — see Notes, and see the three m02 briefs, which have to compensate for it.
What the chapter gives you to practise formulating on (Part II pp. 399–404). Verified by reading every printed item on the page images: the five worked Examples all open by naming an objective in symbols and listing symbolic constraints, and all ten items of Exercise 12.1 do the same. Not one item in the chapter asks a student to formulate anything. The furniture dealer is the only worded situation in the chapter and it is formulated for them. This is why the second half of the explanation has no printed practice behind it, and section 12 must say so to the student — see Notes.

Figures to have open

The stated paragraph, typeset for screen with the six numbers separately addressable so each can be lit in turn, over an illustration of the dealer's situation: a counter with a stack of money labelled fifty thousand, a shelf with sixty slots, and two price tags and two profit tags. The chapter's own content, Part II p. 394; the treatment is added here.
A three-row trial table for section 2, columns as listed above. Built with the repo's DataTable component. The chapter runs the three plans in prose on Part II p. 395 and tabulates nothing.
A nested-region diagram for section 3, optimisation problems containing linear programming problems containing this one example. Not in the book; the chapter states the containment in words only.
A pair of small graphs for section 4, one straight and one bending, each annotated with the real-world assumption that makes it so. Not in the book.
A persistent two-line legend giving each letter and what it counts, pinned from section 5 to the end. Not in the book; the chapter states the meanings once and never repeats them.
A side-by-side of the money constraint before and after the division, with one sample purchase evaluated against both. The two lines are the chapter's own, Part II p. 395; the check is added here.
A three-column read-back audit for section 10 — the paragraph's number, where it lands in the programme, and the unit the term carries. Entirely added here; the chapter never writes a unit beside a term.
A four-clause layout of the printed definition for section 11. The content is the chapter's own Part II p. 396 sentence, restructured; do not set it as a quotation.
A four-stop timeline for section 12 built from the Historical Note, Part II pp. 404–405. Spell both surnames correctly — see Notes.
The portrait on Part II p. 394 is the textbook's own. Use a caption card giving the surname; the printed caption supplies no dates, so do not invent any. Ignore the QR code.
No drawn figure of the chapter's is used by the explanation. Part II pp. 394 to 396 carry only the portrait; the first drawn graph in the chapter is Fig 12.1 on Part II p. 397 and belongs to the m02 module.

Where this sits in the book

NCERT Class 12 Mathematics, Chapter 12 "Linear Programming", §12.1 Introduction, including the stated situation and the genus-and-species framing, Part II p. 394
§12.2, the three trial plans, Part II p. 395
§12.2.1 Mathematical formulation of the problem — the two letters, the non-negative pair, and the investment and storage constraints with the chapter's own labels, Part II p. 395
§12.2, the objective function, the finished programme in its printed layout, the formal definition and the three bold-lead definitions of objective function, constraints and optimisation problem, Part II p. 396
§12.2.2, the sentence restricting the chapter to the graphical method, Part II p. 397
Examples 1 to 5, all stated symbolically, Part II pp. 399 to 403
Exercise 12.1, all ten items, all stated symbolically, Part II pp. 403–404
Summary, the single bullet, Part II p. 404
Historical Note, Part II pp. 404–405