Linear Programming, SPM Additional Mathematics Form 5

Linear Programming is the chapter of Add Math most openly about decision-making. Given a situation with limited resources, money, time, materials, space, and a quantity you want to make as large or as small as possible, it gives you a systematic way to find the best possible choice.

In SPM Additional Mathematics the whole method is carried out graphically: you draw the constraints on a set of axes, shade the region of choices that are actually allowed, and then read the optimum answer off that region.

This is the seventh chapter of Form 5, and it sits in the applied, problem-solving side of the course. It is built from just two content standards taken straight from the KSSM DSKP: Linear Programming Model and Application of Linear Programming.

The first teaches you to turn words into a mathematical model, a set of linear inequalities drawn on a graph. The second teaches you to use that model to solve the real question: find the maximum or minimum value of an objective function.

The workflow of the chapter is short and repeatable. First you decide what the two unknowns are and name them $x$ and $y$.

Then you read the problem line by line, turning every condition, "at least", "at most", "not more than twice as many", into a linear inequality. You draw the boundary line of each inequality, decide which side to keep, and shade the region that satisfies all of them together.

Finally you form the objective function $k=ax+by$ and find the corner of the region where it reaches its best value.

Linear Programming is unusual in two ways, and both are worth knowing from the start. First, the SPM formula list supplies you with nothing for this chapter, every inequality and the objective function are things you build yourself from the words of the question.

Second, it is the one chapter that regularly asks for a clean, accurately scaled graph, so a sharp pencil, a ruler and a sensible scale matter as much as the algebra.

Our teachers enjoy teaching Linear Programming because it rewards careful reading rather than heavy computation. There is no long formula to memorise and no calculator gymnastics, just a disciplined translation from English into inequalities, a tidy graph, and a clear-headed check of the corners.

Students who slow down at the reading stage and speed up at the drawing stage tend to score very well here, which makes it a satisfying chapter to end the Form 5 syllabus with.

Linear Programming is organised into two content standards. These codes and titles come directly from the DSKP, and seeing them helps you tell the two halves of the chapter apart, building the model, then using it.

The two standards are a pair, meant to be read in order. Standard 7.1 is the translation and drawing stage, it produces a shaded region but does not yet answer anything.

Standard 7.2 is where the region earns its keep: you lay the objective function over it and extract the single best decision. A question almost always tests both together, so a shaky model in 7.1 quietly caps the marks you can reach in 7.2.

Two decision variables carry the whole problem. Every question hides two quantities you are free to choose, the number of tables and chairs to make, the litres of two drinks to mix, the hours spent on two tasks.

Name them $x$ and $y$ at the very start and write down exactly what each one stands for. Everything else in the chapter is built on top of these two letters, so define them clearly before anything else.

Every condition becomes a linear inequality. The core skill is translation.

"At least" and "not less than" give $\ge$; "at most" and "not more than" give $\le$; "more than" and "fewer than" give strict $>$ or $<$. Comparisons need care: "there are at least twice as many $x$ as $y$" becomes $x \ge 2y$, not $2x \ge y$.

Read each sentence slowly and turn it into exactly one inequality.

$$ax + by \le c \quad\text{or}\quad ax + by \ge c$$

Quantities cannot be negative. Because $x$ and $y$ usually count real things, almost every problem carries the silent constraints $x \ge 0$ and $y \ge 0$.

Forgetting them leaves your region open on the left or below and can send the optimum to an impossible negative answer, so write them in even when the question does not spell them out.

Draw each boundary as a line, then choose a side. Turn every inequality into its boundary line by replacing the inequality sign with $=$, and draw it.

Use a solid line for $\le$ or $\ge$ (the boundary is included) and a dashed line for strict $<$ or $>$ (the boundary is excluded). To decide which side to keep, substitute a test point, the origin $(0,0)$ is easiest when the line does not pass through it, and see whether the inequality is true there.

The feasible region is where all the constraints overlap. The region $R$ that satisfies every inequality at the same time is the intersection of all the chosen sides.

Shade it clearly and label it $R$; this convex polygon is the set of every allowed decision. Decide once whether you are shading the region you keep or shading out the region you reject, then be consistent, an unlabelled or half-shaded graph is where many marks quietly leak away.

The objective function is the quantity you optimise. The thing the question actually wants, profit, cost, total number, time, is written as $k=ax+by$, a linear expression in your two variables.

You are asked to make $k$ as large as possible (maximise) or as small as possible (minimise) while staying inside $R$. Forming this function correctly is half of Standard 7.2.

$$k = ax + by$$

The optimum always sits at a corner. Because both the region and the objective are linear, the largest and smallest values of $k$ are found at the vertices of $R$.

Two reliable methods reach them. In the moving-line method you draw the line $ax+by=k$ for a convenient trial $k$ and slide it parallel, keeping its gradient $-\tfrac{a}{b}$, until it last touches $R$; that final contact point is the optimum.

In the testing-vertices method you find the coordinates of every corner and simply evaluate $k$ at each, then pick the best.

$$y = -\frac{a}{b}\,x + \frac{k}{b}$$

Answers are often whole numbers. When $x$ and $y$ count indivisible things, people, chairs, complete trips, the optimum must be a point in $R$ with integer coordinates.

If the corner is not a whole-number point, search the nearby lattice points that still lie inside the region and choose the best of those, rather than reporting a fractional "half a chair" answer.

Linear Programming appears in the written papers. Paper 1 (3472/1) lasts 2 hours and carries 80 marks: Section A has 12 questions worth 64 marks that you answer all of, and Section B has 3 questions worth 16 marks from which you answer 2.

Paper 2 (3472/2) lasts 2 hours 30 minutes and carries 100 marks across Section A (7 questions, 50 marks, answer all), Section B (4 questions, 30 marks, answer 3) and Section C (4 questions, 20 marks, answer 2).

Because linear programming asks you to draw an accurate graph, shade a region and read an optimum from it, the chapter sits naturally among the longer, applied items of Paper 2, where a single question can carry a whole modelling task from wording to answer. The items across both papers are limited-response subjective and structured questions, they are marked with analytic scoring, and you sit them with a non-programmable scientific calculator.

Overall the papers are balanced across difficulty in a low-to-medium-to-high ratio of 5 : 3 : 2, so expect a straightforward set-up before the question turns to the sharper optimisation. We do not predict how many marks any single chapter carries, because that varies from year to year.

Analytic scoring means your method earns marks step by step, so make the working visible: state the two variables, list each inequality, draw and label the boundary lines and the region $R$, write the objective function, and show either the sliding line or the value of $k$ at each vertex. Our lessons are taught in English while SPM papers are set bilingually in Malay and English, so you will meet the key terms, constraint, feasible region, objective function, maximum, minimum, in both languages and can recognise the task however a question is phrased.

Most marks lost in Linear Programming come from the translation and the graph rather than from any hard mathematics. Each slip below is easy to avoid once you can name it, so read them before every practice set until checking for them is automatic.

• Choosing the wrong inequality direction. "At least" is $\ge$ and "at most" is $\le$, mixing them flips the region and the answer. Underline the comparison words in each sentence and decide the sign deliberately before writing anything down.
• Mistranslating a comparison between the variables. "Twice as many $x$ as $y$" means $x = 2y$, and "$x$ is at least twice $y$" means $x \ge 2y$, not $2x \ge y$. Test your inequality with a simple pair of numbers to confirm it says what the sentence says.
• Shading the wrong side, or shading inconsistently. Decide once whether you keep the region that satisfies the inequalities or shade out the region you reject, then apply that convention to every line. Always test a point such as the origin instead of guessing which side to keep.
• Using a solid line for a strict inequality. A $\le$ or $\ge$ boundary is drawn solid because it is included; a strict $<$ or $>$ boundary is drawn dashed because it is not. Getting this wrong can wrongly include or exclude the very corner that gives the optimum.
• Forgetting $x \ge 0$ and $y \ge 0$. When the variables count real quantities they cannot be negative, so the region must be confined to the first quadrant. Leaving these out opens the region and can produce an impossible negative optimum.
• Reading the optimum off the wrong point. The best value lies at a vertex of $R$, so test every corner (or slide the objective line to its last contact) rather than eyeballing a point in the middle, and when the answer must be whole numbers, check the nearest lattice points inside the region.

None of these is about talent, each is a small habit of careful reading and tidy drawing. Students who score well in this chapter are simply the deliberate ones: comparison words underlined, inequalities checked with a test pair, a consistent shading convention, solid and dashed lines used correctly, and every corner examined before the answer is written.

Treat each slip as feedback, and your accuracy climbs fast.

Linear Programming rewards a steady routine, define the variables, translate each condition, draw and shade the region, then optimise at a corner. Work through the steps below, then use the resources that follow to revise and test yourself.

Keep your sessions short and frequent rather than one long push, and always work on graph paper so the drawing becomes second nature. The two skills that decide your marks, turning words into inequalities and reading the optimum off a vertex, are built by repetition, so a couple of full modelling questions a day beat an occasional marathon.

When translation feels automatic, spend your extra time on clean, accurately scaled graphs.

1. 1

   Drill the translation of words into inequalities

   Take short conditions, "at least", "at most", "twice as many as", "no more than half", and turn each into a single inequality, checking it with a test pair of numbers until the wording never fools you.

2. 2

   Practise drawing and shading the feasible region

   For a full set of constraints, draw each boundary line (solid or dashed), test a point to choose the side, and shade the region $R$ with one clear, consistent convention.

3. 3

   Master both optimisation methods

   On the same region, find the optimum twice, once by sliding the objective line $k=ax+by$ to its last contact point, and once by testing the value of $k$ at every vertex, so you can pick whichever is quicker in the exam.

4. 4

   Handle whole-number answers with care

   When $x$ and $y$ must be integers, learn to search the lattice points nearest the optimal corner that still lie inside the region, and to justify the whole-number answer you choose.

5. 5

   Do full Paper 2-style questions under time

   Tackle complete modelling questions on graph paper, from wording to final decision, with a clock running, then review the worked examples afterwards to tighten your method and your scale choices.

• Linear Programming, Revision Notes
• Linear Programming, Common Mistakes
• Linear Programming, Practice Questions
• Linear Programming, Paper 2 Answering Guide
• Linear Programming, Key Terms
• Linear Programming, Worked Examples (Easy)
• Linear Programming, Worked Examples (Medium)
• Linear Programming, Worked Examples (KBAT)