Linear programming: turning words into a graph

Linear programming is one of the few chapters in Add Math where almost all the marks come from reading carefully rather than from clever algebra. The maths itself is gentle, straight lines and simple inequalities.

The challenge is translation: turning an everyday situation, written in ordinary sentences, into mathematics you can draw.

Every linear programming question is built from two ingredients. First, a set of constraints, the limits you must respect, such as how much material you have, how much time, or a required ratio.

Second, an objective, the one quantity you are trying to make as big as possible (a profit, an amount produced) or as small as possible (a cost). Both the constraints and the objective are always linear, which is why the whole picture can be drawn with straight lines.

Start by naming your two unknowns clearly, almost always $x$ and $y$, and writing down exactly what each one counts. This single habit prevents more lost marks than anything else, because every inequality afterwards depends on it.

If $x$ is "the number of type A items" and $y$ is "the number of type B items", write that sentence down first.

Then work through the passage one sentence at a time. Each restriction becomes one inequality.

The trick is knowing which phrase turns into which symbol, and here the everyday English is very consistent:

Two quiet constraints are almost always present even when the passage never mentions them: you usually cannot have a negative amount of anything, so $x \geq 0$ and $y \geq 0$. Forgetting these is a classic way to draw the wrong region.

Finally, write the objective function, the thing to be optimised, as a separate line, for example a return of $k = 4x + 6y$.

$$\text{maximise (or minimise)}\quad k = ax + by$$

Each inequality describes a half of the graph, everything on one side of a straight line. To draw it, first draw the line as if it were an equation (replace the inequality sign with $=$).

For $x + y \leq 12$ you draw the line $x + y = 12$; the easiest way is to find where it crosses the axes: when $x = 0, y = 12$, and when $y = 0, x = 12$.

Then decide which side you want. A quick, safe method is to test a point that is not on the line, the origin $(0,0)$ is ideal when the line does not pass through it.

If the point makes the inequality true, the region you want is the side containing that point; if it makes it false, you want the other side.

Do this for every inequality on the same axes. The feasible region is the area where all of the inequalities are satisfied at the same time, the overlap of all the wanted sides.

Shade it clearly and label it $R$. Whichever shading convention your school uses, make sure it is obvious to a marker exactly which region is your answer.

Here is the fact that makes linear programming reliable: when the objective is linear, its greatest and least values over a region always occur at a corner (vertex) of that region. You never have to search the inside.

This means the whole optimisation shrinks to a short checklist, find the corners, test each one.

1. 1

   Find the vertices

   Read the corner coordinates off the graph, or solve the two lines that cross at each corner as simultaneous equations.

2. 2

   Evaluate the objective

   Put each corner's $x$ and $y$ into $k = ax + by$.

3. 3

   Pick the best

   The largest value is the maximum; the smallest is the minimum. State the point and the value in a sentence.

There is also the "moving line" method, where you draw one line of the objective and slide it across the region until it last touches a corner, useful, but the corner-testing checklist above is the most dependable under exam pressure.

Most lost marks in this chapter are not maths errors at all, they are reading and drawing errors. A short mental checklist catches almost all of them before they happen.

• Mixing up $\leq$ and $\geq$. Re-read the phrase and match it to the table above; "not more than" is $\leq$, not $<$.
• Forgetting $x \geq 0$ and $y \geq 0$, which quietly changes the whole region.
• Reading a ratio the wrong way round: "B is at most twice A" is $y \leq 2x$, not $x \leq 2y$.
• Leaving the region unclear, so the marker cannot tell which area is $R$.
• Testing only some corners, or misreading a corner's coordinates off the graph.

Linear programming is a chapter where a student can understand every line of the theory and still lose marks, simply because the translation step, words into inequalities, is a skill that grows with feedback. Working one-to-one, a teacher can watch how you read a passage, catch the exact sentence you misread, and show you the habit of naming variables and testing corners until it becomes automatic.

Our teachers are experienced; lessons are online and taught in English, while the SPM papers are set bilingually in Bahasa Melayu and English.

If this chapter feels slippery, you are welcome to begin with a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience). We won't promise a grade, but we can help you turn a wordy, intimidating question into a calm, step-by-step routine you trust.