Singapore Math guide to linear law, the upper secondary Additional Mathematics topic that turns curved relationships into straight lines, for families and teachers outside Singapore.
A straight line is the easiest graph to read.
Its gradient and intercept can be measured with a ruler.
In Singapore Mathematics, students in G3 Additional Mathematics (Secondary 3 to 4) learn to transform relationships such as y = axn and y = kbx into straight-line form. From the line, they find the unknown constants.
“Linear law” is the common Singapore name for this skill. The syllabus describes it as the transformation of given relationships to linear form.
This guide is useful for families and teachers who want to:
- understand exactly what the syllabus asks students to do;
- teach the method as one idea, Y = mX + c, rather than a set of cases;
- prepare a student who is moving into a Singapore secondary school; or
- support a student using Singapore Math materials in another country.
Quick facts
Level: Secondary 3 to 4 (G3 Additional Mathematics, an elective)
Strand: Geometry and Trigonometry (coordinate geometry in two dimensions)
Official source: SEAB G3 Additional Mathematics, Syllabus K341 (SEC, from 2027). The 2026 O-Level Additional Mathematics syllabus 4049 has the same content.
Comes after: Exponential and Logarithmic Functions and Linear Graphs and Gradient
Leads to: Differentiation and Mathematical Modelling
What Linear Law Means in Singapore Mathematics
Many real relationships are curves. A pendulum’s period, a cooling drink or a growing population do not give straight-line graphs of y against x.
Linear law changes the variables so the relationship becomes Y = mX + c. Here Y and X are new quantities built from x and y, such as log y or y/x.
Once the points lie on a straight line, the gradient m and the intercept c can be read from the graph. These then give the constants in the original relationship.
The method also tests a model. If the transformed points do not lie close to a straight line, the suggested relationship is probably wrong.
Where Linear Law Sits in the Singapore Syllabus
Linear law is part of coordinate geometry in two dimensions, in the Geometry and Trigonometry strand of G3 Additional Mathematics.
The syllabus wording is: transformation of given relationships, including y = axn and y = kbx, to linear form to determine the unknown constants from a straight line graph.
In practice, students:
- choose the variables Y and X that make a given relationship linear;
- plot Y against X from a table of values, and draw the line of best fit;
- find the gradient and the intercept from the line; and
- convert these back into the unknown constants.
| Relationship | Plot Y | Against X | Gradient | Y-intercept |
|---|---|---|---|---|
| y = axn | log y | log x | n | log a |
| y = kbx | log y | x | log b | log k |
| y = ax² + bx | y/x | x | a | b |
| y = a/x + b | y | 1/x | a | b |
Here log means log10. The natural logarithm ln works in the same way. Linear law depends on the laws of logarithms from the same course.
How Singapore Teaches Linear Law
1. Recall Y = mX + c
Students revisit gradient and intercept from lower secondary work on straight-line graphs. The letters Y and X are placeholders for whatever is plotted.
2. Transform by algebra first
For y = axn, take logs: log y = n log x + log a. Then match it to Y = mX + c term by term.
3. Name what each part means
Students write Y = log y, X = log x, m = n and c = log a before touching the graph paper.
4. Plot and draw the best line
They compute a new table, plot the points with sensible scales and draw a single straight line of best fit.
5. Read and convert
The gradient comes from two points on the line that are far apart. The intercept is read where X = 0, or calculated from c = Y − mX.
6. Use the line
Students estimate values from the graph and comment on whether the model fits.
Worked Examples
Finding a and n for y = ax^n
Question: The table shows values of x and y that satisfy y = axn. x: 1, 4, 9, 16. y: 3, 24, 81, 192. Find a and n.
Step 1. Take logs: log y = n log x + log a. Plot Y = log y against X = log x.
Step 2. New table, to 3 decimal places. log x: 0, 0.602, 0.954, 1.204. log y: 0.477, 1.380, 1.908, 2.283.
Step 3. Gradient = (2.283 − 0.477) ÷ (1.204 − 0) = 1.806 ÷ 1.204 = 1.5, so n = 1.5.
Step 4. Y-intercept = 0.477 = log a, so a = 100.477 = 3.00.
Check: y = 3x1.5 gives 3 × 41.5 = 3 × 8 = 24 when x = 4, which matches the table.
Finding k and b for y = kb^x
Question: Variables x and y satisfy y = kbx. When log y is plotted against x, the straight line passes through (1, 1.0) and (5, 2.2). Find k and b.
Step 1. Take logs: log y = (log b)x + log k. The gradient is log b and the intercept is log k.
Step 2. Gradient = (2.2 − 1.0) ÷ (5 − 1) = 0.3, so log b = 0.3 and b = 100.3 = 2.00, correct to 3 significant figures.
Step 3. Intercept: 1.0 = 0.3 × 1 + c, so c = 0.7. Then log k = 0.7 and k = 100.7 = 5.01.
Check: At x = 1, y = 5.01 × 2.00 is about 10.0, and log 10 = 1.0, which matches the point (1, 1.0).
Common Mistakes
- Reading the intercept as the constant. In the second example the intercept is 0.7, but k is 100.7. Fix: write c = log k before reading the graph.
- Plotting the wrong X. For y = kbx, the horizontal axis is x, not log x. Fix: match the logged equation to Y = mX + c term by term.
- Using table points for the gradient. Experimental points are scattered. Fix: use two points on the drawn line, far apart.
- Reading an intercept from a shifted axis. If the X-axis does not start at 0, the line’s crossing point is not the intercept. Fix: calculate c = Y − mX from a point on the line.
- Rounding logs too early. Two-decimal logs can give a noticeably different constant. Fix: keep three decimal places in the table and full values in calculations.
- Undoing the logs wrongly. Writing n = 101.5 instead of n = 1.5, or a = 0.477 instead of 100.477. Fix: say aloud what each of m and c stands for.
Teaching Linear Law Outside Singapore
- The name. “Linear law” is the usual Singapore name. Other books may call it reducing to linear form, or use the terms log-log graph and semi-log graph.
- Graph paper and scales. Reading constants from a line depends on accurate plotting and sensible scales. Practise on real graph paper, not only on screens.
- Timing. Linear law sits in the upper secondary elective and depends on logarithms. The Singapore Math Grade Levels guide explains why direct conversions are unreliable.
- Prerequisites. Students need the laws of logarithms, the meaning of gradient and intercept, and confident algebraic rearranging.
- Calculators. An approved calculator may be used in both papers, which helps with the log tables. The plotting and reading still need care.
How to Practise Linear Law at Home
- Take four relationships and write each in the form Y = mX + c, naming Y, X, m and c.
- Make a table of values from y = 2x3 and check that log y against log x gives gradient 3.
- Plot one data set by hand on graph paper with a line of best fit.
- Read the gradient from two points on the line, then convert m and c into constants.
- Use the line to estimate one value of y, and check it with the original relationship.
- Once a week, try a data set that does not fit the model and describe what the graph shows.
Accurate plotting takes practice, so allow time for it.
Try These
- To make y = ax² + bx linear, what should be plotted against x? What do the gradient and intercept give?
- A graph of log y against log x is a straight line with gradient 2 that passes through (0, 0.301). Find the relationship between x and y.
- A graph of log y against x is a straight line with gradient 0.477 and log y-intercept 1. Find k and b in y = kbx.
- For y = a/x + b, the graph of y against 1/x passes through (0, 4) and (2, 10). Find a and b.
Answers
- Plot y/x against x. Dividing by x gives y/x = ax + b, so the gradient is a and the intercept is b.
- y = 2x². Here n = 2 and log a = 0.301, so a = 100.301 = 2.00.
- k = 10 and b = 3.00. The intercept gives log k = 1, and the gradient gives b = 100.477.
- a = 3 and b = 4. The gradient is (10 − 4) ÷ 2 = 3, and the intercept is 4.
What Comes Next
Linear law connects algebra, graphs and data.
- Differentiation extends the idea of gradient from straight lines to curves.
- Mathematical Modelling describes how a model is proposed, tested and improved.
- Exponential and Logarithmic Functions is worth revisiting whenever the log steps feel shaky.
A student who can read a straight line confidently has a skill that serves well beyond the examination.
Frequently Asked Questions
When do Singapore students learn linear law?
In G3 Additional Mathematics, the upper secondary elective taken in Secondary 3 to 4, once logarithms are secure.
Is “linear law” the official syllabus name?
It is the common Singapore name. The syllabus item itself describes transforming given relationships to linear form to find unknown constants from a straight-line graph.
Which relationships must students know?
The syllabus names y = axn and y = kbx as included examples. Other relationships may be given in a question, and students transform whatever they are given.
Should students use log or ln?
Either works, as long as the same base is used throughout. The constants come out the same.
Do students have to draw the graph by hand?
The syllabus asks for constants to be found from a straight-line graph, so plotting points and drawing a line of best fit by hand are worth practising on graph paper.
Why are my child’s constants slightly different from the answer key?
Small differences usually come from reading the line. Answers close to the key are normal, but large gaps suggest a wrong transformation.
Helpful Reading
- Singapore Mathematics | International Edition: Exponential and Logarithmic Functions
- Singapore Mathematics | International Edition: Linear Graphs and Gradient
- Singapore Mathematics | International Edition: Coordinate Geometry of Circles
- Additional Mathematics Synthesis Guide 18: Linear Law Transformations and Parameter Recovery
- Additional Mathematics Synthesis Guide 2: Exponentials, Logarithms, Linear Law and Modelling
- Why Do Graphs Feel Hard in Secondary Mathematics?
- How to Make Sense of the SEC G3 Additional Mathematics Syllabus
- SEAB G3 Additional Mathematics Syllabus K341
Linear Law in Singapore Mathematics
Curves are hard to read.
Lines are easy.
Transform, then match Y = mX + c.
Name each part before plotting.
Undo the logs at the end.
Taught this way, linear law stops being a list of cases.
It becomes one idea applied with care.
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Singapore Mathematics | International Edition is a Bukit Timah Tutor guide series for families and teachers outside Singapore. It is not a textbook and is not affiliated with MOE, SEAB or any publisher. Syllabus details were checked against MOE and SEAB documents in October 2026. Always confirm current requirements with your school.
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