Linear law is the A-Maths topic students underestimate — it is worth steady marks every year, and the whole topic rests on one idea: rewrite a non-linear relationship in the straight-line form Y = mX + c so its constants can be read from a graph's gradient and intercept. Once you can spot what plays the role of Y, X, m and c, every linear-law question follows the same route. This guide is from Ancourage Academy, whose secondary A-Maths tuition teaches linear law method-first in small groups of 3–6 at Bishan and Woodlands.
This is a single-topic deep-dive — a sibling to our A-Maths indices and logarithms guide, on which linear law depends. If you are still deciding whether to take A-Maths, read E-Maths vs A-Maths first.
If linear law feels confusing, Ancourage Academy's Sec 3 A-Maths programme teaches the conversion technique directly — book a trial class (usually $18) for a diagnostic assessment.
What Is Linear Law in A-Maths?
In O-Level / SEC A-Maths, linear law covers transforming a given relationship into linear form and using a straight-line graph to determine unknown constants. The SEAB A-Maths syllabus (4049) lists it under coordinate geometry as the transformation of given relationships — including y = axⁿ and y = kbˣ — to linear form, to determine the unknown constants from a straight-line graph. From 2027 the same content carries over to the SEC G3 A-Maths syllabus (K341) unchanged.
Why Do We Convert to Straight-Line Form?
A straight line is the only graph whose constants — gradient and y-intercept — can be measured directly, so converting a curve to a line lets you find unknown constants experimentally. Plotting a curve and trying to read constants off it is unreliable; a straight line is not. The whole topic is built on matching a rearranged equation to the template Y = mX + c, where Y and X are expressions you can compute and plot, m is the gradient, and c is the vertical intercept.
How Do You Rewrite an Equation in Linear Form?
The method is to algebraically manipulate the given equation until it matches Y = mX + c, then identify which expression is the vertical variable (Y), which is the horizontal variable (X), and what the gradient and intercept represent.
| Original relationship | Linear form (Y = mX + c) | Plot |
|---|---|---|
| y = axⁿ | lg y = n lg x + lg a | lg y against lg x |
| y = abˣ | lg y = (lg b)x + lg a | lg y against x |
| y = ax + bx² | y/x = b x + a | y/x against x |
| y = a/x + b | y = a(1/x) + b | y against 1/x |
For relationships where the unknown is in a power (like y = axⁿ or y = abˣ), the conversion uses logarithms — which is why linear law builds directly on indices and logarithms. Taking lg (or ln) of both sides and applying the log laws is the standard first move.
How Do You Find the Unknown Constants?
Once the data is plotted as a straight line, the gradient gives one constant and the vertical intercept gives the other — you match them to the m and c of your linear form.
- Plot Y against X using the converted variables (e.g., lg y against lg x).
- Draw the line of best fit through the plotted points.
- Measure the gradient using two points far apart on the line — this equals m.
- Read the vertical intercept where X = 0 — this equals c.
- Solve back for the original constants, e.g. if c = lg a then a = 10^c.
The final "solve back" step is where marks are most often lost: when the intercept equals lg a (not a itself), you must undo the logarithm to recover a.
The Most Common Linear Law Mistakes
In our A-Maths classes at Ancourage Academy, a handful of recurring linear-law errors cause most avoidable mark loss.
| Mistake | Why it happens | How to fix it |
|---|---|---|
| Reading the intercept as the constant | Forgetting the intercept is lg a, not a | Undo the logarithm: if c = lg a, then a = 10^c |
| Wrong choice of Y and X | Not matching the rearranged equation to Y = mX + c | Write the linear form first, then label which expression is Y and which is X |
| Gradient from close points | Picking two nearby points on the line | Use two points far apart on the line of best fit for accuracy |
| Forgetting log laws in conversion | Not splitting lg(axⁿ) correctly | lg(axⁿ) = lg a + n lg x — apply the product and power laws |
| Plotting raw data, not converted | Plotting y against x instead of the transformed variables | Plot the converted variables (e.g., lg y against lg x), not the originals |
How Does Linear Law Connect to the Rest of A-Maths?
Linear law is the practical pay-off of indices and logarithms, and it reinforces the straight-line skills used across the syllabus.
- Indices and logarithms: power and exponential relationships are linearised with logs — the direct application of the log laws.
- Coordinate geometry: gradient and intercept are the same straight-line tools from coordinate geometry.
- Real-world modelling: linear law mirrors how scientists fit experimental data, a connection that reappears in JC and beyond.
A Study Plan for Mastering A-Maths Linear Law
Master linear law in order: secure logarithms first, then conversion, then reading constants off a graph.
- Week 1 — prerequisites: revise the log laws so lg(axⁿ) = lg a + n lg x is automatic.
- Week 2 — conversion: practise rewriting many relationship types into Y = mX + c form and identifying Y, X, m and c.
- Week 3 — graphs and constants: plot converted data, draw the line of best fit, and extract constants, including the "solve back" step.
- Week 4 — mixed practice: tackle full linear-law questions under timed conditions, checking units and the final unlogging.
Ancourage Academy's Sec 3 and Sec 4 A-Maths programmes work through linear law on exactly this progression in small groups of 3–6. If your child got stuck here, our A-Maths survival guide covers the wider recovery plan — book a trial class (usually $18) for a diagnostic, or WhatsApp us with any questions.
Common Questions About A-Maths Linear Law
What is linear law in A-Maths?
Linear law is the technique of transforming a non-linear relationship into the straight-line form Y = mX + c so that unknown constants can be found from a graph's gradient and vertical intercept. For example, y = axⁿ becomes lg y = n lg x + lg a, which is a straight line when lg y is plotted against lg x. The gradient gives n and the intercept gives lg a.
How do you convert y = abˣ to linear form?
Take logarithms of both sides: lg y = lg(abˣ) = lg a + x lg b. This matches Y = mX + c with Y = lg y, X = x, gradient m = lg b, and intercept c = lg a. Plotting lg y against x gives a straight line, from which b = 10^(gradient) and a = 10^(intercept). The conversion relies directly on the laws of logarithms.
Why must you plot the converted variables, not the raw data?
The original relationship is a curve, and you cannot reliably measure constants from a curve. Converting to linear form means plotting transformed quantities — such as lg y against lg x, or y/x against x — which produces a straight line. Only then can the gradient and intercept be measured accurately and matched to the constants. Plotting the raw y against x defeats the purpose.
What is the most common linear-law mistake?
Reading the intercept as the constant itself. When the linear form is lg y = n lg x + lg a, the intercept equals lg a, not a — you must undo the logarithm with a = 10^(intercept) to recover the actual constant. A close second is taking the gradient from two points that are too near each other; always use widely spaced points on the line of best fit.
Is linear law the same under SEC from 2027?
Yes. Linear law moves from the O-Level A-Maths syllabus (4049) to the SEC G3 A-Maths syllabus (K341) from 2027 with no change to the transformation-to-linear-form and graph requirements. The "O-Level / SEC" dual reference reflects this transition.
Related: A-Maths Indices & Logarithms · A-Maths Coordinate Geometry · A-Maths differentiation and integration · Coping with A-Maths · The E-Maths/A-Maths decision · A-Maths Polynomials & Partial Fractions