Common Mistakes in IGCSE Add Maths Linear Law

Q: What is linear law in IGCSE additional mathematics?

Linear law in IGCSE Additional Mathematics is the technique of rewriting a non-linear equation (such as y equals ax to the power n, or y equals ab to the power x) into the form Y equals mX plus c, so that a straight-line graph can be plotted. By comparing the rearranged equation to Y equals mX plus c, students identify what Y, X, m, and c each represent, then use the gradient and intercept of the plotted line to find the unknown constants.

Q: How do you reduce y equals ax to the power n to linear form?

To reduce y equals ax to the power n to linear form, take logarithms of both sides: lg y equals lg a plus n lg x. Compare this to Y equals mX plus c: Y is lg y, X is lg x, m (the gradient) is n, and c (the intercept on the Y-axis) is lg a. Plot lg y against lg x to get a straight line; the gradient gives n and the Y-intercept gives lg a, from which a equals 10 to the power of (Y-intercept).

Q: How do you find a and b from a linear law graph?

After plotting the straight-line graph and reading the gradient and Y-intercept, substitute back into the expressions identified in the comparison step. For example, if the comparison gives gradient equals n and Y-intercept equals lg a, then n is read directly from the gradient, and a equals 10 raised to the power of the Y-intercept value. Never report the Y-intercept or gradient as the final answer for a or b without this conversion step.

Q: What is the difference between lg and ln in IGCSE add maths linear law?

In IGCSE Additional Mathematics, lg means log base 10 (common logarithm) and ln means log base e (natural logarithm, where e is approximately 2.718). For linear law, use lg when the equation has the form y equals ax to the power n (power equations) and ln when the equation has the form y equals ae to the power bx (natural exponential equations). Using the wrong base gives a wrong intercept value when converting back to find the constants.

Short answer

What are the most common mistakes in IGCSE Add Maths linear law?

The five most common IGCSE Additional Mathematics linear law mistakes are: swapping Y and X when comparing the rearranged equation to Y = mX + c, applying logarithm rules incorrectly when taking log of both sides, reading gradient or intercept from the wrong axis on the plotted graph, forgetting to convert the found gradient or intercept back into the original constants, and using the wrong log base (lg vs ln) for the type of equation given. Each is a method-step error, not a concept gap.

What linear law does

The core idea: turn a curve into a straight line

A non-linear equation such as y = axⁿ or y = ab^x cannot be analysed using a straight-line graph directly. Linear law rearranges the equation algebraically (often by taking logarithms) into the form Y = mX + c, where Y and X are expressions involving x and y, and m and c are constants related to a, b, or n. Once in this form, a plot of Y against X gives a straight line whose gradient is m and whose Y-axis intercept is c. From those two values, the unknown constants in the original equation are recovered.

The two most common equation types in 0606 linear law questions are power equations (y = axⁿ, reduce by taking lg of both sides) and exponential equations (y = ab^x or y = ae^bx, reduce by taking lg or ln of both sides). Each has a specific rearrangement, and mixing up the steps for the two types accounts for a large portion of linear law errors.

Mistake 1

Swapping Y and X in the comparison to Y = mX + c

After rearranging the equation into linear form, the next step is to identify Y, X, m, and c by comparing it to Y = mX + c. Students who swap Y and X plot the graph with the axes reversed, which gives a gradient that is the reciprocal of the correct one and an intercept on the wrong axis. Everything that follows in the question then uses these wrong values.

Example. Given y = axⁿ, taking lg of both sides gives lg y = lg a + n lg x. Compare to Y = mX + c: Y = lg y, X = lg x, m = n, c = lg a. If a student writes Y = lg x and X = lg y instead, they plot the graph with the axes swapped. The gradient they read from this graph is 1/n rather than n, and they recover the wrong value for the exponent.
The weak step underneath. Students see two “log” expressions on both sides and are unsure which one is Y and which is X. The distinction is that the left-hand side of the rearranged equation must match the left-hand side Y of the linear form. In the equation lg y = n lg x + lg a, the left-hand side is lg y, so Y = lg y and X = lg x (the variable multiplied by the gradient).
The fix. After rearranging, write a clear comparison table: Y = [expression], X = [expression], m = [expression], c = [expression]. This is four lines, one per unknown, and it takes under thirty seconds. The table forces the correct pairing and prevents the swap error. Many students skip this step to save time and lose far more marks than the four lines cost.

Mistake 2

Applying logarithm rules incorrectly when rearranging

Taking logarithms of both sides of y = ab^x gives lg y = lg a + x lg b. Students who mis-apply the log product or power rule arrive at a rearranged form that looks similar but is algebraically wrong, and every comparison that follows inherits the error.

The two rules that are most often misapplied. First, lg(ab^x) must be split as lg a + lg b^x, then lg b^x = x lg b. Students who write lg(ab^x) = lg(a) × lg(b^x) (multiplying instead of adding) or lg(ab^x) = (lg a)(x)(lg b) produce a wrong linear form that cannot be matched to Y = mX + c in any meaningful way. Second, for y = axⁿ, taking lg of both sides gives lg y = lg a + n lg x. Students who write n = lg x × lg a are treating the product of logs as the log of a product, which is a different rule.
The three log rules to have fluent. 1. lg(AB) = lg A + lg B (product rule). 2. lg(A/B) = lg A − lg B (quotient rule). 3. lg(Aⁿ) = n lg A (power rule). The weak step is the power rule: students who have not internalised lg(Aⁿ) = n lg A often try to use it as a calculator operation instead of an algebraic identity.
The fix. Before rearranging, state which log rules apply. For y = ab^x, write: “take lg: lg y = lg a + lg b^x (product rule), then lg y = lg a + x lg b (power rule).” Labelling the rule used at each step makes the working clear and confirms it is being applied correctly.

Mistake 3

Reading gradient or intercept from the wrong axis

Once the straight-line graph has been plotted, the gradient is found by choosing two clear points on the line and computing (change in Y)/(change in X). The Y-intercept is read where the line crosses the Y-axis. Both of these operations are reversed by students who read the intercept from the X-axis, or who compute (change in X)/(change in Y) instead of the other way round.

Why this happens. Students who plot the graph with axes swapped (Mistake 1) naturally read off the intercept from the “vertical axis,” which is in fact the X-axis. Even students who have the axes correct sometimes compute the gradient as run-over-rise (X-change over Y-change) instead of rise-over-run (Y-change over X-change), especially when the line slopes downward and both differences are negative, causing a sign error.
Example. Two points on the line: (1, 3) and (3, 7). Gradient = (7 − 3)/(3 − 1) = 4/2 = 2. A student who reads this as (3 − 1)/(7 − 3) = 2/4 = 0.5 gets the reciprocal of the correct gradient and recovers the wrong constant from the comparison. For a negative-gradient line, the same swap also loses the sign.
The fix. When identifying two points on the line for the gradient calculation, write them as (X₁, Y₁) and (X₂, Y₂) with the labels explicit, then write m = (Y₂ − Y₁)/(X₂ − X₁). This template prevents the axis-swap error and keeps the sign correct. Always use two points that lie exactly on the drawn line, not two data points from the table (the plotted line is a best-fit line, not a point connector).

Mistake 4

Reporting gradient and intercept as the final answer without back-converting

The gradient and Y-intercept of the plotted straight line are the values of m and c in the linear form Y = mX + c. They are not necessarily the values of a, b, or n in the original equation. When the comparison gives “c = lg a,” the value found from the graph is lg a, and a must be recovered by computing 10 raised to the power of that value. Students who write their graph-reading directly as the answer for a lose the back-conversion marks.

Example. Equation: y = axⁿ. After rearranging: lg y = n lg x + lg a. Comparison: gradient = n, Y-intercept = lg a. From the graph: gradient = 2.5, Y-intercept = 0.6. Therefore n = 2.5 (direct read, no conversion needed). But a = 10^0.6 = 3.98, not 0.6. A student who writes a = 0.6 has read the intercept correctly but skipped the anti-log step.
When back-conversion is needed vs. when it is not. The gradient is sometimes a direct read (gradient = n when the comparison is m = n). The intercept is almost never a direct read when logarithms are involved: if c = lg a, then a = 10^c; if c = ln a, then a = e^c. For non-logarithmic reductions (e.g. the equation is rewritten as Y = mX + c with no logs), the intercept may be a direct constant and no anti-log is needed.
The fix. After reading the gradient and intercept from the graph, go back to the comparison table and substitute into each expression. State clearly: “gradient = n = 2.5” (direct) and “Y-intercept = lg a = 0.6, so a = 10^0.6 = 3.98.” The extra line showing the anti-log is the back-conversion step that earns the final mark.

Mistake 5

Using lg when the equation calls for ln (or vice versa)

When a question gives an equation of the form y = ae^bx (with a natural exponential base e), taking ln of both sides gives ln y = ln a + bx. Students who take lg instead of ln get the rearrangement wrong at the intercept: the intercept is lg a (not ln a), and when they back-convert they use 10 raised to the power of the intercept value to find a, which is correct for lg but wrong for the original e-based equation. Conversely, for y = ab^x, the correct log base is lg (or any consistent base), and using ln produces a valid rearrangement but gives a different intercept value that must be used consistently throughout.

The rule of thumb. When the equation contains e (the natural base), use ln throughout. When the equation is a power or general exponential not involving e, use lg. The key is consistency: whichever base is chosen, the back-conversion must use the same base.
Example. y = 3e^2x. Taking ln: ln y = ln 3 + 2x. Comparison: Y = ln y, X = x, gradient = 2 = b, intercept = ln a = ln 3 ≈ 1.099. Back-convert: a = e^1.099 ≈ 3. A student who uses lg instead gets intercept = lg 3 ≈ 0.477, then reports a = 10^0.477 ≈ 3 (coincidentally correct magnitude because lg and ln give the same anti-log when the original constant is a nice number), but the gradient step differs and the working is invalid for a question that gives y = ae^bx.
The fix. Before taking any log, identify the base of the exponential. Write at the top of the working: “equation contains e: use ln throughout,” or “equation is y = ab^x: use lg throughout.” This one-line label prevents log-base confusion across the rest of the question.

What this means for your child

Linear law marks are won in the comparison step and the back-conversion step

Linear law is a chapter where a student who understands the idea of “making a straight line” can still lose most of the marks by rushing through the comparison table or skipping the back-conversion. Both of those steps are short, explicit lines of working that are also mark-earning lines. Skipping them to save time costs marks instead.

In class, Teacher Au teaches linear law by separating it into three visible stages: rearrange (write the log rules you are using), compare (fill in the four-cell table of Y, X, m, c), and recover (back-convert from graph readings to original constants). Students who practise this three-stage method consistently do not confuse which expression is Y and do not forget the anti-log, because each stage has a visible checkpoint.

Linear law is one of the topics where a session of targeted practice on just the comparison step and the back-conversion step produces a measurable improvement in marks, without having to review the entire chapter again.

Common mistakes in IGCSE Add Maths logarithms

Linear law uses the logarithm rules directly. If log rules feel uncertain, see the five most common logarithm mistakes in IGCSE Add Maths and how each one is corrected.

Read the logarithm mistakes guide

What is the hardest topic in IGCSE Add Maths?

Linear law is a mid-difficulty chapter. See the full topic difficulty ranking and which chapters carry the most marks across Paper 1 and Paper 2.

Read the topic difficulty guide

Questions parents ask

Frequently asked questions

What are the most common mistakes in IGCSE add maths linear law?

The five most common IGCSE Add Maths linear law mistakes are: swapping Y and X when comparing to Y equals mX plus c, applying logarithm rules wrong when rearranging, reading gradient or intercept from the wrong axis, not converting the intercept back into the original constant (forgetting the anti-log step), and using lg when the equation contains a natural exponential base e (should use ln instead).

What is linear law in IGCSE additional mathematics?

How do you reduce y equals ax to the power n to linear form?

How do you find a and b from a linear law graph?

What is the difference between lg and ln in IGCSE add maths linear law?

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