How to reduce a relation to linear form

Linear Law is about turning a curved relationship between two variables into a straight line. A straight line is easy to draw through data points and its gradient and vertical-axis intercept are easy to measure, so if you can rewrite a relation in the form $Y = mX + c$, you can read the unknown constants straight off the graph.

This method rearranges relations such as $y = ax^{n}$ or $y = ab^{x}$, usually by taking logarithms of both sides, until they match the straight-line shape.

For a power relation, taking $\lg$ of both sides does the job:

Comparing with $Y = mX + c$, you plot $\lg y$ against $\lg x$, read the gradient as $n$, and the intercept as $\lg a$.

Reach for this method when a question gives a relation containing unknown constants together with either experimental data or a described straight-line graph of transformed variables. Phrases such as 'reduce to linear form', 'when $\lg y$ is plotted against $\lg x$ a straight line is obtained', or 'the graph of $\frac{y}{x}$ against $x$ is a straight line' are direct signals.

Choose the transformation from the shape of the relation. A power law $y = ax^{n}$ or an exponential $y = ab^{x}$ calls for taking logarithms so the exponent comes down.

A relation like $y = ax + bx^{2}$ is better handled by dividing through by $x$ to reach $\frac{y}{x} = a + bx$. Deciding the transformation first tells you exactly what to plot on each axis.

1. 1

   Read off the relation and constants

   Note the non-linear relation and which constants you must find, for example $a$ and $n$ in $y = ax^{n}$.

2. 2

   Transform to Y = mX + c

   Take $\lg$ of both sides for a power or exponential relation, or rearrange (for example divide by $x$) so the equation becomes linear.

3. 3

   Match the parts

   Identify $Y$, $X$, the gradient $m$, and the intercept $c$ by comparing your equation with $Y = mX + c$.

4. 4

   Read the graph

   From the straight-line graph, take the gradient and the vertical-axis intercept.

5. 5

   Form equations in the constants

   Set the gradient equal to its expression and the intercept equal to its expression, then solve for each constant.

6. 6

   Undo any logarithm

   If the intercept was $\lg a$, recover $a = 10^{c}$; state the constants and, if asked, the original relation.

• Reading the intercept as $a$ instead of $\lg a$; you must undo the logarithm with $a = 10^{c}$.
• Swapping the axes, being unsure which transformed variable is $X$ and which is $Y$, which flips gradient and intercept.
• Splitting $\lg(ax^{n})$ incorrectly; it equals $\lg a + n\lg x$, not $\lg a \times n\lg x$.
• Forgetting to bring the power down with the power law, leaving $\lg x^{n}$ instead of $n\lg x$.
• Mixing up gradient and intercept when matching the equation to $Y = mX + c$.

The step that trips students most is the very last one, reading the intercept as $a$ rather than $\lg a$, and forgetting to write $a = 10^{c}$. In one-to-one lessons our teachers pause exactly at the comparison line, where you match your equation to $Y = mX + c$, and make sure you label $Y$, $X$, the gradient and the intercept before touching the numbers.

That single habit removes most Linear Law errors at once. We also check that you take $\lg$ of the whole side, not just part of it.

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