7.4 Linear Law Transformations

📖 Main Content

Part 1 — Why Do We Transform Equations?

In real life and in experiments, many relationships between two variables are not linear — meaning if you plot them on a graph, you get a curve, not a straight line.

Curves are much harder to work with than straight lines. It is difficult to read precise values from a curve or to find the equation of a curve from a graph.

The clever solution is this: change what you plot on each axis so that the curve becomes a straight line. This process is called linearisation or applying the linear law.

Once you have a straight line, you can:

• Find the gradient $m$ (the slope of the line)
• Find the $Y$-intercept $c$ (where the line crosses the vertical axis)
• Use these two values to work out the unknown constants in the original equation

Part 2 — The Key Idea: The Standard Linear Form

Every straight line can be written as:

$Y = mX + c$

where:

• $Y$ is what you plot on the vertical axis (it may be a function of $y$, not just $y$ itself)
• $X$ is what you plot on the horizontal axis (it may be a function of $x$, not just $x$ itself)
• $m$ is the gradient — how steeply the line rises or falls
• $c$ is the $Y$-intercept — the value of $Y$ when $X = 0$

Part 3 — Transforming Non-Linear Equations TO Straight-Line Form

3.1 Equations of the Form $y = Ax^n$ (Power Relationships)

This type of equation involves $x$ raised to a power $n$. To linearise it, take logarithms (base 10, written $\lg$) of both sides.

Step-by-step:

$y = Ax^n$

Take $\lg$ of both sides:

$\lg y = \lg(Ax^n)$

Use log rules: $\lg(AB) = \lg A + \lg B$ and $\lg x^n = n\lg x$:

$\lg y = \lg A + n\lg x$

Rearrange to match $Y = mX + c$:

$\lg y = n\lg x + \lg A$

Now compare:

Finding the constants:

• Read the gradient → that gives you $n$ directly
• Read the intercept $c = \lg A$ → solve $A = 10^c$