Linear Law is one of the most predictable topics in the O-Level Additional Mathematics syllabus. Almost every paper has a question on it, the structure barely changes from year to year, and the marks are very gettable once you know the routine. The catch is that many students lose marks not because the algebra is hard, but because they rush the conversion step or read values off the wrong points on the graph.
Here is the method our tutors use, step by step.
Why Linear Law Exists
A straight line is the only graph whose gradient and intercept you can measure reliably by hand. So if a relationship between two variables is not linear, we rearrange it until it looks like:
Y = mX + c
where Y and X are expressions built from the original variables, m is the gradient, and c is the vertical intercept. Once the graph is straight, you can find the unknown constants from the gradient and intercept.
The golden rule: X and Y must contain only the variables, never the unknown constants. The constants must live inside m and c.
The Three Question Types You Will Meet
1. Equations with a power of x
Example: y = ax^n, where a and n are constants.
Take lg of both sides:
lg y = lg a + n lg x
So lg y = n(lg x) + lg a.
- Y = lg y
- X = lg x
- Gradient m = n
- Intercept c = lg a, so a = 10^c
2. Equations with x in the exponent
Example: y = ab^x.
lg y = lg a + x lg b
- Y = lg y
- X = x
- m = lg b, so b = 10^m
- c = lg a, so a = 10^c
Notice the difference from type 1. If x is the power, you plot x itself on the horizontal axis, not lg x. Mixing these two up is the single most common Linear Law error we see.
3. Equations you rearrange by dividing
Example: xy = ax + b.
Divide every term by x:
y = a + b(1/x)
- Y = y
- X = 1/x
- m = b
- c = a
Another common one: y = ax^2 + bx. Divide by x to get y/x = ax + b, so plot y/x against x.
And one with a surd: y = a√x + b/√x. Multiply by √x to get y√x = ax + b, so plot y√x against x.
A Reliable Five-Step Routine
- Rearrange the given equation so that all constants sit as coefficients or as the standalone term.
- Label clearly: write "Y = ...", "X = ...", "m = ...", "c = ..." on your working. Examiners award method marks for this.
- Build the table. Compute the transformed values to at least 3 significant figures, or as the question asks. Round consistently.
- Plot and draw the line of best fit with a ruler and a sharp pencil. Use as much of the grid as your scale allows.
- Read off two points that lie ON YOUR LINE, not from the original data table, and compute the gradient. Then substitute back to find the constants.
Mark Scheme Details That Cost Marks
- Use the line, not the data. Cambridge markers expect gradient calculations to come from points on your drawn line. Circle the two points you used and show the subtraction clearly.
- Spread your points out. Choosing two points close together magnifies reading error and can push your answer outside the accepted range.
- Answer in the right variable. If the question says "estimate the value of y when x = 5", you often need to convert back. Reading lg y = 1.4 off the graph means y = 10^1.4, not 1.4.
- Watch lg versus ln. If the question uses ln, then a = e^c, not 10^c.
- Do not force the line through the origin unless the algebra shows c = 0.
- Check the axes labels. If you plotted lg y against lg x, label the axes exactly that way. A missing label is a soft mark lost.
Quick Self-Check Before You Move On
Ask yourself these three questions after every Linear Law attempt:
- Does my expression for Y contain any unknown constants? If yes, rearrange again.
- Is my gradient positive or negative, and does the sign match my drawn line?
- Have I converted my final answers back to the original constants the question asked for?
How to Practise
Work through the Linear Law question in five to six past papers back to back in one sitting. You will notice the same three shapes repeating. Once you can spot the type in ten seconds and write down Y, X, m and c before touching the graph paper, you have essentially banked those marks.
If your conversions are shaky, or if your graph readings never quite land in the accepted range, our A Math tutors can walk you through it live and mark your working the way a Cambridge examiner would.
Ready to try it? Message us on WhatsApp at https://wa.link/dsgbkf to book a free trial class with Dojo Education.