Linear law means rewriting a curved relation so that it looks like Y = mX + c. Once it does, a graph of Y against X is a straight line, and its gradient and intercept carry the unknown constants.
This lesson opens the chapter on linear law in SPM Additional Mathematics. The next lesson, choosing axes for a straight-line graph, turns the rearrangement into points you can plot.
What must the rearranged equation look like?
It needs one expression on the left that you can calculate from data (this is Y), one expression multiplied by a constant m (this is X), and a constant c added on. The letters x and y in the question may sit inside Y or X, but the unknown constants such as a and b may not.
Think of it as a test with three checks:
- Is the left side made only of variables?
- Is there exactly one constant multiplying the X expression?
- Is the remaining term a constant with no variable in it?
How do the common relations convert?
Most SPM questions use a short list of relations. The last two rows need logarithms to base 10, written lg.
| Relation | Rearranged form | Y | X | m | c |
|---|---|---|---|---|---|
| y = ax² + b | y = a(x²) + b | y | x² | a | b |
| y = a/x + b | y = a(1/x) + b | y | 1/x | a | b |
| y = ab^x | lg y = (lg b)x + lg a | lg y | x | lg b | lg a |
| y = ax^n | lg y = n(lg x) + lg a | lg y | lg x | n | lg a |
Worked example: a relation with two variable terms
Given y = hx² + kx, where h and k are constants, reduce it to linear form.
There is no constant term on its own, so the equation is not yet in the right shape. Divide every term by x:
y/x = hx + k
Now Y = y/x, X = x, the gradient is h and the intercept is k. A graph of y/x against x is a straight line.
The tempting wrong move
A tempting move is to spot the x² and write Y = y, X = x², as in the first table row. Then the kx term is left over and the equation cannot be read as Y = mX + c.
The fix is to ask what is left after you choose X. If anything with x in it remains besides m·X, change the division or multiplication until only a constant is left.
Worked example: constants in an index
A colony is modelled by y = p·q^x, where p and q are constants. The constants sit in a base and a constant multiplier, so take lg of both sides:
lg y = lg(p·q^x) = lg p + x lg q
Reorder: lg y = (lg q)x + lg p. So Y = lg y, X = x, m = lg q and c = lg p. Notice that m is lg q, not q, which matters when you find the constants later.
Check yourself
Reduce y = k·x^n, where k and n are constants, to the form Y = mX + c. State Y, X, m and c.
Answer
Take lg of both sides: lg y = lg k + n lg x.
Reorder: lg y = n(lg x) + lg k.
So Y = lg y, X = lg x, m = n, c = lg k. The gradient is the power n itself, and the intercept is lg k, not k.
What to study next
Go to choosing axes for a straight-line graph to practise picking Y and X from a table of data. You can also try your own relations in the linear law axes explorer.
If reductions like these keep stalling on unfamiliar relations, see online one-to-one Additional Mathematics tuition.