Linear versus non-linear relations
A relation between two variables x and y is linear if its graph is a straight line. Such a relation can always be written as y = mx + c, where m is the gradient and c is the y-intercept. If the graph is a curve, the relation is non-linear — for example y = ax² + b, y = axn, or y = abx.
Many experimental laws are non-linear, but they can be reduced to the straight-line form Y = mX + c by a suitable substitution. Here X and Y are expressions in the original variables, chosen so that plotting Y against X gives a straight line.
Choosing X and Y
Rewrite the equation so one side is linear in a new pair (X, Y). For y = ax² + b, let X = x² and Y = y; then Y = aX + b, a straight line of gradient a and intercept b.
Worked example
The relation y = 3x² + 5 is non-linear. Put X = x² and Y = y. When x = 2, X = 4 and Y = 3(4) + 5 = 17; when x = 3, X = 9 and Y = 3(9) + 5 = 32. The points (4, 17) and (9, 32) give gradient (32 − 17)/(9 − 4) = 15/5 = 3, matching the coefficient a = 3, and the intercept is 5. So Y = 3X + 5 is a straight line.
Key formula
Linear form: Y = mX + c
gradient m = (Y₂ − Y₁)/(X₂ − X₁)
Y-intercept c = value of Y when X = 0
Remember
- A straight-line graph confirms a linear relation between X and Y.
- m and c are constants read from the line, not from the raw curve.
- Always state clearly what X and Y represent.