The idea

Gradient and intercept

Why gradient is rise over run (in that order, signs included), what m and c each control in y = mx + c, and how to test whether a point lies on a line.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Algebra · Straight-line graphs

Gradient and intercept

Why gradient is rise over run (in that order, signs included), what m and c each control in y = mx + c, and how to test whether a point lies on a line.

Why it works

A constant rate of climb

A straight line has one defining property: it climbs at a constant rate. The gradient measures that rate:

m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}

Through

(1, 2)

and

(4, 11)

: rise

= 9

, run

= 3

, so

m = 3

— each unit right climbs 3 up.

Order and signs are the whole game

Subtract the coordinates in the same order top and bottom. Through

(-2, 7)

and

(4, -5)

m = \frac{-5 - 7}{4 - (-2)} = \frac{-12}{6} = -2

— a downhill line has a NEGATIVE gradient, and mixing the order (

\frac{-5-7}{-2-4}

) silently flips the sign. Run-over-rise (

\frac{x}{y}

) is the other classic inversion; the word order "rise over run" is the antidote.

Keep reading — free

The rest of the explanation, plus 2 worked examples you step through move by move.

Start free

Takes a minute — no card.