Straight lines

The gradient of a line as a constant rate of change, the three forms of a line's equation — y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0 — and how to move between them.

Pure · Coordinate geometry

Straight lines

The gradient of a line as a constant rate of change, the three forms of a line's equation — y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0 — and how to move between them.

Why it works

A line is straight precisely because its steepness never changes. Pick any two points on it,

(x_1, y_1)

and

(x_2, y_2)

, and the gradient

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

comes out the same every time. That single number — change in

y

per unit change in

x

— is the line. Get the order consistent: whichever point's

y

you put first on top, its

x

must go first on the bottom.

Once you have a gradient and one point, the line is pinned down. The cleanest way to write it is point–gradient form:

y - y_1 = m(x - x_1).

This is just the gradient definition rearranged: any other point

(x, y)

on the line satisfies

\frac{y - y_1}{x - x_1} = m

, and multiplying up gives the form above. It works from any point on the line — you don't need the

y

-intercept.

Tidy it and you get $y = mx + c$ , where

c

is the

y

-intercept (the value at

x = 0

). Multiply out and collect everything on one side and you get the general form $ax + by + c = 0$ with integer coefficients — the form exam mark schemes usually want, and the only one that can describe a vertical line (

x = k

), which has no gradient and so no "

y = mx + c

".

To read the gradient straight off

ax + by + c = 0

, rearrange to

y = -\frac{a}{b}x - \frac{c}{b}

: the gradient is

-\frac{a}{b}

, not

a

. A horizontal line

y = k

has gradient

0

; a vertical line

x = k

has an undefined gradient (the run is zero, so you'd divide by

0