The equation of a straight line

By the end of this lesson

By the end you can find the equation of a straight line from a point and a gradient or from two points, and move between the three forms Cambridge uses.

One line, three outfits

A straight line can wear three different equations. Same line, different outfit for different occasions:

Form	Best for	What you read off
$y = mx + c$	sketching, comparing gradients	gradient $m$ , $y$ -intercept $(0, c)$
$y - y_1 = m(x - x_1)$	writing the equation	passes through $(x_1, y_1)$ with gradient $m$
$ax + by + c = 0$	final answers ("integers, please")	nothing directly: rearrange first

The middle one is the workhorse. It's just the gradient formula rearranged: $\dfrac{y - y_1}{x - x_1} = m$ .

One kind of line can't wear the $y = mx + c$ outfit: a vertical line. Every point on it has the same $x$ -coordinate, so its equation is $x = k$ , and its gradient is undefined. (A horizontal line is $y = k$ , gradient 0.)

Why y − y₁ = m(x − x₁) works, with numbers

Say a line goes through $(1, 2)$ with gradient 3. Which points $(x, y)$ are on it?

A point is on the line exactly when the gradient from $(1, 2)$ to it is 3. Using rise ÷ run from $(1, 2)$ to $(x, y)$ :

$\frac{y - 2}{x - 1} = 3.$

Multiply both sides by $(x - 1)$ : $\;y - 2 = 3(x - 1)$ . (This multiplied-out form also includes the point $(1, 2)$ itself, where the fraction would have been $\tfrac00$ .) That's the whole formula: here $x_1 = 1$ , $y_1 = 2$ (the known point) and $m = 3$ (the gradient).

Test it on a point you can find by hand. From $(1, 2)$ go 1 across and 3 up to $(2, 5)$ . Put it in: $5 - 2 = 3$ and $3(2 - 1) = 3$ ✓. Now a point that's not on the line, $(2, 4)$ : $4 - 2 = 2$ but $3(2 - 1) = 3$ . They don't match, so it isn't on the line ✓.

Tidying up, $y - 2 = 3x - 3$ , so $y = 3x - 1$ : the same line in $y = mx + c$ form.

Worked example: A point and a gradient

Find the equation of the line through $(3, -2)$ with gradient 4.

Have a go first?

Quick check

Write an expressionQuestion 1 · Recall

Multiple choiceQuestion 2 · Recall

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