Equation through two points

Objective. Find gradient and equation from two coordinates.

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2. Worked example
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Learn: the key idea

The equation of a line through two known points is found using the gradient formula m = (y2−y1)/(x2−x1), then substituting into y−y1 = m(x−x1). For points (1,2) and (4,11), m = (11−2)/(4−1) = 9/3 = 3, so using (1,2): y−2 = 3(x−1), giving y = 3x−1. Always choose one of the two given points to substitute into the point-gradient form after finding m. You can check your final equation by substituting the other point in to confirm it also satisfies the equation. This method works for any two distinct, non-vertical points on a straight line.

Worked example

Find the equation of the straight line passing through the points (2, 5) and (6, 13).

1 $Gradient m = (13-5)/(6-2) = 8/4 = 2$
2 $Use point (2,5) in y-y1=m(x-x1): y-5=2(x-2)$
3 $Expand: y-5=2x-4$
4 $Rearrange: y=2x+1$

Answer: y = 2x + 1

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Common mistake. Pupils compute the gradient the wrong way round, as (x2−x1)/(y2−y1) instead of (y2−y1)/(x2−x1). For (1,2) and (4,11) this wrongly gives 3/9=1/3 instead of 3. Always put the change in y over the change in x.

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