Equations of straight lines

Maths: Analysis and Approaches · Functions. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A straight line is fixed by its gradient and one point on it. The gradient m measures steepness: the change in y divided by the change in x between any two points on the line. A line that falls from left to right has a negative gradient.

The same line can be written in three forms. y = mx + c shows the gradient and the y-intercept. y − y₁ = m(x − x₁) is the quickest to write when you know a point. ax + by + d = 0 is the general form.

Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to give −1.

Rules to remember

gradient = change in y ÷ change in x (m = (y₂ − y₁)/(x₂ − x₁))
y = mx + c: gradient m, y-intercept (0, c)
Through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁)
From ax + by + d = 0: gradient = −a/b; rearrange to y = ... to check
x-intercept: put y = 0. y-intercept: put x = 0
Parallel: m₁ = m₂. Perpendicular: m₁ × m₂ = −1, so m₂ = −1/m₁
Midpoint of two points: ((x₁ + x₂)/2, (y₁ + y₂)/2)

Common mistake

Reading the gradient straight from ax + by + d = 0 as a. Rearrange to y = mx + c first: 3x + 2y − 5 = 0 has gradient −3/2, not 3.

Worked example

Find the equation of the line through (−1, 4) and (3, −2) in the form ax + by + d = 0.

m = (−2 − 4)/(3 − (−1)) = −6/4 = −3/2.
y − 4 = −(3/2)(x + 1). Multiply by 2: 2y − 8 = −3x − 3.
Rearrange: 3x + 2y − 5 = 0. Check with (3, −2): 9 − 4 − 5 = 0.

Answer: 3x + 2y − 5 = 0

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Topic 2: Functions

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