Equation of a Straight Line — Pure Mathematics 1 (9709) — Mathematics

Introduction

The ability to find the equation of a straight line is one of the most frequently examined skills in CAIE 9709 Pure Mathematics 1. It underpins nearly every topic in coordinate geometry — from circles and transformations to calculus, where tangents and normals are straight lines whose equations must be found. Marks are regularly awarded (and lost) on this single skill, so mastering it cleanly and efficiently is essential.

Core Concept

A straight line is uniquely determined whenever you know sufficient information about it. "Sufficient information" in the 9709 context means:

Two points on the line, or
One point on the line and its gradient.

From either of these, you can write down the equation of the line.

The Gradient

The gradient $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

m = \frac{y_2 - y_1}{x_2 - x_1}

The gradient measures the rate of change of $y$ with respect to $x$ . A positive gradient slopes upward left-to-right; a negative gradient slopes downward.

Connecting Gradient to the Equation

Once $m$ is known, together with any point $(x_1, y_1)$ on the line, the equation follows immediately from the point–gradient form (see Key Formulae below). All other standard forms are simply rearrangements of this single relationship.

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Introduction

Core Concept

The Gradient

Connecting Gradient to the Equation

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