Forms of Linear Equations

A straight line can be written in several equivalent forms, each highlighting different features of the line. Knowing when to use each form is a key skill in IB Maths AA.

Gradient-Intercept Form

$y = m x + c$
where $m$ is the gradient and $c$ is the $y$ -intercept. This is the most commonly used form and makes it easy to read off the gradient and $y$ -intercept directly.

General Form

$a x + b y + d = 0$
where $a$ , $b$ , and $d$ are constants. (The letter $d$ is used here to avoid confusion with the $y$ -intercept $c$ in gradient-intercept form.) The gradient is $- \frac{a}{b}$ and the $y$ -intercept is $- \frac{d}{b}$ .

Point-Gradient Form

$y - y_{1} = m (x - x_{1})$
where $(x_{1}, y_{1})$ is any known point on the line and $m$ is the gradient. This form is extremely useful when you are given a point and a gradient but do not know the $y$ -intercept.

Exam Tip

Think of the three forms as tools in a toolkit. Gradient-intercept is best for reading features; point-gradient is best for constructing a line; general form is often required as a "neat" final answer or in problems involving simultaneous equations.

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SL 2.1—Equations of straight lines, parallel and perpendicular

Forms of Linear Equations

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