Formulas/maths/Straight Lines/Intercept Form

Intercept Form

Line with x-intercept a and y-intercept b. Fails when the line passes through the origin.

Derivation

A line cuts the $x$ -axis at $A(a, 0)$ and the $y$ -axis at $B(0, b)$ , where $a \neq 0$ and $b \neq 0$ .

Apply the two-point form with $P_1 = (a, 0)$ and $P_2 = (0, b)$ :

\frac{y - 0}{b - 0} = \frac{x - a}{0 - a}

\frac{y}{b} = \frac{x - a}{-a} = -\frac{x}{a} + 1

Rearranging:

\boxed{\frac{x}{a} + \frac{y}{b} = 1}

The intercepts $a$ and $b$ appear directly as denominators — the form is named for this.

Warning

This form fails when the line passes through the origin ($a = 0$ or $b = 0$), since the intercepts vanish. Use slope-intercept or general form instead.

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