Intercept Form

This section specializes the two-point form of Section 7 to the case where the two given points are the line's own x- and y-intercepts — the two places it crosses the coordinate axes.

Setting up. Suppose a line meets the x-axis at $(a,0)$ and the y-axis at $(0,b)$, where $a$ and $b$ (both non-zero) are respectively called the x-intercept and y-intercept of the line.

Derivation. Apply the two-point form of Section 7 with $(x_1,y_1)=(a,0)$ and $(x_2,y_2)=(0,b)$:

Multiply both sides by $a$: $ay = -b(x-a) = -bx+ab$. Rearranging: $bx+ay=ab$. Finally divide throughout by $ab$ (valid since $a,b\ne0$):

— the intercept form. It is the fastest form to write down whenever the two axis-intercepts are already known, and equally the fastest form from which to read off the intercepts once the equation is already in this shape.

Sign convention. $a$ and $b$ carry sign: a negative x-intercept means the line crosses the x-axis on the negative side of the origin, and likewise for $b$. So a line making intercepts "$4$ and $-6$" crosses the axes at $(4,0)$ and $(0,-6)$ — not at two points both taken as positive lengths.

Converting a general equation to intercept form. Given $Ax+By+C=0$ ($A,B,C$ all non-zero), move the constant across and divide by it: $Ax+By=-C \implies \dfrac{x}{-C/A}+\dfrac{y}{-C/B}=1$, giving $a=-C/A$ and $b=-C/B$ directly, without needing to substitute $y=0$ and $x=0$ separately. …