Mathematics and Statistics · Ch 5 — Locus and Straight Line

The General Form of a Line

Every straight line in the plane — including vertical lines that have no slope — can be written in one common form:

Note

General Form of a Line

$ax + by + c = 0$

where $a$ , $b$ , $c$ are real constants and $a$ , $b$ are not both zero.

This is the form in which lines are most often stated in problems, so it is essential to be able to read useful information straight out of it. Assuming $b \neq 0$ (the line is not vertical), rearrange to slope–intercept form:

by = -ax - c \quad\Rightarrow\quad y = -\dfrac{a}{b}\,x - \dfrac{c}{b}

Comparing with $y = mx + c'$ shows at once:

Note

Reading Slope and Intercepts from $ax + by + c = 0$

$\text{slope } m = -\dfrac{a}{b}, \qquad y\text{-intercept} = -\dfrac{c}{b}, \qquad x\text{-intercept} = -\dfrac{c}{a}$

So for the line $3x + 4y - 12 = 0$ , the slope is $-\tfrac{3}{4}$ , the y-intercept is $-\tfrac{-12}{4} = 3$ , and the x-intercept is $-\tfrac{-12}{3} = 4$ . These same three numbers could be read from its intercept form $\tfrac{x}{4} + \tfrac{y}{3} = 1$ , confirming the two descriptions agree. …

Definition 1General form

$ax + by + c = 0$ with $a, b$ not both zero — the single form that represents every line, includ …

Definition 2Slope from general form

For $ax+by+c=0$ with $b\neq 0$ , the slope is $m = -\dfrac{a}{b}$ ; the y-intercept is $-\dfrac{c}{b}$ and the x-intercep …

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