Foundations — Equations of a line — slope-intercept, point-slope, two-point, standard (ax+by+c=0)

Before you can read a single line-equation, you need to be fluent in the little symbols hiding inside it. This page builds each one from absolute zero, in the order they depend on each other. Nothing here assumes you've met it before.

0. The playground: the coordinate plane

Everything in this topic lives on a flat sheet with two number-lines crossing at right angles.

Why we need it: a line is an infinite collection of points. To talk about those points with numbers, we first need an address system. The plane is that address system.

1. A point and its coordinates $(x,y)$

Look at figure s01: the green dot sits at $(3,2)$ — walk $3$ right, then $2$ up.

Why the topic needs it: an equation of a line is a test. You feed it a point $(x,y)$ and it answers "yes, you're on the line" or "no." Points are the things being tested.

2. The symbol $\Delta$ — "the change in"

Why the topic needs it: steepness is a comparison between two points — "for THIS much sideways, how much up?" That comparison is exactly $\Delta y$ against $\Delta x$.

3. Rise, run, and the ratio $\frac{\text{rise}}{\text{run}}$

Why a ratio and not just the rise? Because rising $2$ over a run of $1$ is twice as steep as rising $2$ over a run of $4$. Only the ratio — up-per-step — measures true steepness independent of how big a step you took. That is the whole reason we divide.

4. Slope $m$

The four cases you must be able to picture (see Slope of a line):

• $m>0$: line climbs to the right (rise and run have the same sign).
• $m<0$: line falls to the right (rise and run have opposite signs).
• $m=0$: flat — no rise at all, so $\frac{0}{\text{run}}=0$.
• $m$ undefined: perfectly vertical — the run is $0$, and dividing by $0$ has no answer. This is the case slope–intercept form can't handle.

5. The angle $\theta$ and $\tan \theta$

Why does an angle turn up in a chapter about lines? Because "steepness" and "angle of tilt" are two words for the same thing — and the bridge between them is $\tan$.

Why $\tan$ and not $\sin$ or $\cos$? $\sin$ and $\cos$ each involve the slanted length (the hypotenuse), which changes when you pick a bigger triangle. But rise-over-run cancels that length out — exactly what slope needs. $\tan \theta$ is defined as opposite-over-adjacent, so it lands precisely on our rise/run ratio. That's why slope $=\tan \theta$ and not any other trig function.

The follow-up idea, Angle between two lines, reuses this exactly.

6. Intercepts — where a line meets an axis

Why the topic needs it: the slope–intercept form $y=mx+c$ literally names the $y$-intercept $c$, and the Intercept form of a line is built entirely from both intercepts.

7. Constants, coefficients, and letters like $a,b,c$

Why it matters here: in $ax+by+c=0$ (the standard form, and the starting point of Distance of a point from a line), the letters $a,b,c$ are frozen numbers picking out one line, while $x,y$ roam over all its points. Telling constants from variables is what lets you "read off" the slope $-a/b$ without confusion.

8. "If and only if" — what an equation of a line really claims

This is the deep reason a first-degree equation always draws a line — explored in Straight line as a first-degree equation.

How these foundations feed the topic

Read top to bottom: the plane hosts points, points let us measure change, change gives rise/run, rise/run is slope, slope is $\tan \theta$ — and slope plus a point plus the membership idea produce every form of the line equation.

Equipment checklist

Cover the right side and see if you can state each before revealing.