Writing Line Equations

Idea

To describe a line you need two pieces of information: how steep it is, and one point it passes through.

Slope alone is not enough, because many parallel lines share it. One point alone is not enough either, because many lines pass through it. Put the two together and only one line is left. The point-slope form writes it down for you.

Rule

If a line has slope $m$ and passes through the known point $(x_0, y_0)$, its equation is

Everything else on this page is a way of finding $m$ and $(x_0, y_0)$ before using this one formula.

How it is used

The formula never changes. Only the way you find $m$ and a point changes.

Two things are worth remembering.

If the line passes through the origin, the constant term is $0$. The equation always has the form $ax + by = 0$.

If the inclination angle is bigger than $90^\circ$, the tangent is negative, so the line falls. For example $\tan 135^\circ = -1$, not $1$.

At the end, clear any fractions and move everything to one side. This gives the standard form $ax + by + c = 0$. Answers are usually written this way.

Explore

Drag A and B. The blue line reaches the edges of the window, because a line has no ends. The green circles mark where it crosses the axes. Move one point and watch the equation change. Try putting B on the origin. Then line the two points up vertically and watch the slope become undefined.