# Equations of a straight line

## General equation

We can establish the general equation of a line from the three-point alignment condition.

Given a straight r, being THE(xTHE, yTHE) and B(xB, yB) known and distinct points of r and P(x, y) a generic point, also of rbeing THE, B and P aligned, we can write:

Doing yTHE - yB = a, xB - xTHE = b and xTHEyB - xByTHE= c, as a and b are not simultaneously null we have:

 ax + by + c = 0

(general equation of line r)

This equation relates x and y to any point P generic straight line. So given the point P(m, n):

• if m + bn + c = 0, P is the point of the line;

• if m + bn + c 0, P It is not the point of the line.

• Let's consider the general equation of the line r that goes through THE(1, 3) and B(2, 4).

Considering one point P(x, y) of the line, we have:

• Let's check if the points P(-3, -1) and Q(1, 2) belong to line r of the previous example. Overriding the coordinates of P at x - y + 2 = 0, we have:

-3 - (-1) + 2 = 0 -3 + 1 + 2 = 0

Since equality is true, so P r.

Overriding the coordinates of Q at x - y + 2 = 0, we get:

1 - 2 + 2 0

Since equality is not true, so Q r.

Next: Segmental Equation