Equation of Line from 2 Points

Try it below:

Enter any two distinct points to calculate and view the equation of the line in the form $ax + by + c = 0$ .

Enter any two distinct points that in the inputs below to see which equation they form.

Point 1

X₁:Y₁:

Point 2

X₂:Y₂:

Method:

Given two distinct points, say $(x_1, y_1)$ and $(x_2, y_2)$ , we can compute the slope:

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Then use point-slope form:

$y - y_1 = m(x - x_1)$

Simplify and rearrange to get the equation in standard form:

$ax + by + c = 0$

If slope is zero because $y_1 = y_2$ , then the equation is simply;

$y - y_1 = 0$

But if slope is infinite or undefined because $x_1 = x_2$ , then the equation is simply;

$x - x_1 = 0$

Theory:

An equation for a line describes all the coordinates for which a given equation is satisfied.

$x + y = 5$

This equation has two variables and describes an infinite number of points on a single straight line where the two variables x and y interact to satisfy the equation.

We are considering a simpler case where we assume the graphing area to be the Cartesian plane, meaning there are only two axes. In a 2 axes system, a linear equation with one or two variables describes a line. However, the same equation describes a plane if we had a 3 axes system.

For a 3 axes system, an equation would need three variables to make a line instead of the two here.

There are multiple ways to define a line in the 2D Cartesian plane. Each method uses a different combination of values:

Two distinct points on the line
One point and the slope
The slope and y-intercept
The slope and x-intercept

All of these can define the same line, just with different types of information, this line can always be written as.

$ax + by + c = 0$