Point-Slope Equation of a Line

Have a play with it (move the point, try different slopes):

Now let's discover more.

What does it stand for?

(x₁, y₁) is a known point

m is the slope of the line

(x, y) is any other point on the line

Making sense of it

It is based on the slope:

Slope m = change in y change in x = y − y₁ x − x₁

Starting with the slope:

we rearrange it like this:

So, it is just the slope formula in a different way!

Now let us see how to use it.

Example 1:

slope "m" = 31 = 3

We know m , and also know that (x₁, y₁) = (3, 2) , and so we have:

That is a perfectly good answer, but we can simplify it a little:

Example 2:

m = −3 1 = −3

We can pick any point for (x₁, y₁) , so let's choose (0,0) , and we have:

Which can be simplified to:

Example 3: Vertical Line

What is the equation for a vertical line?
The slope is undefined!

In fact, this is a special case, and we use a different equation, like this:

Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5

What About y = mx + b ?

You may already be familiar with the y=mx+b form (called the slope-intercept form of the equation of a line).

It is the same equation, in a different form!

The "b" value (called the y-intercept) is where the line crosses the y-axis.

So point (x₁, y₁) is actually (0, b)

and the equation becomes: