Point-Slope Equation of a Line

The "point-slope" form of the equation of a straight line is:

y − y1 = m(x − x1)

The equation is useful when we know:

and want to find other points on the line. Let's find how.

What does it stand for?

graph with slope m

(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:


to get this:

  equation of line slope rearrange

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

Now let us see how to use it.

Example 1:

graph with slope m=3

slope "m"  =  31  =  3

y − y1 = m(x − x1)

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

y − 2 = 3(x − 3)

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

y − 2 = 3x − 9

y = 3x − 9 + 2

y = 3x − 7

Example 2:

y=-3x graph

m = −3 1 = −3

y − y1 = m(x − x1)

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

y − 0 = −3(x − 0)

Which can be simplified to:

y = −3x

Example 3: Vertical Line

graph x=2

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:

x = 1.5

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 at (0, b)

and the equation becomes:

Start withy − y1 = m(x − x1)
(x1, y1) is actually (0, b):y − b = m(x − 0)
Which is:y − b = mx
Put b on other side:y = mx + b