Point-Slope Equation of a Line


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

y - y1 = m(x - x1)

Using this formula, when we know:

  • one point on the line
  • and the slope of the line,

we can find other points on the line.

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:

graph

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

 

So this is the slope:

and we can rearrange it like this:

 

to get this:

  m

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"  = 
3
1
= 3

y - y1 = m(x - x1)

We know m now, 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

slope "m"  = 
3
-1
= -3

y - y1 = m(x - x1)

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

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 with y - 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

And that is called the "slope-intercept" form of the equation of a line.