# Point-Slope Equation of a Line

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

y − y_{1} = m(x − x_{1})

The equation is useful when we know:

and want to find other points on the line.

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

Now let's discover more.

## What does it stand for?

(x_{1}, y_{1}) 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 = \frac{change in y}{change in x} = \frac{y − y_{1}}{x − x_{1}}

Starting with the slope: we rearrange it like this:
to get this: |

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

### Now let us see how to use it.

### Example 1:

slope "m" = \frac{3}{1} = 3

y − y_{1} = m(x − x_{1})

We know m, and also know that (x_{1}, y_{1}) = (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:

m = \frac{−3}{1} = −3

y − y_{1} = m(x − x_{1})

We can pick any point for (x_{1}, y_{1}), 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

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**,

thatâ€™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 (x_{1}, y_{1}) is actually at (0_{}, b_{})

and the equation becomes:

_{1}= m(x − x

_{1})

_{1}, y

_{1}) is actually (0

_{}, b

_{}):y − b

_{}= m(x − 0

_{})

_{}= mx

**y = mx + b**