# Equation of a Line from 2 Points

First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow.

## The Points

We use Cartesian Coordinates to mark a point on a graph by **how far along** and **how far up** it is:

Example: The point **(12,5)** is 12 units along, and 5 units up

## Steps

There are 3 steps to find the Equation of the Straight Line :

- 1. Find the slope of the line
- 2. Put the slope and one point into the "Point-Slope Formula"
- 3. Simplify

## Step 1: Find the Slope (or Gradient) from 2 Points

What is the slope (or gradient) of this line?

We know two points:

- point "A" is (6,4) (at x is 6, y is 4)
- point "B" is (2,3) (at x is 2, y is 3)

The slope is the **change in height** divided by the **change in horizontal distance**.

Looking at this diagram ...

Slope m = \frac{change in y}{change in x} = \frac{y_{A} − y_{B}}{x_{A} − x_{B}}

In other words, we:

- subtract the Y values,
- subtract the X values
- then divide

Like this:

m = \frac{change in y}{change in x} = \frac{4−3}{6−2} = \frac{1}{4} = 0.25

It doesn't matter which point comes first, it still works out the same. Try swapping the points:

m = \frac{change in y}{change in x} = \frac{3−4}{2−6} = \frac{−1}{−4} = 0.25

Same answer.

## Step 2: The "Point-Slope Formula"

Now put that **slope** and **one point** into the "Point-Slope Formula"

Start with the "point-slope" formula (**x _{1}** and

**y**are the coordinates of a point on the line):

_{1}y − y_{1} = m(x − x_{1})

We can choose **any point** on the line for **x _{1}** and

**y**, so let's just use point (2,3):

_{1}y − 3 = m(x − 2)

We already calculated the slope "m":

**m** = \frac{change in y}{change in x} = \frac{4−3}{6−2} = \frac{1}{4}

And we have:

y − 3 = \frac{1}{4}(x − 2)

**That is an answer**, but we can simplify it further.

## Step 3: Simplify

And we get:

y = \frac{x}{4} + \frac{5}{2}

Which is now in the Slope-Intercept (**y = mx + b**) form.

### Check It!

Let us confirm by testing with the second point (6,4):

**y** = **x**/4 + 5/2 = **6**/4 + 2.5 = 1.5 + 2.5 = **4**

Yes, when x=6 then y=4, so it works!

## Another Example

### Example: What is the equation of this line?

Start with the "point-slope" formula:

y − y_{1} = m(x − x_{1})

Put in these values:

- x
_{1}= 1 - y
_{1}= 6 - m = (2−6)/(3−1) = −4/2 = −2

And we get:

y − 6 = −2(x − 1)

Simplify to Slope-Intercept (**y = mx + b**) form:

y − 6 = −2x + 2

y = −2x + 8

DONE!

## The Big Exception

The previous method works nicely except for one particular case: a **vertical line**:

A vertical line's gradient is undefined (because we cannot divide by 0): m = \frac{y_{A} − y_{B}}{x_{A} − x_{B}} = \frac{4 − 1}{2 − 2} = \frac{3}{0} = undefined But there is still a way of writing the equation: use x = 2 |