First, let"s check out it in action. Right here are 2 points (you can drag them) and the equation that the line through them. Explanations follow.

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## The Points

We usage Cartesian collaborates to mark a suggest on a graph by **how far along** and also **how far up** the is:

**Example: The suggest (12,5)**is 12 devices along, and also 5 units up

## Steps

There space 3 actions to discover the Equation that the right Line :

1. Uncover the steep of the line2. Placed the slope and also one point into the "Point-Slope Formula"3. Simplify## Step 1: find the steep (or Gradient) from 2 Points

What is the steep (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 in ~ this chart ...

Slope m = *change in y***change in x** = *yA − yB***xA − xB**

In other words, we:

subtract the Y values, subtract the X values then divideLike this:

m = *change in y* **change in x** = *4−3***6−2** = *1***4** = 0.25

It doesn"t issue which suggest comes first, that still functions out the same. Try swapping the points:

m = *change in y* **change in x** = *3−4* **2−6** = *−1* **−4** = 0.25

Same answer.

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

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

** **

**Start through the "point-slope" formula (x1** and also **y1** room the collaborates of a allude on the line):

y − y1 = m(x − x1)

We can choose **any point** ~ above the line for **x1** and **y1**, for this reason let"s just use point (2,3):

y − 3 = m(x − 2)

We already calculated the slope "m":

**m** = *change in y***change in x** = *4−3***6−2** = *1***4**

And us have:

y − 3 = *1***4**(x − 2)

**That is an answer**, however we have the right to simplify that further.

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## Step 3: Simplify

Start with:y − 3 =

*1*

**4**(x − 2)

Multiply

*1*

**4**and also (x−2):y − 3 =

*x*

**4**−

*2*

**4**

Add 3 to both sides:y =

*x*

**4**−

*2*

**4**+ 3

Simplify:y =

*x*

**4**+

*5*

**2**

And us get:

y = *x***4** + *5***2**

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

### Check It!

Let united state confirm by trial and error with the second point (6,4):

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

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

## Another Example

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

Start with the "point-slope" formula:

y − y1 = m(x − x1)

Put in these values:

x1 = 1y1 = 6m = (2−6)/(3−1) = −4/2 = −2And us get:

y − 6 = −2(x − 1)

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

y − 6 = −2x + 2

y = −2x + 8

DONE!

## The huge Exception

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

A vertical line"s gradient is undefined (because us cannot divide by 0): m = But over there is still a method of composing the equation: usage x = 2 Equation the a straight Line directly Line Graph Calculator Algebra Index |