# Find Equation of Line From 2 Points

### Slope intercept vs Point Slope Form

There are a few different ways to find the equation of line from 2 points.

The first half of this page will focus on writing the equation in slope intercept form like example 1 below.

However, if you are comfortable using the point slope form of a line, then skip to the second part of this page because writing the equation from 2 points is easier with point slope form .

### Example

#### Equation from 2 points using Slope Intercept Form

Find the equation of a line through the points (3,7) and (5,11)

Step 1 Step 2

Substitute the slope for 'm' in the slope intercept form of the equation

y = mx +b
y = 2x +b

Step 3

Substitute either point into the equation. You can use either (3,7) or (5,11)

Step 4

Solve for b, which is the y-intercept of the line

Step 5

Substitute b, -1, into the equation from step 2

### Practice Problems

Step 1
Step 2

Substitute the slope for 'm' in the slope intercept form of the equation

y = mx +b
y = ½x +b

Step 3

Substitute either point into the equation. You can use either (4,5) or (8,7)

Step 4

Solve for b, which is the y-intercept of the line

Step 5

Substitute b, 3, into the equation from step 2

Step 1

Calculate the slope

Step 2

Substitute the slope for 'm' in the slope intercept equation

Step 3

Substitute either point into the equation. You can use either (-6,7) or (-9,8)

Step 4

Solve for b, which is the y-intercept of the line

Step 5

Substitute b, 5, into the equation from step 2

$$y = \frac{1}{3}x +\red{b} \\ y = \frac{1}{3}x +\red{5}$$

Step 1

Calculate the slope

Step 2

Substitute the slope for 'm' in the slope intercept equation

Step 3

Substitute either point into the equation. You can use either (-3,6) or (15,-6)

Step 4

Solve for b

Step 5

Substitute b, -1, into the equation from step 2

### Example 2

#### Equation from 2 points using Point Slope Form

As explained at the top, point slope form is the easier way to go. Instead of 5 steps, you can find the line's equation in 3 steps, 2 of which are very easy and require nothing more than substitution! In fact, the only calculation, that you're going to make is for the slope.

The main advantage, in this case, is that you do not have to solve for 'b' like you do with slope intercept from.

Find the equation of a line through the points (3,7) and (5,11)

Step 1

Calculate the slope from the 2 points

Step 2

Substitute the slope for 'm' in the point slope equation

y − y1 = m(x −x1)
y − y1 = 2(x −x1)

Step 3

Substitute either point as x1, y1 in the equation. You can use either (3,7) or (5,11)

using (3,7):
y − 7 = 2(x− 3)

using (5,11):
y − 11 = 2(x − 5)

### Practice Problems

Step 1
Step 2

Substitute the slope for 'm' in the point slope equation

y − y1 = m(x −x1)
y − y1 = ½(x −x1)

Step 3

Substitute either point into the equation. You can use either (4,5) or (8,7)

using (4,5):
y − 5 = ½(x − 4)

using (5,11) :
y − 11 = ½(x − 5)

Step 1
Step 2

Substitute the slope for 'm' in the point slope equation

y − y1 = m(x −x1)
y − y1 = (x −x1)

Step 3

Substitute either point into the equation. You can use either (-6,7) or (-9,8)

using (-6,7):
y − 7 = (x + 6)

using (-9, 8):
y − 8 = (x +9)

Step 1
Step 2

Substitute the slope for 'm' in the point slope equation

y − y1 = m(x −x1)
y − y1 = (x −x1)

Step 3

Substitute either point into the equation -3,6) and (15,-6)

using (-3, 6):
y − 6 = (x + 3)

using (15, -6):
y + 6 = (x − 15 )

If you read this whole page and looked at both methods (slope intercept form and point slope, you can see that it's substantially quicker to find the equation of line through 2 points by means of point slope