Find Equation of Line From 2 Points

Video Tutorial

on Finding the Equation of a 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.

Slope Intercept Form

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

Point Slope Form

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 3

Step 4

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

Step 4 the y intercept

Step 5

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

Final Equation of Line

Practice Problems

Problem 1

Find the equation of a line through the following 2 points: (4,5) and (8,7)

Step 1

step 1 find slope

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 3

Step 4

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

Step 4 the y intercept

Step 5

Substitute b, 3, into the equation from step 2

Final Equation of Line

Problem 2

Find the equation of a line through the following the points: (-6,7) and (-9,8)

Step 1

Calculate the slope

step 1 find slope

Step 2

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

equation of line

Step 3

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

Step 3

Step 4

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

Step 4 the y intercept

Step 5

Substitute b, 5, into the equation from step 2

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

Problem 3

Find the equation of a line through the following the 2 points: (-3,6) and (15,-6)

Step 1

Calculate the slope

step 1 find slope

Step 2

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

equation of line

Step 3

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

Step 3

Step 4

Solve for b

Step 4 the y intercept

Step 5

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

Final Equation of Line

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 1 find slope 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

Problem 1

Find the equation of a line through the following 2 points: (4,5) and (8,7)

Step 1

step 1 find slope

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)

Problem 2

If a line goes through the following 2 points, what is the line's equation? (-6,7) and (-9,8)

Step 1

step 1 find slope

Step 2

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

y − y1 = m(x −x1)
y − y1 = one third(x −x1)

Step 3

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

using (-6,7):
y − 7 = one third(x + 6)

using (-9, 8):
y − 8 = one third(x +9)

Problem 3

Find the equation of a line through the following the 2 points: (-3,6) and (15,-6)

Step 1

step 1 find slope

Step 2

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

y − y1 = m(x −x1)
y − y1 = one third(x −x1)

Step 3

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

using (-3, 6):
y − 6 = one third(x + 3)

using (15, -6):
y + 6 = one third(x − 15 )

If you read this whole page and looked at both methods (lope intercept fro 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

Equation from 2 points Calculator