How to Find the Equation in Point Slope Form﻿

Written by: Miguel Santiago (The Vegan Math Guy)

You might’ve had this experience some of my students have:

You found It REALLY hard to find the equation in point slope form in algebra class.

No matter how many hours they’d flip through their notes, they couldn’t get to the answer. In short, the next best bet is getting really effective algebra textbook survival guides and an amazing tutor (I’m a little biased :).

All jokes aside- No matter if you’re in high school or college algebra, finding the slope of a line is possible if you explain it as it is: easy.

In this post you’re going to learn what point slope form is, the equation of a line and a slope, and two examples to illustrate just how it can be.

What is point slope form?

Let’s discuss what point slope form (also called point-gradient form) is and in what situation we use it.

Simply put, it’s used to find the equation of a straight line. It’s really useful because all you need is the slope of the line and a point to find the equation.

The Equation of a Line

When you find the equation of a straight line, this is the form you can put it in:

yy1=m(xx1)

In the equation you can see the variables x and y. There’s also x1 and y1 which represent the point of a line on the x and y axis. Lastly we have the m value which shows the slope of the line.

So if we put this into a simple context, let’s take a look at this coordinate plane:

Imagine there’s a line that passes through the point (1,3) with a slope of -⅕ which is represented by the vertical blue line (-1) and horizontal one (5). Hypothetically, that’s how it would be expressed on a coordinate plane. Putting it in point slope form, it would like this:

y3=-⅕(x1)

Step by Step Examples: Find the equation in point slope form

I’m going to show you two typical examples of how to find the equation in point slope form. Check out the video below to see them step by step and read below it to see it in written form:

Let’s take a look at the two different scenarios:

Example Problem #1 : Using point slope form, find the equation of the line with slope m=⅓ and passing through (2,-1).

This is most likely the easiest scenario that you’re going to see in a point slope question. Given that we know point slope form is yy1=m(xx1), we need two things to solve this: the point that a line passes through and the slope:

1)Right away we know two key things:

(x1, y1)= (2,-1) and m=⅓.

2)Plug in those key values into the point slope equation  yy1=m(xx1):

y-(-1)=⅓(x-2)

3)Simplify the left side of the equation:

y-(-1) becomes y+1

4) The final simplified point slope equation is: y+1=⅓(x-2).

Example Problem #2 : Using point slope form, find the equation of the line passing through (-5,13) and (3,-3).

In this case you can see there is no slope given but two points in which the line is passing through.

1) We know our (x1, y1). So we can choose any of the two points to plug it into the equation: (x1, y1)= (-5/13).

2)We don’t know the slope but we can plug in numbers in this equation to find it:
m=rise = y2-y1
run  x2-x1

In this case the equation of a slope would look like this with the numbers plugged in:

m= (-3-13)
3-(-5)

3) Let’s simplify m= (-3-13)
3-(-5)

(-3-13)=-16=-2
3-(-5) 8

Now you have your slope (-2) and (x1, y1).

4) Lastly, let’s plug the numbers into yy1=m(xx1) and simply the point slope equation:

y-13=-2(x-(-5))

In this case all we need to simplify is the right side:

y-13=-2(x+5)

And that is our equation in point slope form!

Conclusion: Practice Always Makes Perfect

Understanding the core concepts of point slope form might be pretty easy. But as the old saying goes, practice makes perfect.

If you’re in a course that uses linear algebra, things can get pretty confusing fast without great study materials.