Standard, Point-Slope and Slope-Intercept Line Equations

Welcome to Algebra, a world where letters and numbers intermingle together to... maybe make your head spin in class once in a while. Crazy, right?! One of the most fundamental concepts in Algebra is the equation of a line. Technically speaking, it has three forms: the point-slope form, the slope-intercept form (y = mx + b), and the standard form (Ax + By = C). In this guide, we'll be focusing on the first (point-slope) form; you'll be able to approach the point-slope form and appreciate how easy it actually is! 📈

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The Equation

The Point-Slope Form looks like this:

y-y1=m(x-x1)

Many people prefer the point-slope form because it’s super easy to use and the potential of making an error is very minimal, which is good (aka less chances to mess up on your quiz or test)! ❌

This form is also super advantageous because you only need the slope and a point of the line or just two points in the Cartesian plane in order to form the equation! How fun is THAT? 😆

With this form, you can convert to slope-intercept form, standard form, or pretty much any linear equation form of your choice with a little bit of algebra! Doesn’t that sound so fun? Let’s see how we can use it! 🎠

How to Use the Point-Slope Form

Using the equation isn’t that hard, though at first you might scratch your head and wonder, “why are there two x and ys?” 🤔 Don’t get intimidated just because there are two variables (x and y). Looking back to the formula above, you'll plug in numbers for y1and x1and end up with the form you're used to. Just remember: don’t mess with x and y! Only mess with the ones that have a 1 attached to it. Let's repeat that one more time: don't mess with x and y; only mess with x1and  y1! 🧩

When you’re doing these problems, you’ll usually be given the slope of the line and a point. 

(1) What you’ll do with that information is you’ll first plug in the x-coordinate for x1and the y-coordinate for y1. Remember that since the equation already contains a subtraction, if your x-coordinate is positive, you should leave it as a subtraction expression, for example, x-3. If your x-coordinate is negative, it should be x+3. 🔌

(2) Then, you’ll plug in the slope for m because "m" in the equation represents the slope of the line (why? well, m supposedly stands for either modulus or monter, which is a French word, but no one knows for sure. ⛰️

(3) Anyway, aside from the “where did m come from” question, the next step would be to… do nothing. That’s right! Do NOTHING! 😳 Strange as it sounds, you can leave it as it is. For example, you can leave it as y-4=2(x-3) and you’ll be totally fine. Of course, if your teacher tells you to simplify it into a different form, you’ll have to work it out. 

But what does this mean? Well, let’s first look at an example and work our way through. 🙌

Example 1

Find the equation of a line that has the slope of 2 and passes through (3, 4).

So, how would you solve this problem? First of all, let’s look at the equation. 🔎

y-y1=m(x-x1)

(1) You have the right equation. So far, so good! Then, let’s look at the x- and y-coordinates. It’s 3 and 4, respectively. You then plug in 3 for x1 and 4 for y1. You should have an equation that looks something like this afterwards:

y-4=m(x-3)

(2) Then look at the slope. What is it? It’s 2. So plug in 2 for m and you’ll have the equation look like this:

y-4=2(x-3)

Woohoo! We just found the equation! How fun and cool! 😎 If you're allowed (and if the directions say so), you can leave it as it is in this form. But unfortunately, most teachers will ask you to either change it to slope-intercept form or standard form. 

Here’s the process to simplify into slope-intercept form (y = mx + b). Again, here's the equation in its good ol' point-slope form:

y-4=2(x-3)

(1) Distribute the 2 to x and -3. 

y-4=2x-6

(2) Then, add 4 to both sides. 

y=2x-2

And tada! Only a few steps in and you already have the equation converted into slope-intercept form! 🎉

Now if we’re going to convert it into standard form, it’ll take us a little longer. 

y=2x-2

(1) Subtract 2x from both sides. The 2x on the right side will be cancelled out.

y-2x= -2

(2) Then, rearrange the order of x and y. 🚲

-2x+y=2

(3) Almost there! Since we can’t let A from Ax + By = C (standard form equation) be negative, we’ll have to multiply the equation by -1. 💥

2x-y=2

So voila! Final result for standard form! 😍

Wait, What if Slope is Not Given?

Sometimes, teachers won't give you the slope of the line. Don’t worry, it's not the end of the world yet! Instead of the slope, they’ll provide you with two points. It might end up taking a bit more work on your end, but oh well, school is school. 🏫

In this case, you’ll use the slope formula [insert link here] to find the slope. Remember what the slope formula is? The "rise-over-run" formula?

Now with this equation, you’ll just use the two x- and y- coordinates to find the slope! When you’re plugging in your coordinates into the equation, you get the freedom to choose whichever point you want to use! 🌟

Let’s look at an example!

Example 2

Find the equation of a line that passes through (6, 1) and (4, 5).

So in this case, you don’t have a slope. Again, don’t panic! Just plug it into the slope formula and you’ll have it! 😉

(1) With the slope formula, plug in the coordinates for the appropriate variable!

(2) When you subtract, you should get this:

Since 4 is divisible by -2,

Your slope (m) should equal -2! 

(3) Not hard right? Then all you have to do is plug it in for the equation. 

y-1= -2 (x-6)

And that’s it! Hurray! 

If you convert it to slope-intercept form, it should look like:

y-4 = -2x + 12

y = -2x + 16

If you convert it to standard form, it should look like:

y= -2x + 16

2x + y = 16

That’s it for this example. You're done! 😝

Wrapping Up: Point-Slope Who?!

As you can see, using, finding, and applying point-slope form is actually super fun and straightforward with practice! It’s really easy to use; in fact, most people use the point-slope form of an equation to change it to its other forms. ➡️

Feeling hyped up now? Go finish your math homework! ✨

This you?

(Image courtesy of Quote Master)