 How to Write an Equation of the Line

Following hot on the heels of last time, where I showed you how to find the slope of a line, this time I’m going to build on that thought by describing a process to write an equation of the line. As I explained, just figuring out the slope of your line isn’t enough to specifically describe you line. You actually need some way of anchoring this line to a fixed point on the graph. You could take your sloped line and slide it all over the graph anywhere that you’d like, and you could still wind up with the exact same slope but a completely different formula for the line. That’s because it’s not rooted to any particular place. Luckily, there are very few things required to accurately and completely describe a straight line, and they aren’t that hard to obtain.

Based on that introduction, you may already have figured out what you need for your line equation. The first component should be obvious, if you’ve been following along – you need the slope. After that, you only need one more thing: your anchor – any point that lies exactly on your line. Taken together, you can say exactly where you line is seated, and exactly at what angle it is rising or falling. And that is all the info that you need.

Before we get to how to write an equation of a line, first you must know the general form. I will follow up this post with additional information on various forms of the equation (for example: standard form, point-slope form), but for now, I’m going to refer to the form known as slope-intercept form. This is likely the way that it was first demonstrated to you in your math class, and I think that is so because it is simple. Your slope is required, obviously, and the point that is used is the y-intercept. The intercepts are where the line crosses each axis, so each line will have an x-intercept and a y-intercept. This simplifies things a bit because one part of each ordered pair describing each intercept is automatically a zero, so it reduces the amount of math that needs to be done!

Write an Equation of the Line

Here is the general way to write an equation of the line in slope-intercept form: • The x and the y are fairly self-explanatory – together, they represent a full ordered pair coordinate.
• The m represents the slope, as I explained in my last post.
• The b is the new concept here – it represents the y-intercept, or more precisely, the y-value of the y-intercept.
• The equation that describes a specific line typically is written in terms of x and y, meaning that the numerical slope and intercept values are written.

In one of the easiest forms of the question, it is extremely simple to determine the equation of the line. If you are given the slope and y-intercept, this is merely a matter of substituting these values into the above expression, and that is it. Check out the following example.

Suppose you know that you have a line whose slope is 4 and it crosses the y-intercept at the point (0,2). Can you figure out how to write an equation of the line? This is about as simple as it gets. Simply plugging numbers into the expression results in the following equation: Easy, right? Let’s try another, slight harder question.

Find the equation of the line whose slope is 5 and passes through the point (2,8).

If you assess the information that you are given, you can see that once again you have the slope provided, but this time you do not have an intercept. Instead, you have a point that is located somewhere off in the middle of the graph and not on an axis. However, if you are paying attention, you may have noticed that indeed, you are given three of the four variables used in our equation. So, this is only a matter of putting in the numbers that we know, figuring out what the b-value is, and then rewriting in the general form in terms of x and y. In this case, here is what we do: Now that we have solved for the y-intercept, it is easy to once again just plug it and the slope value into the general form to express the equation of the line in terms of x and y, like this: So, that is all I really want to say in this particular post. I have now introduced you to the concepts of slope and intercepts, and shown you how to manipulate one of the more common way of expressing lines by using the slope-intercept form of the equation. With this information, you should be able to determine line equations no matter where they are on the graph or what their slope is. As I noted above, I will soon provide a better explanation of some other methods of expressing the equation of your line – which, essentially, are all just derivations and manipulations of one another. If you are comfortable with one form over another, it is a simple matter of converting and rearranging to wind up with something different, albeit describing the exact same line on the graph.

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