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Graphing Lines

So now we’ll talk about graphing lines. You may well remember from high school that a big topic of the x-y plane has to do with graphing lines and finding equation of lines. So for example we might have a line like this. We might have to find the equation of a line or we might be given an equation and have to produce this line, something along those lines.

In this lesson we will start with the very basics. This in many ways will just be kind of a conceptual introduction to the idea of graphing lines in the plane. First we have to focus on a few big ideas before we can actually get to the mechanics of actually how to graph a particular line. So big idea number one, every possible line in the x-y plane has its own unique equation.

So there’s this, you could say a one to one pairing between a unique line and a unique equation. So that’s a big idea. Every line has its own equation. Big idea number two. For any given line, all the points on the line have x and y coordinates that satisfy the equation of the line.

So that’s a really big idea. That’s a deep idea and people don’t appreciate how deep that idea is. On any line, there’s an infinite number of points. All infinity of those points, every single one of them, we could pick out any point at all on that line find this x coordinate, y coordinate, plug it in and it would satisfy the equation of the line.

That is absolutely huge. And finally big idea number three. Any linear equation that relates x to the first power to y to the first power, as long as there’s no multiplication or division of variables or you know something odd like square roots or something. So as long as there’s just an ordinary x or an ordinary y and a bunch of numbers, that must be the equation of some line in the x y plane.

So for example, we look at this y is to the first power, x is to the first power. That has to be the, the equation of some line in the x-y plane, and that’s exactly why these are called linear equations. You may remember back in Algebra we were referring to these as linear equations. We were referring to them because every single one of them corresponds to a unique line in the x-y plane.

So let’s look at this particular equation. Suppose the problem gave us that equation. We could find values that satisfy that equations. And these would be points on the line. So for example, we could just plug in. If we plug in x equals 0, then we’d see that’d we’d get 3 y equals 12, so y would equal 4.

So that means that 0,4 has to be one point on the plane. Similarly, we could plug in y equals 0. If y equals 0, then we get negative 4 equals 12. We divide, we get x equals negative 3. So that must be another point on the plane x equals negative 3, y equals 0. So we have two points.

Technically, two points are enough to determine a line. So for example, if we plot those two points and just draw a straight line between them, we get a graph of the line. We were able to draw the line simply by plugging in points. But this is not the most efficient way to plot a line. In the coming lessons of this module, we will learn much more efficient ways to plot the line that represents a particular equation.

So we’ll get to that but first we have to make sure that we understand all the basics here. For now, once again, the really big idea is that any point on the line has to satisfy the equation of the line. This idea can be tested in a variety of contexts. So here’s a relatively easy practice problem, pause the video, and then we’ll talk about this.

Okay. We’re given the equation of the line. And the equation of the line has this variable in it, the variable K. So we don’t know what K is. But we’re told that the line must pass through the point (2, 1). What we know that, that point has to satisfy the equation of the line.

So if we plug in x equals 2 and y equals 1. We’re gonna have to get an equation that works. So we’ll do that. We’ll plug in x equals 2, y equals 1. What we get is 2K plus 3K, 2K plus 3K is 5K, 5K equals 17. And so K must equal seventeen-fifths.

So, that’s the value of K. In summary, every line in the x-y plane has its own unique equation. Every point on the line satisfies the equation of the line. And we can figure out the graph of a line by plotting individual points. This is one option, and we will learn other options in the later videos of this module.