How to Graph Linear Equations
A single linear equation cannot be solved for specific values of x and y, it can only be solved for one variable in terms of the other variable. This leaves you with an infinite set of solutions to satisfy the equation.
Graphing Linear Equations
As I mentioned in the last video, a single linear equation can’t be solved for specific values of x and y. It can only be solved for one variable in terms of the other variable. What that leaves us with, though, is a set of solutions, an infinite set of solutions that will satisfy the equation. Let me show you what I mean by that.
Let’s say we have the linear equation y equals 2x minus 1. Now, we could plug any value of x into this equation and it would generate a value of y, for which the equation would be correct. What you have is, as I mentioned, an infinite set of pairings that satisfy the equation. We can represent that graphically with a line, because a line extends infinitely in both directions.
By graphing a line, we can we can show graphically that any point on that line is a set of values for x and y that will satisfy a given equation. This equation here can be graphed by substituting in for x, a value of x, and seeing what the resultant y value is, and plotting that point, then plugging in a second value for x, getting a second y value, and plotting that point.
Because each line can be defined by two points, that will give us the line that represents this equation. Let’s start out plugging 0 in for x. By plugging 0 in for x, we get 0 minus -1, so y equals -1 when x equals 0. We have our x axis and our y axis. When x equals 0 and y equals -1, that’s this point right here. I’ll go ahead and mark this 0,-1.
That’s one point that’s on this line. It corresponds to x equals 0, y equals -1, which is one set of values that satisfies the equation. Let’s plug in 1 now for x and see what we get. 2 times 1 is 2, minus 1 is 1. When x equals 1, y equals 1. If we plot that point as well, we get x equals 1, y equals 1, and another point that’s on our line is right here.
Now that we have two points, we can plot our line, because two points can only represent, or can only lie on, one single line. This is the line that is the solution set for this linear equation. There are a couple of features that I want to point out about a graphed line that can tell us things about the equation that it represents.
In this equation, for instance, the line crosses the y axis at -1. This is what’s called the “y intercept”. The line intercepts the y axis at a particular point, in this case, -1. The y intercept, when you have an equation of this form, is going to be this part of the equation right here the part of the right side of the equation that doesn’t have an x next to it.
That’s the y intercept. We also have what’s called the “slope”. The slope of a line is defined as the change in the vertical coordinate divided by the change in the horizontal coordinate. For a given interval on this line—it doesn’t matter how big the interval is, but we’re just going to take the smallest one that we have graphed here.
For any given interval, the slope is the change in the vertical divided by the change in the horizontal. In this case, the change in the vertical is 2. We go from -1 to 1. The change in the horizontal is 1. We go from 0 to 1. The slope here is 2 divided by 1, or just 2. The slope, when you have an equation of this form here, is going to be the coefficient of the x term.
This equation here is in what’s called “slope intercept form”, and I’m going to talk a little bit more about that in a later video. These are just some of the key features to note on a graph of a line. You have the y intercept and you have the slope. You can either use the graph of the equation to find the equation, or you can use the equation to graphically show the solution set.