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.
So, let me show you what I mean by that. Let’s say we have linear equation y = 2x – 1.
The transcript is provided for your convenience
Now, we can plug any value of x into this equation, and it would generate a value of y, for which the equation would be correct. And so, what you have is, as I mentioned, an infinite set of pairings that satisfy the equation, and we can represent that graphically with a line, because a line extends infinitely in both directions. And so, by graphing a line, 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.
So, this equation here can be graphed by substituting in for x, a value of x, and seeing what the resulting y value is, and plotting that point, then plugging in a second value for x, getting a second y value, and plotting that point, and because each line can be defined by two points, that will give us the line that represents this equation.
So, let’s start out by plugging 0 in for x. By plugging 0 in for x, we get 0 – 1, or -1, so y = -1 when x = 0. So, we have our x-axis and our y-axis. When x = 0, and y = -1, that’s this point right here. I’ll go ahead and mark this (0,-1).
So, that’s one point that’s on this line, and it corresponds to x = 0, y = -1, which is one set of values that satisfies the equation. But let’s plug in 1 now for x and see what we get. 2 times 1 is 2, minus 1 is 1. So, when x = 1, y = 1. Now, if you plot that point as well, we get x = 1, y = 1, and another point that’s on our line is right here.
So, now that we have two points, we can plot our line because two points can only represent, or can rely on one single line. So, this is the line that is the solution set for this linear equation.
Now, there are a couple of features that we’re going to point out about a graphed line. It can tell us things about the equation that it represents. So, in this equation, for example, the line crosses the y-axis at -1, and this is what’s called the y-intercept. The line intercepts the y-axis at a particular point, in this case, -1.
And the y-intercept, when you have an equation of this form, is going to be this a part of the equation. The part of the right side of the equation that doesn’t have an x next to it. So, that’s the y-intercept. We also have what’s called the slope, and 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, and the change in the horizontal is 1, we go from 0 to 1. So, the slope here is 2/1, or just 2, and the slope, when you have an equation of this form here, is going to be the coefficient of the x term. Therefore, this equation here is in what’s called slope intercept form, 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, and you can either use for graphing the equation, to find the equation, or you can use the equation to graphically show the solution set.