In this video, we're going to look at why the equations for lines on graphs all tend to look like this, rather than any of the other ways that we could have written them. Essentially, it's just for convenience. By always writing the equations in the form of y equals mx plus c, they're all in the same format, which makes it easier to quickly compare them. and imagine what they'd look like when drawn out on a graph.
The important things to know about this form are that m is the gradient of the line, which remember is a measure of how steep the line is, and c is the y-intercept, which is the point where the line crosses the y-axis. For example, if we took this equation, y equals 2x, plus 3, then m would be 2, because that's the number in front of the x, and c would be positive 3. This means that if we had a graph and we wanted to sketch this line, we know that it would cross the y-axis here, because the y-intercept is 3, so it will cross the y-axis at y equals 3. Then, because our gradient is 2, we know that for every 1 place, that our line moves across, it will move up by two places, so it will also go through this point here. And then all we have to do is draw a line between these two points and extend it across the whole graph, and that will be our line.
However, if we had been given our equation in this other form, 2y minus 4x equals 6, then we couldn't have used this technique to sketch it. Instead, we'd have had to rearrange it into the form of y equals mx plus c first. To do this, we'd just add 4x to both sides, to get 2y equals 4x plus 6, and then divide both sides by 2, to get y equals 2x plus 3. And now that it's in the form of y equals mx plus c, we know both the gradient and the y-intercept.
so we can plot it. Let's try doing the same thing for this equation. 4y plus 16 equals 2x.
The first thing we need to do is rearrange it to get it into the form of y equals mx plus c. So we need to subtract 16 from both sides to get 4y equals 2x minus 16 and then divide both sides by so that we only have a single y on the left which will equal 1 half x minus 4 on the right. Then if you compare this to y equals mx plus c, you can see that c will be minus 4, so the line will intercept the y-axis just here at minus 4, and also that m is 1 half, so the gradient of the line will be 1 half.
This means that for every one that the line goes across on the x-axis, it's going to go up by one half. So by plotting a couple of points like this, we can see that the line's going to go through all of these points, and so look like this. Now the last thing I want to mention is that you'll sometimes see equations like these, which might look like they're not in the form of y equals mx plus c, because this one doesn't have a c value and this one doesn't have an m. However, if there's no number after the x term, you can just think of it as a plus zero.
So c is zero, which means that it crosses the y-axis at y equals zero. And the 3x tells us that the line has a gradient of 3. For the other one, if there's no number in front of the x, then it's effectively 1x, so the gradient is 1. And the plus 4 tells us that it intercepts the y-axis at positive 4. Anyway, that's everything for this video, so hope that all made sense, and thanks for watching.