Hey, everyone. So throughout our discussion on lines, we've seen equations and lines in different forms, like slope intercept or point slope form. But you might be given a problem with an equation that kind of looks like this, with all the terms of x and y on the left side. What I'm going to show you in today's video is that this is just another way of writing an equation of a line. This is called the standard form. I'm going to show you what it's helpful for, but most of the time in problems, you're going to be able to take this equation and rewrite it in one of the forms that we already know, like slope intercept. So I'm going to show you that these two equations actually mean the exact same thing. They're just written in slightly different ways. Alright? Let's get started here.
So if you take a look at our problem, what we're going to do is we're going to find the slope and the y-intercept, which we already know how to do. But of this equation over here, −9x + 3y − 120. It doesn't look anything like the forms that we already know. So the standard form, the way that these equations are generally written, is like ax + by + c = 0. So, clearly, we can see here that this is like a, b, and c, and this equals 0. So instead of having m's and b's, these are really just coefficients that stand for the numbers inside of your equation.
Alright? So how do I solve this problem? I have a's, b's, and c's, but I'm asked for the slope and the intercepts. Remember that the slope is just m, so that's what I want to find, and I want to find b over here. How do I find that from this equation over here? Well, basically, what happens is whenever you're given a problem in standard form, like we have here, and you're asked for the slope or the intercept, you're going to have to rewrite this equation. The way that we rewrite the equation is basically just by isolating y and solving for it on the left side of the equation. Alright?
So I'm going to take this equation over here and, to convert it into y=mx+b, I'm going to have to rewrite it so that y is on the left side. Alright? So I'm going to take this −9x, and I have to move it over to the right side and add 9x, and I have to take the −120 and also move it over and add 120. So I have to take everything and move it to the right side. What happens is the 3y just stays behind on the left side. I get 3y equals, and then both of these things become positive. So I have 9x + 120. Alright? Now, we're not done yet because this still doesn't look like y equals mx + b. I have to get rid of the 3, and the way that I do that is by dividing it out. So I have to divide each number in the equation by 3. The 3 goes away on the left side, and all I'm left with is y. And then what here happens with 9 over 3, it just turns to 3, and then 120 over 3 just becomes 40. So this is the equation now that I've ended up with, y = 3x + 40. And if you look at this, this actually is in slope intercept form. Right? I've got y = mx + b. So these are the two numbers. I've got 3 and 40, and I'm basically done here. Those are the numbers I'm looking for. My slope is equal to 3 and my y-intercept is equal to 40. So that's the answer to this problem. These equations over here and this one mean the exact same thing. They're just written in slightly different ways. Alright? Okay.
So that's one of the cases where you use standard form. It's basically when you're just asked to rewrite it in a different form. But what I'm also going to show you right now is it's also really helpful in finding the x and y intercepts. So let's go ahead and take a look at our next example over here.
So you might be given a line that's written in standard form like this: 3x+2y−6=0, and you might have to find the intercepts without first converting it back to slope intercept form. Alright? So I'm going to show you how to do that. To graph a line in standard form, you can find these intercepts very quickly without rewriting it in slope intercept. And basically, let's just sort of recall what these intercepts actually mean. Remember that the intercept, the x-intercept, is where it crosses the x-axis, and it's where the y value is equal to 0. And the opposite happens for the y-intercept. That's where the x value is 0. Right? So that's where it crosses the y-axis, and that's where the x value is equal to 0. So what we're going to do here in this problem is we're going to take this equation over here, and we're going to set the y and x equal to 0 and solve.
Let's go ahead and do this for the x-intercept. Right? So, if I want the x-intercept, what I'm going to do is I'm going to rewrite this equation: 3x+2y−6=0, and I'm going to set y equal to 0, and then I'm just going to solve for x. So, in other words, I'm going to take this 3x, and I'm just going to replace y with 0, in which that whole term now just goes away. And now all I have to do is solve for x. So how do I do that? Well, I'm going to bring the 6 over to the other side. So this just becomes 3x = 6 and x = 2. So that's my x-intercept. Right? I just solved for x. What that means is, if I go over my graph here, I can say that I know this graph crosses the x-axis at x = 2, but that's not enough to graph the line because it's only one point. So now we have to do the exact same thing, but now for the y-intercept.
And once we have these two points, then we can go ahead and connect them and form our line. So with the y-intercept, I'm going to do the exact same thing, 3x+2y−6=0. But now what I'm going to do is do the opposite. I'm going to set x equal to 0 over here, and I'm going to solve for y. So what I have to do is I have to replace the x with 0, and that term just goes away. So this changes to 2y−6=0. And now I just move the 6 over like I did on the other side. So, that's plus 6, and I end with 2y=6. And if you figure this out, this is going to be y = 3. So these are my x-and y-intercepts. I've got x = 2 and y = 3. So that's this point over here. Now we have the two points of our line, so we can actually just connect them, and that's going to form our line segment. So, I just connect these two points using a ruler or a straight edge or something like that, and that's going to be the equation of my line. Just going to move it over a little bit. There we go.
So this is the equation of my line, and I got that without having to convert it to slope intercept form. Alright? So, hopefully, that made sense. Thanks for watching.