Writing slope-intercept equations
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In this video I'm going to do a bunch of examples of finding the equations of lines in slope-intercept form. Just as a bit of a review, that means equations of lines in the form of y is equal to mx plus b where m is the slope and b is the y-intercept. So let's just do a bunch of these problems. So here they tell us that a line has a slope of negative 5, so m is equal to negative 5. And it has a y-intercept of 6. So b is equal to 6. So this is pretty straightforward. The equation of this line is y is equal to negative 5x plus 6. That wasn't too bad. Let's do this next one over here. The line has a slope of negative 1 and contains the point 4/5 comma 0. So they're telling us the slope, slope of negative 1. So we know that m is equal to negative 1, but we're not 100% sure about where the y-intercept is just yet. So we know that this equation is going to be of the form y is equal to the slope negative 1x plus b, where b is the y-intercept. Now, we can use this coordinate information, the fact that it contains this point, we can use that information to solve for b. The fact that the line contains this point means that the value x is equal to 4/5, y is equal to 0 must satisfy this equation. So let's substitute those in. y is equal to 0 when x is equal to 4/5. So 0 is equal to negative 1 times 4/5 plus b. I'll scroll down a little bit. So let's see, we get a 0 is equal to negative 4/5 plus b. We can add 4/5 to both sides of this equation. So we get add a 4/5 there. We could add a 4/5 to that side as well. The whole reason I did that is so that cancels out with that. You get b is equal to 4/5. So we now have the equation of the line. y is equal to negative 1 times x, which we write as negative x, plus b, which is 4/5, just like that. Now we have this one. The line contains the point 2 comma 6 and 5 comma 0. So they haven't given us the slope or the y-intercept explicitly. But we could figure out both of them from these coordinates. So the first thing we can do is figure out the slope. So we know that the slope m is equal to change in y over change in x, which is equal to-- What is the change in y? Let's start with this one right here. So we do 6 minus 0. Let me do it this way. So that's a 6-- I want to make it color-coded-- minus 0. So 6 minus 0, that's our change in y. Our change in x is 2 minus 2 minus 5. The reason why I color-coded it is I wanted to show you when I used this y term first, I used the 6 up here, that I have to use this x term first as well. So I wanted to show you, this is the coordinate 2 comma 6. This is the coordinate 5 comma 0. I couldn't have swapped the 2 and the 5 then. Then I would have gotten the negative of the answer. But what do we get here? This is equal to 6 minus 0 is 6. 2 minus 5 is negative 3. So this becomes negative 6 over 3, which is the same thing as negative 2. So that's our slope. So, so far we know that the line must be, y is equal to the slope-- I'll do that in orange-- negative 2 times x plus our y-intercept. Now we can do exactly what we did in the last problem. We can use one of these points to solve for b. We can use either one. Both of these are on the line, so both of these must satisfy this equation. I'll use the 5 comma 0 because it's always nice when you have a 0 there. The math is a little bit easier. So let's put the 5 comma 0 there. So y is equal to 0 when x is equal to 5. So y is equal to 0 when you have negative 2 times 5, when x is equal to 5 plus b. So you get 0 is equal to -10 plus b. If you add 10 to both sides of this equation, let's add 10 to both sides, these two cancel out. You get b is equal to 10 plus 0 or 10. So you get b is equal to 10. Now we know the equation for the line. The equation is y-- let me do it in a new color-- y is equal to negative 2x plus b plus 10. We are done. Let's do another one of these. All right, the line contains the points 3 comma 5 and negative 3 comma 0. Just like the last problem, we start by figuring out the slope, which we will call m. It's the same thing as the rise over the run, which is the same thing as the change in y over the change in x. If you were doing this for your homework, you wouldn't have to write all this. I just want to make sure that you understand that these are all the same things. Then what is our change in y over our change in x? This is equal to, let's start with the side first. It's just to show you I could pick either of these points. So let's say it's 0 minus 5 just like that. So I'm using this coordinate first. I'm kind of viewing it as the endpoint. Remember when I first learned this, I would always be tempted to do the x in the numerator. No, you use the y's in the numerator. So that's the second of the coordinates. That is going to be over negative 3 minus 3. This is the coordinate negative 3, 0. This is the coordinate 3, 5. We're subtracting that. So what are we going to get? This is going to be equal to-- I'll do it in a neutral color-- this is going to be equal to the numerator is negative 5 over negative 3 minus 3 is negative 6. So the negatives cancel out. You get 5/6. So we know that the equation is going to be of the form y is equal to 5/6 x plus b. Now we can substitute one of these coordinates in for b. So let's do. I always like to use the one that has the 0 in it. So y is a zero when x is negative 3 plus b. So all I did is I substituted negative 3 for x, 0 for y. I know I can do that because this is on the line. This must satisfy the equation of the line. Let's solve for b. So we get zero is equal to, well if we divide negative 3 by 3, that becomes a 1. If you divide 6 by 3, that becomes a 2. So it becomes negative 5/2 plus b. We could add 5/2 to both sides of the equation, plus 5/2, plus 5/2. I like to change my notation just so you get familiar with both. So the equation becomes 5/2 is equal to-- that's a 0-- is equal to b. b is 5/2. So the equation of our line is y is equal to 5/6 x plus b, which we just figured out is 5/2, plus 5/2. We are done. Let's do another one. We have a graph here. Let's figure out the equation of this graph. This is actually, on some level, a little bit easier. What's the slope? Slope is change in y over change it x. So let's see what happens. When we move in x, when our change in x is 1, so that is our change in x. So change in x is 1. I'm just deciding to change my x by 1, increment by 1. What is the change in y? It looks like y changes exactly by 4. It looks like my delta y, my change in y, is equal to 4 when my delta x is equal to 1. So change in y over change in x, change in y is 4 when change in x is 1. So the slope is equal to 4. Now what's its y-intercept? Well here we can just look at the graph. It looks like it intersects y-axis at y is equal to negative 6, or at the point 0, negative 6. So we know that b is equal to negative 6. So we know the equation of the line. The equation of the line is y is equal to the slope times x plus the y-intercept. I should write that. So minus 6, that is plus negative 6 So that is the equation of our line. Let's do one more of these. So they tell us that f of 1.5 is negative 3, f of negative 1 is 2. What is that? Well, all this is just a fancy way of telling you that the point when x is 1.5, when you put 1.5 into the function, the function evaluates as negative 3. So this tells us that the coordinate 1.5, negative 3 is on the line. Then this tells us that the point when x is negative 1, f of x is equal to 2. This is just a fancy way of saying that both of these two points are on the line, nothing unusual. I think the point of this problem is to get you familiar with function notation, for you to not get intimidated if you see something like this. If you evaluate the function at 1.5, you get negative 3. So that's the coordinate if you imagine that y is equal to f of x. So this would be the y-coordinate. It would be equal to negative 3 when x is 1.5. Anyway, I've said it multiple times. Let's figure out the slope of this line. The slope which is change in y over change in x is equal to, let's start with 2 minus this guy, negative 3-- these are the y-values-- over, all of that over, negative 1 minus this guy. Let me write it this way, negative 1 minus that guy, minus 1.5. I do the colors because I want to show you that the negative 1 and the 2 are both coming from this, that's why I use both of them first. If I used these guys first, I would have to use both the x and the y first. If I use the 2 first, I have to use the negative 1 first. That's why I'm color-coding it. So this is going to be equal to 2 minus negative 3. That's the same thing as 2 plus 3. So that is 5. Negative 1 minus 1.5 is negative 2.5. 5 divided by 2.5 is equal to 2. So the slope of this line is negative 2. Actually I'll take a little aside to show you it doesn't matter what order I do this in. If I use this coordinate first, then I have to use that coordinate first. Let's do it the other way. If I did it as negative 3 minus 2 over 1.5 minus negative 1, this should be minus the 2 over 1.5 minus the negative 1. This should give me the same answer. This is equal to what? Negative 3 minus 2 is negative 5 over 1.5 minus negative 1. That's 1.5 plus 1. That's over 2.5. So once again, this is equal the negative 2. So I just wanted to show you, it doesn't matter which one you pick as the starting or the endpoint, as long as you're consistent. If this is the starting y, this is the starting x. If this is the finishing y, this has to be the finishing x. But anyway, we know that the slope is negative 2. So we know the equation is y is equal to negative 2x plus some y-intercept. Let's use one of these coordinates. I'll use this one since it doesn't have a decimal in it. So we know that y is equal to 2. So y is equal to 2 when x is equal to negative 1. Of course you have your plus b. So 2 is equal to negative 2 times negative 1 is 2 plus b. If you subtract 2 from both sides of this equation, minus 2, minus 2, you're subtracting it from both sides of this equation, you're going to get 0 on the left-hand side is equal to b. So b is 0. So the equation of our line is just y is equal to negative 2x. Actually if you wanted to write it in function notation, it would be that f of x is equal to negative 2x. I kind of just assumed that y is equal to f of x. But this is really the equation. They never mentioned y's here. So you could just write f of x is equal to 2x right here. Each of these coordinates are the coordinates of x and f of x. So you could even view the definition of slope as change in f of x over change in x. These are all equivalent ways of viewing the same thing.