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Find the slope of the line that goes through the ordered pairs 4 comma 2 and negative 3 comma 16. So just as a reminder, slope is defined as rise over run. Or, you could view that rise is just change in y and run is just change in x. The triangles here, that's the delta symbol. It literally means "change in." Or another way, and you might see this formula, and it tends to be really complicated. But just remember it's just these two things over here. Sometimes, slope will be specified with the variable m. And they'll say that m is the same thing-- and this is really the same thing as change in y. They'll write y2 minus y1 over x2 minus x1. And this notation tends to be kind of complicated, but all this means is, is you take the y-value of your endpoint and subtract from it the y-value of your starting point. That will essentially give you your change in y. And it says take the x-value of your endpoint and subtract from that the x-value of your starting point. And that'll give you change in x. So whatever of these work for you, let's actually figure out the slope of the line that goes through these two points. So we're starting at-- and actually, we could do it both ways. We could start at this point and go to that point and calculate the slope or we could start at this point and go to that point and calculate the slope. So let's do it both ways. So let's say that our starting point is the point 4 comma 2. And let's say that our endpoint is negative 3 comma 16. So what is the change in x over here? What is the change in x in this scenario? So we're going from 4 to negative 3. If something goes from 4 to negative 3, what was it's change? You have to go down 4 to get to 0, and then you have to go down another 3 to get to negative 3. So our change in x here is negative 7. Actually, let me write it this way. Our change in x is equal to negative 3 minus 4, which is equal to negative 7. If I'm going from 4 to negative 3, I went down by 7. Our change in x is negative 7. Let's do the same thing for the change in y. And notice, I implicitly use this formula over here. Our change in x was this value, our endpoint, our end x-value minus our starting x-value. Let's do the same thing for our change in y. Our change in y. If we're starting at 2 and we go to 16, that means we moved up 14. Or another way you could say it, you could take your ending y-value and subtract from that your starting y-value and you get 14. So what is the slope over here? Well, the slope is just change in y over change in x. So the slope over here is change in y over change in x, which is-- our change in y is 14. And our change in x is negative 7. And then if we want to simplify this, 14 divided by negative 7 is negative 2. Now, what I want to show you is, is that we could have done it the other way around. We could have made this the starting point and this the endpoint. And what we would have gotten is the negative values of each of these, but then they would've canceled out and we would still get negative 2. Let's try it out. So let's say that our start point was negative 3 comma 16. And let's say that our endpoint is the 4 comma 2. 4 comma 2. So in this situation, what is our change in x? Our change in x. If I start at negative 3 and I go to 4, that means I went up 7. Or if you want to just calculate that, you would do 4 minus negative 3. 4 minus negative 3. But needless to say, we just went up 7. And what is our change in y? Our change in y over here, or we could say our rise. If we start at 16 and we end at 2, that means we went down 14. Or you could just say 2 minus 16 is negative 14. We went down by 14. This was our run. So if you say rise over run, which is the same thing as change in y over change in x, our rise is negative 14 and our run here is 7. So notice, these are just the negatives of these values from when we swapped them. So once again, this is equal to negative 2. And let's just visualize this. Let me do a quick graph here just to show you what a downward slope would look like. So let me draw our two points. So this is my x-axis. That is my y-axis. So this point over here, 4 comma 2. So let me graph it. So we're going to go all the way up to 16. So let me save some space here. So we have 1, 2, 3, 4. It's 4 comma-- 1, 2. So 4 comma 2 is right over here. 4 comma 2. Then we have the point negative 3 comma 16. So let me draw that over here. So we have negative 1, 2, 3. And we have to go up 16. So this is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. So it goes right over here. So this is negative 3 comma 16. Negative 3 comma 16. So the line that goes between them is going to look something like this. Try my best to draw a relatively straight line. That line will keep going. So the line will keep going. So that's my best attempt. And now notice, it's downward sloping. As you increase an x-value, the line goes down. It's going from the top left to the bottom right. As x gets bigger, y gets smaller. That's what a downward-sloping line looks like. And just to visualize our change in x's and our change in y's that we dealt with here, when we started at 4 and we ended at-- or when we started at 4 comma 2 and ended at negative 3 comma 16, that was analogous to starting here and ending over there. And we said our change in x was negative 7. We had to move back. Our run we had to move in the left direction by 7. That's why it was a negative 7. And then we had to move in the y-direction. We had to move in the y-direction positive 14. So that's why our rise was positive. So it's 14 over negative 7, or negative 2. When we did it the other way, we started at this point. We started at this point, and then ended at this point. Started at negative 3, 16 and ended at that point. So in that situation, our run was positive 7. And now we have to go down in the y-direction since we switched the starting and the endpoint. And now we had to go down negative 14. Our run is now positive 7 and our rise is now negative 14. Either way, we got the same slope.

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Find the slope of the line that goes through the ordered pairs 4 comma 2 and negative 3 comma 16. So just as a reminder, slope is defined as rise over run. Or, you could view that rise is just change in y and run is just change in x.

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The slope of a line is rise over run. Learn how to calculate the slope of the line in a graph by finding the change in y and the change in x.

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Sal analyzes what it means for a slope to be positive or negative (spoiler: it affects the direction of the line!)

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- [Voiceover] As we start to graph lines, we might notice that they're differences between lines. For example, this pink or this magenta line here, it looks steeper than this blue line. And what we'll see is this notion of steepness, how steep a line is, how quickly does it increase or how quickly does it decrease, is a really useful idea in mathematics. So ideally, we'd be able to assign a number to each of these lines or to any lines that describes how steep it is, how quickly does it increase or decrease? So what's a reasonable way to do that? What's a reasonable way to assign a number to these lines that describe their steepness? Well one way to think about it, could say well, how much does a line increase in the vertical direction for a given increase in the horizontal direction? So let's write this down. So let's say if we an increase increase, in vertical, in vertical, for a given increase in horizontal for a given increase a given increase in horizontal. So, how can this give us a value? Well let's look at that magenta line again. Now let's just start at an arbituary point in that magenta line. But I'll start at a point where it's going to be easy for me to figure out what point we're at. So if we were to start right here, and if I were to increase in the horizontal direction by one. So I move one to the right. To get back on the line, how much do I have to increase in the vertical direction? Well I have to increase in the vertical direction by two. By two. So at least for this magenta line, it looks like our increase in vertical is two, whenever we have an increase in one in the horizontal direction. Let's see, does that still work if I were to start here, instead of increasing the horizontal direction by one, if I were increase in the horizontal direction... So let's increase by three. So now, I've gone plus three in the horizontal direction, then to get back on the line, how much do I have to increase in the vertical direction? I have to increase by one, two, three, four, five, six I have to increase by six. So plus six. So when I increase by three in the horizontal direction, I increase by six in the vertical. We were just saying, hey, let's just measure how much to we increase in vertical for a given increase in the horizontal? Well two over one is just two and that's the same thing as six over three. So no matter where I start on this line, no matter where I start on this line, if I take and if I increase in the horizontal direction by a given amount, I'm going to increase twice as much twice as much in the vertical direction. Twice as much in the vertical direction. So this notion of this increase in vertical divided by increase in horizontal, this is what mathematicians use to describe the steepness of lines. And this is called the slope. So this is called the slope of a line. And you're probably familiar with the notion of the word slope being used for a ski slope, and that's because a ski slope has a certain inclination. It could have a steep slope or a shallow slope. So slope is a measure for how steep something is. And the convention is, is we measure the increase in vertical for a given in increase in horizontal. So six two over one is equal to six over three is equal to two, this is equal to the slope of this magenta line. So let me write this down. So this slope right over here, the slope of that line, is going to be equal to two. And one way to interpret that, for whatever amount you increase in the horizontal direction, you're going to increase twice as much in the vertical direction. Now what about this blue line here? What would be the slope of the blue line? Well, let me rewrite another way that you'll typically see the definition of slope. And this is just the convention that mathematicians have defined for slope but it's a valuable one. What is are is our change in vertical for a given change in horizontal? And I'll introduce a new notation for you. So, change in vertical, and in this coordinate, the vertical is our Y coordinate. divided by our change in horizontal. And X is our horizontal coordinate in this coordinate plane right over here. So wait, you said change in but then you drew this triangle. Well this is the Greek letter delta. This is the Greek letter delta. And it's a math symbol used to represent change in. So that's delta, delta. And it literally means, change in Y, change in Y, divided by change in X, change in X. So if we want to find the slope of the blue line, we just have to say, well how much does Y change for a given change in X? So, the slope of the blue line. So let's see, let me do it this way. Let's just start at some point here. And let's say my X changes by two so my delta X is equal to positive two. What's my delta Y going to be? What's going to be my change in Y? Well, if I go by the right by two, to get back on the line, I'll have to increase my Y by two. So my change in Y is also going to be plus two. So the slope of this blue line, the slope of the blue line, which is change in Y over change in X. We just saw that when our change in X is positive two, our change in Y is also positive two. So our slope is two divided by two, which is equal to one. Which tells us however much we increase in X, we're going to increase the same amount in Y. We see that, we increase one in X, we increase one in Y. Increase one in X, increase one in Y. >From any point on the line, that's going to be true. You increase three in X, you're going to increase three in Y. It's actually true the other way. If you decrease one in X, you're going to decrease one in Y. If you decrease two in X, you're going to decrease two in Y. And that makes sense from the math of it as well Because if you're change in X is negative two, that's what we did right over here, our change is X is negative two, we went two back, then your change in Y is going to be negative two as well. Your change in Y is going to be negative two, and negative two divided by negative two, is positive one, which is your slope again.

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Intro to slope | Algebra (video) | Khan Academy

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Voiceover] As we start to graph lines, we might notice that they're differences between lines. For example, this pink or this magenta line here, it looks steeper than this blue line. And what we'll see is this notion of steepness, how steep a line is, how quickly does it increase or how quickly does it decrease, is a really useful idea in mathematics.

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What Does Negative Slope Mean? | Virtual Nerd

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Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

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What's the Formula for Slope? | Virtual Nerd

# data[].attributes.attachment_link.description
Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.