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as we start to graph lines we might notice that there are 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 put assign a number to each of these lines or to any line that describes how steep it is how 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 how much does a line increase for how much does it 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 give an increase in horizontal so how could this give us how could this give us a value well let's look at that let's look at that magenta line again now let's just start at an arbitrary point of that magenta line 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 store 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 and we could let's see does that apply let's see does that still work if I were to go if I were to start here instead of increasing the horizontal direction by one if I were to increase in the horizontal direction so let's increase by three so now I've gone plus three in the horizontal direction then again 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 where you're just saying hey let's just measure how much we increase in vertical for a given increase in horizontal well 2 over 1 is just 2 and that's the same thing as 6 over 3 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 increase in vertical / 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 say a key a ski slope and that's because the 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 increase in horizontal so 6 2 over 1 is equal to 6 over 3 is equal to 2 this is equal to the slope of this magenta line so slope so let me write this down so this slope right over here the slope of that line is going to be equal to 2 and one way to interpret that if you 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 you'll typically see the definition of slope and this is just the convention that mathematicians have defined for slope but it's a it's a valuable it's what 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 it's going to be the vertical is our y-coordinate divided by our change in horizontal and X is our horizontal coordinate in this in this X in this coordinate plane right over here and you might say wait you said change in when you drew this triangle well this triangle 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 given change in X so the slope of the blue line so let's see let's see actually let me do this let me do it this way let me have a whene if I let's just start at some point here and let's say if my X changes by 2 so my Delta X is equal to positive 2 what's my Delta Y going to be what's going to be my change in Y well if I go to the right by 2 to get back on the line I have to increase my Y by 2 so my change in Y is also going to be plus 2 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 2 our change in Y is also positive 2 so our slope is 2 divided by 2 which is equal to 1 which tells us how ever much we increase in X we're going to increase that same amount in Y and you see that the increase 1 and X you increase 1 and Y increase 1 and X increase 1 and Y from any point on the line that's going to be true you increase 3 and X you're going to increase 3 in Y it's actually true the other way if you decrease 1 in X you're going to decrease 1 and Y if you decrease 2 and X you're going to decrease 2 and Y and that makes sense from from the the math of it as well because if your change in X is negative 2 that's what we did right over here we change our change in X was negative 2 we went to back then your change in Y is going to be negative 2 as well your change in Y is going to be negative 2 and negative 2 divided by negative 2 is positive 1 which is your slope again