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there's a lot of different ways that you can represent a linear equation so for example if you had the linear equation y is equal to 2x plus 3 that's one way to represent it but I can represent this in an infinite number of ways I could let's see I could subtract 2x from both sides I could write this as negative 2x plus y is equal to 3 I could I could manipulate it in ways where I get it to and I'm not gonna do it right now but this is another way of writing that same thing Y minus 5 is equal to 2 times X minus 1 you could actually simplify this and you could get either this equation here or that equation up on top these are all equivalent you can get from one to the other with logical algebraic operations so there's an infinite ways to there's an infinite number of ways to represent a given linear equation but what I want to focus on in this video is this representation in particular because this one is a very useful representation of a linear equation and we'll see in future videos this one and this one can also be useful depending on what you are looking for but we're gonna focus on this one and this one right over here it's often called slope intercept form slope slope intercept form and hopefully in a few minutes it will be obvious why it is called slope intercept form and before I explain that to you let's just try to graph this thing and I'm gonna try to graph it I'm just gonna plot some points here so X comma Y and I'm gonna pick some X values where it's easy to calculate the Y values so maybe the easiest is if X is equal to 0 if X is equal to 0 then 2 times 0 is 0 that term goes away and you're only left with this term right over here Y is equal to 3 y is equal to 3 and so if we were to plot this actually let me start plotting it so that is my y-axis and let me do the x-axis so that can be my X that's not as straight as I would like it so that looks pretty good alright that is my ax access and let me let me mark off some hash marks here so this is x equals one x equals two x equals three this is y equals let me do this y equals one y equals two y equals three and obviously I can keep going I can keep going this would be y is equal to negative one this would be X is equal to negative one negative two negative three so on and so forth so at this point right over here zero comma 3 this is X is zero Y is three well the point that represents when X is equal to zero and y equals three this is this is we're right on the y axis if there have a line going through it and this line contains this point this is going to be the y intercept so one way to think about it the reason why this is called slope intercept form is it's very easy to calculate the y intercept the y intercept here is going to happen when it's written in this form it's going to happen when X is equal to zero and Y is equal to 3 it's going to be this point right over here so it's very easy to figure out the intercept the y intercept from this form now you might be saying oh well it's a slope intercept form it must also be easy to figure out the slope from this form and if you made that conclusion you would be correct and we're about to see that in a few seconds so let's plot some more points here and I'm just going to keep increasing X by 1 so if you increase X by 1 so we could write that our Delta X our change in X Delta greek-letter this triangles Greek letter Delta and represents change in change in X here is 1 we just increased X by 1 what's going to be our corresponding change in Y well it's going to be our change in Y so let's see when X is equal to 1 if 2 times 1 plus 3 is going to be 5 so our change in Y is going to be 2 let's do that again let's increase our X by 1 change in X is equal to 1 so then if we go from if we're gonna increase by 1 we're gonna go from x equals 1 to x equals 2 well what's a corresponding change in Y well when X is equal to 2 2 times 2 is 4 plus 3 is 7 well our change in Y our change in Y is equal to two we went from five when X went from 1 to 2 y went from 5 to 7 so for every one that we increase X Y is increasing by two so for this linear equation our change in Y over change in X is always going to be our change in Y is two when our change in X is 1 or it's equal to 2 or we could say that our slope is equal to 2 and let's just let's just graph this to make sure that we we understand this so when x equals 1 Y is equal to 5 and actually we're gonna have to graph 5 up here so when X is equal to 1 Y is equal to and actually this is a little bit higher this let me clean this up a little bit so this one let me erase that a little bit just like that so that's y is equal to 4 and this is y is equal to 5 so when X is 1 Y is equal to 5 so that's that point right over there so our line is going to look you only need two points to define a line our line is going to look like let me do this in this color right over here our line is going to look like is going to look is going to look something like it is going to look let me see if I can I didn't draw it completely at scale but it's going to look something like this this is the line this is the line y is equal to 2x plus 3 and we already figured out that it's slope is equal to 2 our change are when our change in X is 1 when our change in X is 1 our change in Y is 2 if our change in X was negative 1 if our change in exit was negative 1 our change in Y is negative 2 and you could see that if from 0 we went we went down 1 if we went to negative 1 then what's our Y going to be 2 times negative 1 is negative 2 plus 3 is 1 so we see that the point 1 or the point negative 1 comma 1 is on the line as well so the slope here our change in Y over change in X if we're going from from between any two points on this line is always going to be two but where did you see two in this original equation well you see the two right over here and when you write something in slope-intercept form where you explicitly solve for y y is equal to some constant times X to the first power plus some other constant the second one is going to be your intercept your Y and to be a way to figure out the y-intercept the intercept itself is this point the point at which the line intersects the y axis and then this two is going to represent your slope and that makes sense because every time you increase X by one you're going to multiply that by two so you're gonna increase Y by 2 so this is just a kind of a again get your feet wet with the idea of slope intercept form but you'll see at least for me this is the easiest form for me to think about what the graph of something looks like because if you were given another if you're given another linear equation let's say Y is equal to negative x negative x plus 2 well immediately you say okay look my y intercept is going to be the point 0 comma 2 so I'm going to intersect intersect the y-axis right at that point and then I have a slope of the coefficient here is really just negative 1 so I have a slope of negative 1 so as we increase X by 1 we're going to decrease Y by 1 increase X by 1 you're going to decrease Y by 1 if you increase X by 2 you're gonna decrease Y by 2 and so our line is going to look something like this let me see if I can draw it relatively neatly it's going to look something it's let me I can do it a little bit better than that yeah it's good my graph paper is hand-drawn it's not ideal but I think you get you get the point it's gonna look something like that so from slope intercept form very easy to figure out what the y-intercept is and very easy to figure out the slope the slope here slope here is negative 1 that's this negative 1 right over here and the y-intercept y-intercept is the point 0 comma 2 very easy to figure out because essentially that gave you the information right there