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CCSS.Math: , ,

let's say we have a linear equation and we know when X is equal to four that not that Y is equal to nine and we've plotted that point here on our XY plane I forgot to label the X the x axis right over there and let's say we also know we also know that when X is equal to 6 y is equal to 1 and we've plotted that point there and so this green line represents all of the solutions to this linear equation now what I want to do in this video is I want to say well can we find that linear equation and can we express it in both point-slope form and in slope intercept form and I encourage you like always pause the video and see if you can do it so let's first think about point-slope form point point-slope pones point-slope form in point-slope form is very easy to generate if you know a point on the line or if you know a point that satisfies where the x and y coordinates satisfy the linear equation and if you were to know the slope of the of the line that represents a solution set of that linear equation now for sure we actually hit we're given two points that are solutions that represent solutions to the linear equation to fully apply point-slope were to apply point-slope easily we just have to figure out the slope and what we could do is we could just evaluate what's the slope between the two points that we know and we just have to remind ourselves that slope slope is equal to slope is equal to change in Y over change in X sometimes people say rise over run and what's that going to be well if we say if we if we say that this the second point right over here if we say if we say this is kind of our if we starting at this point and we go to that point then our change in Y going from this point to that point is going to be it's going to be equal to 1 minus 1 minus 9 1 minus 9 this point right over here is the point 6 comma 1 so we started at y equals 9 we finish at y equals 1 our change in Y is going to be 1 minus 9 we have a negative 8 change in Y which makes sense we've gone down 8 so this is going to be equal to this is going to be equal to negative 8 that's our change in Y and what's our change in X well we go from x equals 4 to x equals 6 so we end up at x equals 6 and we started at x equals 4 we started at x equals 4 so our change in X is 6 minus 4 which is equal to 2 which is equal to 2 and you could have even done it visually to go from this point to this point your change in Y your change in Y is you went down 8 so your change in let me write this so your change in Y is equal to negative 8 and what was your change in X to get them to get to this point well your change in X is positive 2 so your change in X is equal to 2 and so what's your slope change in Y over change in X negative 8 over 2 is equal to negative 4 so now that we have a now that we know the slope and we know a point and we know a and we know a point we actually know two points on the line we can express this in point-slope form and so let's do that and the way I like to do is I always like to just take it straight from the definition of what slope is we know that the slope between any two points on this line is going to be negative 4 so if we take an arbitrary Y that sits on this line and if we find the difference between that Y and let's focus on this point up here so if we find the different that Y and this Y and 9 and it's over over the difference between some X on the line and this X and 4 & 4 this is going to be the slope between any XY on this line and this point right over here and the slope between any two points on a line are going to have to be constant so this is going to be equal to the slope of the line it's going to be equal to negative for and we're not in point-slope form or classic point-slope form just yet to do that we just multiply both sides times X minus 4 so we get Y minus 9 we get Y minus 9 is equal to our slope negative 4 times X minus 4 times X minus 4 and this right over here is our classic this right over here is our classic point-slope form we have the point sometimes they even put parentheses like this but we could figure out the point from this point slope form the point that sits on this line would things that make both sides of this equation equal to 0 so it would be x equals 4 y equals 9 which we have right up there and then the slope is right over here it's negative 4 now from this can we now express this this this linear equation in y intercept form and y-intercept form just as a bit of a reminder it's y is equal to MX plus B where this coefficient is our slope and this constant right over here is allows us to figure out our y-intercept and to get this and to get this in this form we just have to simplify a little bit of this algebra so you have Y minus 9 y minus 9 is equal to well let's distribute this negative 4 and I'll just switch some colors let's distribute this negative 4 negative 4 times X is negative 4x negative 4 times negative 4 is plus 16 and now if we just want to isolate the Y on the left hand side we can add 9 to both sides so let's do that let's add 9 let's add 9 to both sides let's add 9 to both sides on the left hand side we're just left with Y and on the right hand side we're left with negative 4x and then 16 plus 9 is plus 25 and there you have it we have the same linear equation but it's now represented in slope-intercept form once again we see the slope right over here and now we can figure out what the y-intercept is the y-intercept when X is equal to 0 Y is going to be equal to 25 my axis right here haven't drawn it high enough but if I if I made it even taller and taller and you see that this line is going to intersect the y axis when y is equal to y is equal to 25 so there you go we wrote it in point-slope form that is that right over there and we wrote it in Y I'm sorry we wrote it in slope we wrote it in slope-intercept form point-slope and slope-intercept hopefully you enjoyed that