Intro to point-slope form

so what I've drawn here in yellow is a line and let's say we know two things about this line we know that it has a slope of M and we know that the point a comma B is on this line so the question that we're going to try to answer is can we easily come up with an equation for this line using this information well let's try it out so any point on this line would or any X comma Y on this line would have to satisfy the condition that the slope between that point so let's say that this is some point X comma Y it's an arbitrary point on the line the fact that it's on the line tells us that the slope between a comma B and X comma Y must be equal to M so let's use that knowledge to actually construct an equation so what is the slope between a comma B and X comma Y well our change in Y remember slope is just change in Y over change in X let me write that slope is equal to change in Y over change in X this little triangle character that's the Greek letter Delta shorthand for change in our change in Y let's see if we start at y equals we're starting at y is equal to B and if we end up at y equals this arbitrary Y right over here this change in Y right over here is going to be Y minus B Y - let me write in that those same colors so this is going to be Y minus my little orange B and that's going to be over our change in X and the exact same logic we start at x equals a we finish at x equals this arbitrary X whatever X we happen to be at so that change in X is going to be that ending point minus our starting point minus a and we know this is the slope between these two points that's the the slope between any two points on this line and that's going to be equal to M so this is going to be equal to M and so we've already done here what we've already done right here is actually create an equation that describes this line it might not be in any form that you're used to seeing but this is an equation that describes any X Y that satisfies this equation right over here will be on the line because any X Y that satisfies this the slope between that X Y and this point right over here between the point a a b is going to be equal to M so let's actually now convert this in two forms that we might recognize more easily so let me paste that so two to simplify this expression a little bit or at least to get rid of the X minus a in the denominator let's multiply both sides by X minus a let me do that same so if we multiply both sides by X minus a so X minus a on the left hand side X minus a on the left and X minus a on the right X minus a on the right we put some parentheses around it so we're going to multiply by both sides by X minus a the whole point of that is you have X minus a divided by X minus a which is just going to be equal to one and then on the right hand side you just have M times X minus a so this whole thing has simplified to Y minus B y minus B is equal to is equal to M times X minus a and right here this is a form that people that mathematicians have categorized as point-slope form so this right over here is the point point-slope form of the equation that describes this line now why is it called point-slope form well it's very easy to inspect this and say okay well look this is the slope of the line in green that's the slope of the line and I can put the two points in if my if the the point a comma B is on this line I'll have the slope times X minus a is equal to Y minus B now let's let's see why this is useful why people like to use this type of thing let's not use just a B in a slope of M anymore let's make this a little bit more concrete let's say that we have let's say that someone tells you that I'm dealing with some line where the slope is equal to two and it goes through the point it goes through the point I don't know let's say it goes through the point negative seven comma negative seven comma five so very quickly you could use this information and the fact that of on your knowledge of point-slope form to write this in this form you would just say well an equation that contains this that contains this point it has this slope would be Y minus B which is 5y minus the y coordinate of the point that this line contains is equal to is equal to my slope is equal to my slope x times X minus the x coordinate that this line contains so X minus negative 7 and just like that we have written an equation that can that has a slope of 2 and that contains this point right over here and if we don't like the X minus negative 7 right over here we could obviously rewrite that as X plus 7 but this is kind of the purist point-slope form if you want to simplify it a little bit you could write it as Y minus 5 is equal to 2 times X plus 7 and if you want to see that this is just one way of expressing the equation of this line there are many others and the one that we're most familiar with is y-intercept form this can easily be converted to y-intercept form to do that we just have to distribute this two so we get Y minus 5 is equal to 2 times X plus 2 times 7 so it's equal to 14 and then we can get rid of this negative 5 on the left by adding 5 to both sides of this equation and then we are left with on the left hand side y and on the right hand side 2 X plus 19 so this right over here is slope-intercept form you have your slope and your y-intercept so this is slope-intercept form and this right up here is point-slope form