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CCSS.Math:

in this video I'm going to do a bunch of examples of finding the equations of lines in slope-intercept form and just as a bit of review that means equations of lines in the form of y is equal to MX plus b where m is the slope and b is the y-intercept so let's just do a bunch of these problems so here they tell us that a line has a slope of negative 5 so m is equal to negative 5 and it has a y-intercept of 6 so B is equal to 6 so this is pretty straightforward the equation of this line is y is equal to negative 5 negative 5x plus 6 that wasn't too bad let's do this next one over here the line has a slope of negative one and contains the point 4/5 comma 0 so they're telling us the slope slope of negative 1 so we know that M is equal to negative 1 but we're not 100% sure about where the y-intercept is just yet so we know that this equation is going to be of the form Y is equal to the slope negative 1 X plus B where B is the y-intercept now we can use this coordinate information the fact that it contains this point we can use that information to solve for B the fact that the line contains this point means that the value X is equal to 4/5 Y is equal to 0 must satisfy this equation so let's substitute those in Y is equal to 0 when X is equal to 4/5 so 0 is equal to negative 1 times 4/5 plus B let me scroll down a little bit so let's see we get 0 is equal to negative 4/5 plus B we can add 4/5 to both sides of this equation so we could add a 4/5 there we could add a 4/5 to that side as well and I the whole reason I did that is so that cancels out with that and you get B is equal to 4/5 B is equal to 4/5 so we now have the equation of the line y is equal to negative 1 times X we could write as negative x plus B which is for this and just like that now we have this one the line contains the point 2 comma 6 and 5 comma 0 so they haven't given us the slope or the y-intercept explicitly but we could figure out both of them from these coordinates the first thing we can do is figure out the slope so we know that the slope M is equal to change in Y over change in X which is equal to what is the change in Y let's start with this one right here so we do 6-0 6-0 let me do it this way so that's a 6 I want to make it color-coded minus 0 so 6 minus 0 that's our change in Y and then our change in X is 2 2 minus 2 minus 5 and the reason why I color-coded it is I wanted to show you when I change when I use this Y term first I use the 6 up here that I have to use this X term first as well so I wanted to show you this is the coordinate 2 comma 6 this is the coordinate 5 comma 0 I couldn't have done I couldn't have swapped the 2 and the 5 then then I would have gotten the negative of the answer but what do we get here this is equal to this is equal to 6 minus 0 is 6 2 minus 5 is negative 3 so this becomes negative 6 over 3 which is the same thing as negative 2 so that's our slope so so far we know that the line must be Y is equal to the slope then an orange negative 2 times X plus our y-intercept now we can do exactly what we did in the last problem we can use one of these points to solve for B we can use either one both of these are on the line so both of these must satisfy this equation I'll use the five comma zero because it's always nice when you have a zero they're things the math is a little bit easier so let's put the five comma zero there so Y is equal to zero when X is equal to five so Y is equal to zero when you have negative 2 times 5 when X is equal to 5 plus B and so you get 0 is equal to negative 10 plus B if you add 10 to both sides of this equation let's add 10 to both sides these two cancel out and you get B is equal to 10 plus 0 or 10 so you get B is equal to 10 and now we know the equation for the line the equation is y let me do it in a new color y is equal to negative 2x plus B plus 10 and we are done let's do another one of these let's do another one of these all right the line contains the points 3 comma 5 and negative 3 comma 0 just like the last problem we start by figuring out the slope the slope which we will call M is the same thing as the rise over the run which is the same thing as the change in Y over the change in X if you were doing this for your homework you wouldn't have to write all this I just want to make sure that you understand that these are all the same things and then what is our change in Y over change in X this is equal to let's start with this side first just to show you I could have picked either of these points so let's say it's 0 0 minus 5 minus 5 just like that so I'm using this coordinate first I'm kind of viewing it as the endpoint 0 minus 5 and remember I was when I first learned this I was always be tempted to do the X in the numerator no you use the Y's in the numerator so that's the second of the coordinates and then that is going to be over negative 3 negative 3 negative 3 minus 3 minus three this is a coordinate negative three zero negative three zero this is the coordinate three five we're subtracting that so what are we going to get this is going to be equal to this is a neutral color this is going to be equal to the numerator is negative five over negative three minus three is negative six so the negatives cancel out you get five over six so we know that the equation is going to be of the form y is equal to five six x five six x plus B and now we can substitute one of these coordinates in to solve for B so let's do I always like to use one that has a zero in it so Y is zero when five when x is when X is negative 3 when X is negative three plus B so all I did is I substitute a negative 3 for X 0 for Y and I know I can do that because this is on the line this must satisfy the equation of the line and let's solve for B so we get 0 is equal to well if we divide negative 3 by 3 that becomes a 1 if you divide 6 by 3 that becomes a 2 so it becomes negative 5 negative 5 halves plus B we can add 5 halves to both sides of the equation plus 5 halves plus 5 halves I'd like to change my notation just so you get familiar with both and so the equation becomes 5 halves is equal to that's a 0 is equal to B B is 5 halves so the equation of our line is the equation of our line is y is equal to 5 6 X plus B which we just figured out is 5 halves plus 5 halves and we are done let's do another one we have a graph here let's figure out the equation of this graph this is actually on some level a little bit easier what's the slope slope is change in Y over change in X so let's see what happens when we move in X when our change in X is 1 so that is our change in X so change in X is 1 I'm just deciding to change my X by 1 increment by 1 what is the change in Y it looks like Y changes exactly by 4 it looks like my Delta Y my change in Y is equal to 4 when my Delta X when my Delta X is equal to 1 so change in Y over change in X change in Y is 4 when change in X is 1 so the slope is equal to 4 and now what's its y-intercept well here we can just look at the graph it looks like it intersects the y-axis at y is equal to negative 6 or at the point 0 negative 6 so we know that B is equal to negative 6 B is equal to negative 6 so we can we know the equation of the line the equation ax line is y is equal to the slope the slope times X plus the y-intercept or next row actually I should write that so or minus 6 that is plus negative 6 so that is the equation of our line let's do one more of these so they tell us that f of 1 point 5 is negative 3 f of negative 1 is 2 or da you know what what is that well all this is telling you this is just a fancy way of telling you that the point when X is 1.5 when you put 1.5 into the function the function evaluates as negative 3 so this tells us that the coordinate 1 point 5 negative 3 is on the line and then this tells us that the point when X is negative 1 f of X is equal to 2 this is just a fancy way of saying that both of these two points are on the line nothing unusual I think the point of this problem is to get you familiar with function notation for you to not get intimidated if you see something like this if you evaluate the function at 1.5 you get negative 3 so that's the coordinate Y if you imagine that Y is equal to f of so this would be the y-coordinate be equal to negative 3 when X is 1 point 5 anyway I've said it multiple times let's figure out the slope of this line the slope which is change in Y over change in X is equal to let's start with CO 2 2 minus this guy negative 3 right these are the Y values over all of that over negative 1 negative 1 minus negative 1 minus this guy we write it this way negative 1 minus that guy - 1.5 and I do the colors because I want to show you that the negative 1 and the 2 are both coming from this that's why both I use both of them first what if I use these guys first I would have to be use both the X in the Y's first if I use the 2 first I have to use the negative 1 first that's why I'm color coding it so this is going to be equal to 2 minus negative 3 that's the same thing as 2 plus 3 so that is 5 1 minus knit 1 minus 1 point for our negative 1 minus one point five is negative two point five negative two point five and five divided by two point five is equal to two so the slope of this line is negative two actually I'll take a little side to show you it doesn't matter what order I do this in as long as I use if I use this coordinate first and have to use that coordinate first let's do it the other way if I date it as negative 3 negative 3 minus 2 minus 2 over over 1.5 - minus negative 1 this should be minus the 2 over 1.5 minus the negative 1 minus the negative 1 this should give me the same answer this is equal to what negative 3 minus 2 is negative 5 over 1.5 minus negative 1 that's 1.5 plus 1 so it's over 2.5 so once again this is equal to negative 2 so I just want to sell it doesn't matter which one you pick is the starting at the end point as long as you do it you're consistent if this is the starting why this is the starting X if this is the finishing why this has to be the finishing X but anyway we know that the slope is negative 2 now so we know that the equation is y is equal to negative 2x plus some y-intercept let's use one of these coordinates I'll use this one since it doesn't have a decimal in it so we know that y is equal to make is equal to 2 so y is equal to 2 when X is equal to when X is equal to negative 1 when X is equal to negative 1 and of course you have your plus B so 2 is equal to negative 2 times negative 1 is 2 plus B if you subtract 2 from both sides of this equation minus 2 minus 2 we're subtracting it from both sides of this equation you're going to get 0 on the left hand side is equal to B so B is 0 so the equation of our line is just Y is equal to negative 2 X Y is equal to negative 2x or actually if you wanted to write it in function notation it would be that f of X is equal to negative 2x I kind of just assumed that Y is equal to f of X but this is really the equation they never mentioned wise here so you could just write f of X is equal to 2x right here and each of these coordinates are the coordinates of X and f of X or X and f of X so you could even view the definition of slope as change in f of X over change in X these are all equivalent ways of doing the same thing