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CCSS.Math: , ,

so you may or may not already know that any linear equation can be written in the form y is equal to MX plus B where m is the slope of the line the same slope that we've been dealing with the last few videos the rise over run of the line or the inclination of the line and B is the y-intercept y-intercept I think it's pretty easy to verify that B is a y-intercept the way you verify that is you substitute X is equal to 0 if you get X is equal to 0 remember X is equal to 0 that means that's where we're going to intercept the y-axis if X is equal to 0 this equation becomes y is equal to M times 0 plus B M times 0 is just going to be 0 I don't care what M is so then Y is going to be equal to B Y is equal to B so the point 0 B is going to be on that line so 0 B so the line will intercept the y-axis at the point y is equal to B and we'll see that with actual numbers in the next few videos and just to verify for you that M is really the slope let's just try some numbers out we know when x is that we know the point 0 B is on the line what happens when x is equal to 1 when x is equal to 1 you get Y is equal to M times 1 or it's equal to M plus B so we also know that the point 1 M plus B is also on the line right this is just the Y value so what's the slope between that point and that point well let's take this as the end point so you have M plus B our change in Y M plus B minus B over our change in X over over 1 minus 0 all right this is our change in Y over change in X we're using two points that's our end point that's our starting point so if you simplify this B minus B is 0 1 minus 0 is 1 so you get M over 1 or it gets equal to M so hopefully you're satisfied I don't let it confuse you by staying in the abstract with all of these variables here but this is definitely going to be the slope and this is definitely going to be the y-intercept now given that what I want to do in this exercise is look at these graphs and then use the already drawn graphs to figure out the equation so we're going to look at these figure out the slopes figure out the y-intercepts and then know the equation so let's do equal this line a first let's do line a right here so what is a slope what is a slope so let's start at some arbitrary point let's say let's let's start right over there and then let us see and we want to get to even numbers so let's see if we run one two three so if our Delta X is equal to 3 right 1 2 3 our Delta Y and I'm just doing because I want to hit a even number here our Delta Y our Delta Y is equal to we go down by 2 it's equal to negative 2 so for a change in Y for a change in X when our change in X is 3 our change in Y is negative 2 so our slope is negative 2/3 when we go when we go over by 3 we're going to go down by 2 or if we go over by 1 we're going to go down by 2/3 you can't exactly see it there but you definitely see it when you go over by 3 so that's our slope we've essentially done half of that problem now we have to figure out the y-intercept so that right there is our M now what is our B our y-intercept well where does this intersect the y-axis well we already said the slope is 2/3 so this is the point y is equal to 2 when we go over by one to the right we would have gone down by 2/3 so this right here must be the point 1 in 1/3 or another way to say we could say it's 4/3 that's the point y is equal to 4/3 right there right a little bit more than 1 but 1 put 1 in 1/3 so say B is equal to 4/3 and so we'll know that the equation is is y is equal to M negative 2/3 X plus B plus 4/3 that's equation a let's do a Quay ssin B hopefully we won't have to deal with as many fractions here equation B let's figure out its slope first let's start at some reasonable point let's see we could start at that point and then if we let me do it right here B equation B when our Delta X is equal to let me write it this way Delta X so our Delta X could be 1 when we move over 1 to the right what happens to our Delta Y we go up by 3 Delta X Delta Y our change in Y is 3 so Delta Y over Delta X when we go to the right our change in X is 1 our change in X is 1 our change in Y is positive 3 so our slope is equal to 3 what is our y-intercept well when X is equal to 0 Y is equal to 1 so B is equal to 1 so this was a lot easier here the equation is y is equal to 3x plus 1 let's do that last line there line C line C all right so let's do the y-intercept first you see immediately the y-intercept when X is equal to 0 Y is negative 2 so B is equal to negative 2 and then what is the slope M is equal to change in Y over change in X so let's see if we start at that y-intercept and if we go over to the right by 1 2 3 4 so our change in X is equal to 4 what is our change in Y our change in Y is positive 2 change in Y is equal to 2 so change in Y is 2 when change in X is 4 or the slope is equal to 1/2 2 over 4 so the equation here is y is equal to 1/2 X that's our slope minus 2 and we're done now let's go the other way let's look at some equations of line knowing that this is the slope and this is the y-intercept that's the M that's the B and actually graph them so let's do this first line I already started circling it in orange the y-intercept is 5 when X is equal to 0 Y is equal to 5 you can verify that on the equation so when X is equal to 0 Y is equal to 1 2 3 4 5 that's the y-intercept and then the slope is 2 that means when I move one in the X direction I move up 2 in the Y direction so if I move one in the X direction I move up 2 in the Y direction if I move one in the X direction I move up 2 in the Y direction if I move back 1 in the X direction I move down 2 in the Y direction if I move back 1 in the X direction I move down 2 in the Y direction I keep doing that so this line is going to look I can't draw lines too neatly but this is going to be my best shot it's going to look something like that and I'll just keep going on on and on and on so that's our first line I mean I can just keep it just keep going down like that now let's do this second line y is equal to negative 0.2 X plus 7 so let me write that y is equal to negative zero point two X plus seven so it's always easier to think infraction so point two is the same thing as 1/5 so we could write y is equal to negative 1/5 X plus 7 so we know its y-intercept it's 7 so it's one two three four five six that's our y-intercept when X is equal to zero and this tells us that for every 5 we move to the right we move down one so let's we could view this as negative one over five the Delta Y over Delta X is equal to negative one over five for every five we move to the right we move down one so every 5 1 2 3 4 5 we moved five to the right that means we must move down one we move five to the right one two three four five we must move down one if we go backwards if you move five backwards so if you view instead of this you view this as 1 over negative 5 these are obviously equivalent numbers so if you go back 5 that's negative 5 1 2 3 four five then you move up one you move up one you go back five one two three four five you move up one so the line is going to look like this so just connect the dots I think you get the idea I just have to connect those dots I could have drawn a little bit straighter now let's do this one Y is equal to negative x y is equal to negative x where's the B term I don't see any B term you remember we're saying Y is equal to M X plus B where is the B well the B is zero you could view this as plus zero so here is B is zero when x is 0 Y is zero so it's that's our y-intercept right there at the origin and then the slope once again you see a negative sign that you could view that as negative one X plus zero so the slope is negative one when you move to the right by one when change in X is one change in Y is negative one when you move up by one and X you go down by one and Y or if you go down by one and X are going to go up by one and Y x and y are going to have opposite signs they go in opposite direction so the line is going to look like that it's going to look like that you can almost imagine it's splitting the second and fourth quadrants now I'll do one more let's do this last one right here Y is equal to three point seven five so now you're saying gee you know you know we're looking for y is equal to MX plus B where is this X term it's completely gone well the reality here is this could be rewritten as Y is equal to zero X plus three point seven five now it makes sense the slope is zero no matter how much we change our X Y does not change Delta Y over Delta X is equal to zero I don't care how much you change your X so our y-intercept is three point seven five so one two three point seven five is right around there just want to get close three and three fourths and then as I change X Y will not change Y is always going to be three point seven five so it's going to look it's just going to be a horizontal line at Y is equal to three point seven five five anyway hopefully you found this useful