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I've written some example relationships between two variables in this case between m and n between a and B between x and y and what I want to do in this video is see if we can identify whether the relationships are a direct relationship whether they vary directly or maybe they vary inversely or maybe it is neither so let's explore it a little bit so over here we have M over N is equal to 1/7 so let's see how we can manipulate this if we multiply both sides by n what are we going to get and in general you want to separate them so that the two variables on different sides of the equation so you can see is it going to meet is it going to be the pattern let me write it this way M is equal to K times n this would be direct variation or is it going to be the pattern M is equal to K times 1 over n this is inverse variation and you see in either one of these they're on different sides of the equal sign so let's take this first relationship right now let's multiply both sides by n multiplied both sides by N and you get M because these cancel out is equal to 1/7 times n so this actually meets the direct variation pattern it's some constant times n m is equal to some constant times n so this right over here is direct that they vary directly this is direct variation let's see a B is equal to negative 3 so if we want to separate them and we could do it with the either variable we could divide both sides I don't know let's divide both sides by a we're going to done it with by B if we divide both sides by a we get B is equal to negative 3 over a or we could also write this as B is equal to negative 3 times 1 over a and once again this is this pattern right here one variable is equal to a constant times 1 over the other variable in this case our constant is negative 3 so the over here they vary inversely they vary inversely this is an inverse relationship let's try this one over here I'll try it in that same color XY is equal to 1/10 once again let's try to separate the variables isolate them on either side of the equation let's divide both sides by X you could divide by Y because you're really just trying find a inverse or direct relationship so divide both sides by X you get Y is equal to one tenth over X which is the same thing as 1 over 10 X which is the same thing as 1 over 10 times 1 over X so y is equal to some constant some constant times 1 over X once again this is an inverse where this isn't it the y and x vary inversely let's do this one over here 9 times M go to that same orange color 9 times 1 over m is equal to n so this one's actually already done for us and it might be a little bit clearer if we just flip this around if we just flip the left on the right hand side we get n is equal to 9 times 1 over m n is equal to some constant some constant times 1 over m so n varies inversely with M inverse and remember if I say that n varies inversely with M that also means that M varies inversely with n those two things imply each other now let's try it with this expression over here and this one's a little bit of a trickier one because we've already separated the variables on both hands and then we have this kind of it's not being if this was B is equal to 1/3 times a that we would have direct variation then B would vary directly with a but in this case we should have 1/3 minus a and you say hey maybe there operate opposites or whatever and it actually turns out that this is neither this is neither and to make that point 100% clear let's let's look at two of these examples in direct variation if you scale up one variable in one direction you would scale up the other variable by the same amount so if we have X going if X doubles from 1 to 2 when X is 1 or actually I should do this with M and N that was so M and n so when and the way I've written it here although you could algebraically manipulate it so that one looks more dependent than the other but in this situation where n is n is 1 M is 1/7 M is 1/7 and when n is 7 M is going to be one so you have the situation that if n is scaled up by seven then M is also scaled up by seven or vice-versa sits more of a relationship I could have expressed n in terms of M but when you scale one variable up by seven you also have to scale up the other variable by seven or when you scale it up by some amount you have to scale the other variable by the same amount so this is direct direct variation let's take the inverse or when two variables vary inversely this situation right over here let's take a and B when a is equal to 1 B is equal to negative 3 and we could do it explicitly right over here we can even go to the original one when a is equal to 1 we have 1 B is equal to negative 3 B is equal to negative 3 now when a if I were to take a and if I were to let's say I were to triple it so if I were to multiply it by 3 so now a is 3 we have 1 over 3 times negative 3 so now B is negative 1 notice we didn't multiply B times 3 here now we divided B times 3 or we should we divided B by 3 I should say or in other ways we multiply it by 1 over 3 so if you scale up a by 3 you're scaling down B by 3 so they're varying inversely what you're going to see in this neither is that neither of these are going to be the case so let's try it out let's try it out I'll do it in that same green color do it in that same green so we have a and B so when a is I don't know when a is 1 what is B 1/3 minus 1 that's 1/3 minus 3/3 that's negative 2/3 and then let's divide just for fun let's divide a by 3 so if a goes to 1/3 so over here we're dividing by 3 or you could say we're multiplying by 1/3 so if a is 1/3 then B is equal to 0 right if a is 1/3 B is equal to 0 so notice if this was direct variation we would be multiplying this by 1/3 as well which clearly we didn't and if this was inverse variation if they vary inversely we would be multiplying by three which clearly we did it we just got some other number we actually ended up the the scaling actually didn't matter what happens is is that things really just got shifted by some amount they got shifted by two thirds so this is neither direct very up these neither very directly nor in verse Li this last one right over here