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determine whether the data in the table is an example of direct inverse or joint variation then identify the equation that represents the relationship so let's just think about what direct inverse or joint variation even means so if you have direct variation direct variation so if Y varies directly with X it literally means that Y is equal to some constant multiple of X or if you divide both sides of this by X it means that Y over X is equal to K so the ratio between y and X is a constant and you can go the other way around you could also say that X is equal to some constant not not going to be the same constant times y or that X over Y is going to be equal to some other constant so these aren't necessarily the same K all I'm just saying is that it's a constant relationship these are all examples of direct variation in or I should say inverse variation is to some degree the opposite depending on how you view the opposite and before I even talk about that let's think about the tell-tale signs of direct variation if x increases Y should increase so if x increases let me do that in the same yellow so the tell-tale signs of direct variation if x increases then Y will increase and vice versa the other telltale sign is is if you increase X by some by some factor so if you have X going to 3x then Y should also increase by that same factor and we can see that with some examples so you can pick a K let's say that let's say that K was 1 so if Y is equal to X if you take if X goes from 1 to 3 then Y is also going to go from 1 to 3 so that's all we're talking about here and let me actually Y should actually go to 3 times y that's what I'm talking about if you triple X you are also going to end up tripling Y inverse variation you have Y being equal to some constant times 1 over X so instead of an X here you have a 1 over X or if you multiply both sides by X you get x times y is equal to some constant and you could switch the X's and Y's around as well for inverse variation now what are the telltale signs well if you increase X if X goes up then what happens to Y if X goes up then this becomes a smaller value because it's 1 over X so then Y will go down then Y will go down and if you take X and if you were to say increase it by a factor of 3 then what's going to happen to Y well if you increase this by a factor of 3 you're actually going to decrease this whole value by a factor of 1 3 so Y is going to go so then you're going to have one-third of Y so that's that these are the tell-tale signs for inverse variation now finally they talk about something called joint variation and this one you won't necessarily see in an introductory algebra course but joint variation deals with more than one variable so if I told you if I told you that area of a rectangle is equal to the width of a rectangle times the length of a rectangle this is an example of joint variation area is proportional to two is the proportional to two different quantities so the main telltale sign here for joint variation frankly is you're going to be dealing with more than two variables joint joint joint variation so when you look at this example they're only giving us two variables so you can rule out joint variation just right from the get-go now let's look at the tell-tale signs so as X is increasing as X goes from one to two what is happening to Y Y is going from 12 to 6 so as X is going up by a factor of two Y is going is is is going by a factor of one half or Y is being multiplied by one half as X goes from 1 to 3 being multiplied by three Y is being multiplied or I guess you could say is is is multiplied by one third so it's definitely not direct variation as x increases Y is decreasing so it's definitely not direct variation and then yearly you could just rule out your since we've ruled out the other two you could probably guess this is going to be inverse variation but we can validate it when x increases Y is decreasing when x increases by a certain factor why is increasing by one over that factor which it's actually decreasing so if you go from one to three if X is being multiplied by three then Y essentially becomes one-third of its original value when X is one y is 12 when X is three Y is 4 so we have inverse variation at play now they asked us identify the equation that represents the relationship well we know with inverse variation the product of x and y need to be equal to some constant so if we take x times y over here so let's just multiply make another column here let me call this the x times y column well 1 times 12 is 12 2 times 6 is 12 3 times 4 is 12 4 times 3 is 12 so clearly in every situation x times y is this is a constant and it is 12 so the equation that represents the relationship it is X y is equal to 12 and that is clearly an inverse relationship