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we're on problem 138 138 and a certain business production index P is directly proportional to efficiency index II so P so we could say that P is equal to some proportionality constant times e because they say P is directly proportional to e fair enough which is in turn directly proportional to investment index I so E is proportional to I so we could say it's equal to some proportionality constant times I and actually we could go even a step further and say well at E if P is proportional to e and E is proportional to I and we could also say that P is proportional to I directly proportional I so it might be some other constant some other constant times I and you could show that you could prove that mathematically right you take this and substitute it for E and you get P is equal to K times e e is equal to some constant times I and so P is equal to KN times I but K and n we're both arbitrary constants so we could just call that M so this is what this is the information that the problem gives us P is proportional to K is proportional to I I'm sorry P is proportional to e e is proportional to I so P is also proportional to I so what are they asking us what is P P if I is equal to 70 so if we knew this proportional constant we'd be all set we could actually solve this problem maybe they're going to go through a or something I don't know so statement number one statement number one e is equal to 0.5 when I is equal to 60 well we can definitely use this information to figure out what n is and but that by itself isn't going to help us because we can figure out what n is and then once we know it n is we can figure out what E is when I is 70 because we'll put 70 here times but whatever we figure out n is will get e but without knowing what K is that still won't help us so statement 1 by itself if I'm right isn't right that won't that won't help us and I'll show it to you let me oh this would be a waste of time if you understood that logic you can say 0.5 is equal to n times 60 this is so we can solve for n so n is equal to 0.5 over 60 that's the same thing as 1 over 120 and then you could say so then that equation boils down to e is equal to AI over 120 right 1 over 120 times I so when I is equal to 70 II would be equal to 70 over 120 which is equal to 7 over 12 but that still doesn't help us is equal to 7 over 12 but we don't know what K is right when I is equal to 70 is 7 over 12 but we still don't know what K is based on just the information in statement 1 so as far as I can tell right now 1 by itself not so useful statement 2 P is 2 P is equal to 2 when I is equal to 50 well this is useful because we already said P is proportional to IP is equal to some constant M times I so we could use this information to solve for M so P is equal to 2 when I is equal to 50 divide both sides by 50 you get M is equal to 1 over 25 and so the question was what is P when I is equal to 70 so we could just say P is equal to M times I well I now a 70 and what's whatever 70 divided by 25 is that's P so statement 2 alone was sufficient to solve this problem and statement 1 didn't help us much next problem 139 139 if X does not equal minus y that's interesting is X minus y over X plus y greater than 1 so why do they say X does not equal minus 1 I'm sorry X does not equal minus y well if X were equal to minus y if this were minus y then you'd have minus y plus y the denominator would be 0 and you'd be undefined so maybe that's why they put the knot there so let me try to let's see I think we could simplify the statement so we could multiply both sides by X plus y you get X minus y is greater than X plus y and so this holds true if what let's see we can subtract X from both sides subtracting X from both sides you get I'll just get rid of the X's you get minus y is greater than Y C we could add Y to both sides of this equation so at the left side if you add Y here you get 0 is greater than if you had a y at this side you get 2y and you could divide both sides of this equation by 2 and you get 0 is greater than Y or Y is less than 0 either way so this fairly hairy looking statement Boyle this statement is true if and only if this statement is true so this is really all we have to test for is y less than 0 if we know that Y is less than 0 we know that this is true or if we know that this is false so we know this is false so statement 1 statement 1 tells us X is greater than 0 well this is useless this actually has no bearing on whether Y is less than 0 X can be anything it doesn't change this so this by itself is useless statement 2 is well stick there you go Y is less than 0 so statement 2 tells us this is true and this is true if and only if this is true so therefore statement 2 alone is sufficient to answer this question very seldom do you get do you boil down the statement to actually one of the statements that they provide you for that notes that was interesting next problem 140 140 in the rectangular coordinate system are the points RS and UV equidistant from the origin ok so they're essentially saying is the distance of the point RS equal distance from the origin equal to the distance for the I'll call it D sub o equals the distance to the origin of point u D and here just to get the intuition of distance I always find it silly that they teach something called the distance formula in high schools because it's really just the Pythagorean theorem and by calling it something DIF and making you memorize a different formula it'll just clutter your head so this is the x-axis and this is the y-axis the point RS will be here what is R and this is s this is the point R s what's its distance from the origin well system's from the origin is the length of this line right there what's the length of that well we can use Pythagorean theorem the height right there is s the base here is R and if we call this distance we spy that Gurren theorem R squared plus s squared is equal to the distance squared right or we could say that the distance the distance is equal to the square root of R squared plus s squared the distance of the origin so this statement up here boils down to that the square root of R squared plus s squared needs to be equal to the square root of U squared plus V squared well just to simplify things let's just square both sides of this equation and then we're left with the question they ask is is R squared plus s squared equal to u squared plus V squared is this true that's what they asked us and that kind of simplifies things make some more concrete so let's see what the statements give us statement 1 R plus s is equal to 1 R plus s is equal to 1 just off the cuff I don't see where that's going to be actually useful I you know it's not like you can just square this R plus s squared is R squared plus 2 R s plus s squared so you can't just you know a lot of people make the mistake that thinking oh R plus s squared is R squared plus s squared notice that's not true you have to distribute all the terms and you end up with three terms so that's not right I don't see an R plus s anywhere up here let's let's try statement number two statement number two U is equal to one minus R and V is equal to one minus s so this this this seems like it could be interesting because there allows us to essentially reduce this question which is a question of four variables and turn it into a question of two variables by substituting u and V with these things so let's do that let's let's turn the statement into a statement of two variables I'll switch colors just to ease the monotony so the left-hand side is R squared plus s squared is equal to u squared well now they're telling us that u squared is 1 minus R squared plus V squared well V is 1 minus s 1 minus s squared let's just keep simplifying R squared plus s squared is equal to 1 minus 2 R plus r squared plus 1 minus 2 s plus s squared see we can get rid of we could subtract R squared from both sides we can subtract s squared from both sides and we're left with 0 is equal to 1/2 2 minus 2r minus 2's and let's see we can bring the R and the S terms over the other side so add 2 R plus 2 s to both sides so you bring them over you get 2 R plus 2 s is equal to 2 divide the whole thing by to both sides you get R plus s is equal to 1 interesting so when you put apply these two constraints on our original question the question gets reduced to this if we know that statement 2 is correct the question that the problem was asking it's reduced to this it's saying if we know that this happens then the question is true if this is true well just with statement 2 alone we don't know that this is true but as you see statement 1 tells us that that is true so if you use both statements together you know that this question is correct this question is true and that was actually pretty interesting and a little bit hairier than normal normally you can identify immediately just by eyeballing it but anyway I'll see you in the next video