1 00:00:00,000 --> 00:00:03,979 I want to talk about direct and inverse variations 2 00:00:03,979 --> 00:00:08,036 I´ll do direct variations on the left over here 3 00:00:08,559 --> 00:00:15,274 And I´ll do inverse variations or two variables that vary inversly on the right hand side over here 4 00:00:15,604 --> 00:00:19,783 So a very simple definition for two variables that vary directly 5 00:00:19,783 --> 00:00:23,266 would be something like this 6 00:00:23,266 --> 00:00:27,260 Y varies directly with X if y is equal to some constant with x 7 00:00:27,260 --> 00:00:29,600 we could rewrite this in kind of english 8 00:00:29,600 --> 00:00:39,267 y varies directly with x 9 00:00:39,267 --> 00:00:43,700 if this constant seems strange to you 10 00:00:43,700 --> 00:00:47,183 remember this could be literally any constant number 11 00:00:47,183 --> 00:00:49,598 so let me give a bunch of examples that would also 12 00:00:49,598 --> 00:00:53,174 or particular examples of y varies directly with x 13 00:00:53,174 --> 00:00:56,285 you cold have Y is equal to X 14 00:00:56,285 --> 00:00:59,954 because of the situation the constant is 1, we didn´t even write it 15 00:00:59,954 --> 00:01:02,694 you could have write Y is equal to 1X, then K is 1 16 00:01:02,694 --> 00:01:05,434 we could write Y is equal to 2X 17 00:01:05,434 --> 00:01:08,452 we could write Y is equal to one half X 18 00:01:08,452 --> 00:01:11,935 we could write Y is equal to negative 2X 19 00:01:11,935 --> 00:01:14,165 we are still variating directly 20 00:01:14,165 --> 00:01:18,112 we could write Y is equal to negative one half X 21 00:01:18,112 --> 00:01:22,617 we could have Y is equal to π times X 22 00:01:22,617 --> 00:01:26,100 we could have Y is equal to negative π times X 23 00:01:26,100 --> 00:01:28,840 You get the point 24 00:01:28,840 --> 00:01:31,951 Any constant times X, we are varying directly 25 00:01:31,951 --> 00:01:35,248 And to understand this maybe a little bit more tanginbly 26 00:01:35,248 --> 00:01:38,592 let´s think about what happens and let´s pick one of these scenarios 27 00:01:38,592 --> 00:01:41,843 I´ll take a positive version and a negative version 28 00:01:41,843 --> 00:01:43,933 just because it might not be completely intuitive 29 00:01:43,933 --> 00:01:46,115 So let´s take the version of Y is equal to 2X 30 00:01:46,115 --> 00:01:51,333 and let´s explore why we say they vary directly with each other 31 00:01:51,333 --> 00:01:54,333 so let´s pick a couple of values for X 32 00:01:54,333 --> 00:01:57,933 and see what the resulting Y value would have to be 33 00:01:57,933 --> 00:02:03,855 so if X is equal to 1, then Y is two times 1, so it´s 2 34 00:02:03,855 --> 00:02:10,264 if X is equal to 2, then Y is 2 times 2, which is going to be equal to 4 35 00:02:10,264 --> 00:02:17,601 So when we doubled X, from 1 to 2, so we doubled X, the same thing happened to Y 36 00:02:17,601 --> 00:02:24,242 We doubled Y, so that´s what it means when something varies directly 37 00:02:24,242 --> 00:02:29,258 If we scale X up by a certain amount, we are going to scale up Y 38 00:02:29,258 --> 00:02:30,698 by the same amount 39 00:02:30,698 --> 00:02:34,924 If we scale down X by a certain amount, we would scale down Y by the same amount 40 00:02:34,924 --> 00:02:37,431 And just to show you it works with all of these 41 00:02:37,431 --> 00:02:40,914 let´s try the situation with Y is equal to negative 2X 42 00:02:40,914 --> 00:02:43,608 I´ll do it in magenta 43 00:02:43,608 --> 00:02:50,689 Y is equal to ..., I´ll do a new example, let´s try Y is equal to negative 3X 44 00:02:50,689 --> 00:02:57,470 so once again, let me do my X and my Y 45 00:02:57,470 --> 00:03:01,789 when X is equal to 1, Y is equal to negative 3 times 1, which is negative 3 46 00:03:01,789 --> 00:03:08,383 when X is equal to 2, negative 3 times 2 is negative 6 47 00:03:08,383 --> 00:03:17,114 so notice, we multiplied, so if we scaled, let me do that in the same green colour 48 00:03:17,114 --> 00:03:20,783 if we scale up X by 2, it´s a different green colour but it serves the purpose 49 00:03:20,783 --> 00:03:24,312 we are also scaling up Y by 2, to go from 1 to 2, we multiply by 2 50 00:03:24,312 --> 00:03:29,467 to go from negative 3 to negative 6, you´re also multiplying by 2 51 00:03:29,467 --> 00:03:34,204 so we grew by the same scaling factor, and if you want to get the other way 52 00:03:34,204 --> 00:03:48,136 let´s say we try, X is 1/3, if X is 1/3 then Y is going to be negative 3 times 1/3, is negative 1 53 00:03:48,136 --> 00:03:53,200 so to go from 1 to 1/3, we divide by 3 54 00:03:53,200 --> 00:03:57,053 to go from negative 3 to negative 1, we also divide by 3 55 00:03:57,053 --> 00:04:00,118 we also scale down by a factor of 3 56 00:04:00,118 --> 00:04:05,458 so whatever direction you scale X in you are going to have the same scaling direction as Y 57 00:04:05,458 --> 00:04:08,477 that´s what it means to vary directly 58 00:04:08,477 --> 00:04:12,378 now it´s not laways so clear 59 00:04:12,378 --> 00:04:13,771 sometimes it would be ofuscated 60 00:04:13,771 --> 00:04:18,043 so you could write, so let´s take this example right over here 61 00:04:18,043 --> 00:04:21,867 Y is equal to negative 3X, (I´m saving this real state for inverse variation in a second) 62 00:04:21,867 --> 00:04:25,200 you could writethis or you could algebraically manipulate it 63 00:04:25,200 --> 00:04:28,307 you could maybe divide both sides of this equation by X 64 00:04:28,307 --> 00:04:34,112 and then you will get Y/X is equal to -3 65 00:04:34,112 --> 00:04:36,944 or maybe you divide both sides by X 66 00:04:36,944 --> 00:04:39,452 and then you divide both sides by Y 67 00:04:39,452 --> 00:04:42,842 so from this you would get, so if you divide both sides by Y now 68 00:04:42,842 --> 00:04:47,933 you could get 1/X is equal to negative 3 times 1/Y 69 00:04:47,933 --> 00:04:52,595 these three statements, these three equations are all the same thing 70 00:04:52,595 --> 00:04:55,938 so sometimes the direct variation isn´t quite in your face 71 00:04:55,938 --> 00:04:59,607 but if you do this, what I did right here with any of these 72 00:04:59,607 --> 00:05:01,604 you would get the exact same result 73 00:05:01,604 --> 00:05:04,948 or you could jus try to manipulate it back, to this form over here 74 00:05:04,948 --> 00:05:07,595 and there´s other ways we could do it 75 00:05:07,595 --> 00:05:09,777 we could divide both sides of this equation by negative 3 76 00:05:09,777 --> 00:05:13,911 and then you would get negative 1/3Y is equal to X 77 00:05:13,911 --> 00:05:17,440 and this is kind of an interesting case here 78 00:05:17,440 --> 00:05:21,109 because here this is X varies directly with Y 79 00:05:21,109 --> 00:05:24,778 or we could say X is equal to some K times Y 80 00:05:24,778 --> 00:05:26,264 and in general that´s true 81 00:05:26,264 --> 00:05:34,112 if Y varies directly with X, though we can also say that X varies directly with Y 82 00:05:34,112 --> 00:05:38,245 it´s not going to be the same constant, it´s going to be essentially the inverse of that constant 83 00:05:38,245 --> 00:05:40,614 but they´re still directly varying 84 00:05:40,614 --> 00:05:43,586 now with that said so about direct variation 85 00:05:43,586 --> 00:05:45,267 let´s explore inverse variation 86 00:05:45,267 --> 00:05:49,576 inverse variation the general form (if we use the same variables) 87 00:05:49,576 --> 00:05:52,502 it always doesn´t have to be Y and X, it could be A and a B 88 00:05:52,502 --> 00:05:55,196 it could be a M and an N 89 00:05:55,196 --> 00:06:05,273 now to inverse varitaion, so if I did it with Ys and Xs 90 00:06:05,273 --> 00:06:07,595 this would be Y is equal to some constant (K) times 1/X 91 00:06:07,595 --> 00:06:14,608 so instead of being some constant times X, it´s some constant times 1/X 92 00:06:14,608 --> 00:06:16,512 so let me show you a bunch of examples 93 00:06:16,512 --> 00:06:18,601 it could be Y is equal to 1/X 94 00:06:18,601 --> 00:06:22,920 it could be Y is equal to 2 times 1/X 95 00:06:22,920 --> 00:06:25,567 which is clearly the same thing as 2/X 96 00:06:25,567 --> 00:06:31,279 it could be Y is equal to 1/3 times 1/X 97 00:06:31,279 --> 00:06:33,927 whis is the same thing as 1/3X 98 00:06:33,927 --> 00:06:39,000 it could be Y is equal to -2/X 99 00:06:39,000 --> 00:06:43,911 and let´s explore this, the inverse variation the same way we explain the direct variation 100 00:06:43,911 --> 00:06:48,276 so let´s pick Y is equal to 2/X 101 00:06:48,276 --> 00:06:55,614 and let me do that same table over here 102 00:06:55,614 --> 00:07:02,766 if X is 1, then Y is 2 103 00:07:02,766 --> 00:07:07,267 if X is 2 then 2 divided by 2 is 1 104 00:07:07,267 --> 00:07:09,453 so if you multiply X by 2 105 00:07:09,453 --> 00:07:12,379 you scale it up by a factor of 2, what happens to Y? 106 00:07:12,379 --> 00:07:15,304 Y gets scaled down by a factor of 2 107 00:07:15,304 --> 00:07:17,301 you divide it by 2 now 108 00:07:17,301 --> 00:07:18,533 notice the difference? 109 00:07:18,533 --> 00:07:22,178 here however we scale X, we sacled up Y by the same amount 110 00:07:22,178 --> 00:07:24,933 now if we scale up X by a factor 111 00:07:24,933 --> 00:07:30,119 when we have inverse variation, we are scaling down Y by the same 112 00:07:30,119 --> 00:07:32,467 so that´s were the inverse is coming from 113 00:07:32,467 --> 00:07:34,716 and we could go the other way 114 00:07:34,716 --> 00:07:36,853 id we made X is equal to 1/2 115 00:07:36,853 --> 00:07:42,704 so if we were to scale down X, we are going to see that this is going to scale up Y 116 00:07:42,704 --> 00:07:44,887 ´cause 2 divided by 1/2 is 4 117 00:07:44,887 --> 00:07:47,441 so here we are scaling up Y 118 00:07:47,441 --> 00:07:49,206 so they are going to do the opposite things 119 00:07:49,206 --> 00:07:55,196 they vary inverse, and you could try it with the negative way of it as well 120 00:07:55,196 --> 00:07:58,215 so here we are multiplying by 2 121 00:07:58,215 --> 00:08:00,537 and once again it is not always neatly written for you like this 122 00:08:00,537 --> 00:08:04,299 it could be rearranged in a bunch different ways 123 00:08:04,299 --> 00:08:09,546 but it will still be inverse variation as long as they are algebraically equivalent 124 00:08:09,546 --> 00:08:13,215 so you could multiply both side of this equation right here by X 125 00:08:13,215 --> 00:08:20,878 and you would get XY is equal to 2, this is also inverse variation 126 00:08:20,878 --> 00:08:22,782 you would get this exact same table over here 127 00:08:22,782 --> 00:08:30,769 you could divide both sides of this equation by Y and you could X is equal to 2/Y 128 00:08:30,769 --> 00:08:33,788 which is also the same thing as 2 times 1/Y 129 00:08:33,788 --> 00:08:36,800 so notice Y varies inversely with X 130 00:08:36,800 --> 00:08:41,868 and you could just manipulate this algebraically to show that X varies inversely with Y 131 00:08:41,868 --> 00:08:50,135 so Y varies inversely with X 132 00:08:50,135 --> 00:09:00,770 this is the same thing (as seen in this example) as saying that X varies inveresely with Y 133 00:09:00,770 --> 00:09:02,116 and there is other things 134 00:09:02,116 --> 00:09:04,531 we could take this and divide both sides by 2 135 00:09:04,531 --> 00:09:07,550 and you would get Y/2 is equal to 1/X 136 00:09:07,550 --> 00:09:09,547 there´s all sort of crazy things 137 00:09:09,547 --> 00:09:13,540 so in general if you see an expression that relates two variables 138 00:09:13,540 --> 00:09:17,209 and they say do they vary inversely or directly or maybe neither 139 00:09:17,209 --> 00:09:20,042 you could either try to make a table like this 140 00:09:20,042 --> 00:09:24,640 if you scale up X by certain amount and Y gets scaled up by the same amount 141 00:09:24,640 --> 00:09:26,544 that is direct variation 142 00:09:26,544 --> 00:09:31,188 if you scale up X, you may want to try it a couple of different times 143 00:09:31,188 --> 00:09:35,878 and you scale down Y, do the opposite with Y, then it is probably inverse variation 144 00:09:35,878 --> 00:09:42,798 a sure way to know what you´re dealing with 145 00:09:42,798 --> 00:09:46,791 it´s to actually algebraically manipulate the equation so it gets back to either this form 146 00:09:46,791 --> 00:09:48,788 that tells you it´s inverse variation 147 99:59:59,999 --> 99:59:59,999 or this form, which will tell you that it is direct variation