1 00:00:00,754 --> 00:00:03,588 Now we mixed up things a little bit more: 2 00:00:03,588 --> 00:00:07,466 on the left side of the scale, not only do we have 3 00:00:07,466 --> 00:00:11,487 these identical unknown masses with mass X, these three blue things, 4 00:00:11,487 --> 00:00:14,087 we also have some of the 1kg masses over here, 5 00:00:14,087 --> 00:00:16,498 actually, we have two of them. 6 00:00:16,498 --> 00:00:18,588 Now, we are going to figure out wat X is. 7 00:00:18,588 --> 00:00:22,687 But before we even do that, I want you to think about a mathematical equation 8 00:00:22,687 --> 00:00:27,554 that can represent what is going on; that equates what we have on the left hand, 9 00:00:27,554 --> 00:00:31,620 with what we have on the right side of the scale. 10 00:00:31,620 --> 00:00:33,354 I will give you a few seconds to think about it... 11 00:00:35,287 --> 00:00:37,620 So let's think about what we have on the left side: 12 00:00:37,620 --> 00:00:46,020 we have 3 masses with mass X, so you can say we have 3x 13 00:00:46,020 --> 00:00:52,154 and then we have 2 masses of 1 kilogram, so in total we have 2 kg. So + 2. 14 00:00:52,154 --> 00:00:57,354 So one way to think about the total mass on the left-hand side is 3x + 2. 15 00:00:57,354 --> 00:01:00,848 Three masses with mass X, plus two kilograms. 16 00:01:00,848 --> 00:01:02,938 That is what we have on the left-hand side. 17 00:01:02,938 --> 00:01:05,585 Now, let us think about what we have on the right-hand side. 18 00:01:05,585 --> 00:01:07,154 We can simply count them: 19 00:01:07,154 --> 00:01:11,855 [counts to 14] 20 00:01:11,855 --> 00:01:19,220 Fourteen blocks, each has a mass of 1 kg, so the total mass will be 14 kg. 21 00:01:19,220 --> 00:01:23,620 And we see that the scale is balanced, not tilting down or upwards. 22 00:01:23,620 --> 00:01:28,666 So this mass over here must be equal to this total mass. 23 00:01:28,666 --> 00:01:34,087 The scale is balanced, so we can write an 'equal'-sign. 24 00:01:34,087 --> 00:01:37,220 (let me do that in a white coulour, I do not like that brown) 25 00:01:37,220 --> 00:01:41,258 Now, what I want you to think about, 26 00:01:41,274 --> 00:01:44,890 and you can think about it either through the symbols or through the scales, is: 27 00:01:44,890 --> 00:01:48,104 how would you go about -- let us think about a few things: 28 00:01:48,104 --> 00:01:54,090 how would you first go about at least getting rid of these little 1kg blocks? 29 00:01:54,090 --> 00:01:56,557 I will give you a second to think about that... 30 00:01:59,023 --> 00:02:01,107 Well, the simpelest thing is: 31 00:02:01,107 --> 00:02:03,940 you can take these 1kg blocks off of the left-hand side, 32 00:02:03,940 --> 00:02:07,098 but remember, if you just took these blocks off of the left-hand side, 33 00:02:07,098 --> 00:02:10,357 and it was balanced before, now the left-hand side will be lighter 34 00:02:10,357 --> 00:02:13,757 and it will move up. But we want to keep it balanced so we can keep saying 'equal'. 35 00:02:13,757 --> 00:02:16,107 That this mass is equal to that mass. 36 00:02:16,107 --> 00:02:21,490 So, if we remove 2 block from the left-hand side, we need to remove 2 from the right-hand side. 37 00:02:21,490 --> 00:02:26,490 So, we can remove two there, and then we can remove two over there. 38 00:02:26,490 --> 00:02:31,340 Mathematically, what we are doing is: we are subtracting 2 kilograms from each side. 39 00:02:31,340 --> 00:02:34,157 We are subtracting 2 from this side, 40 00:02:34,157 --> 00:02:38,909 So on the left-hand side we now have 3x + 2, minus 2 41 00:02:38,909 --> 00:02:42,624 we are left with just 3x, 42 00:02:42,624 --> 00:02:47,557 and on the right-hand side we had 14 and we took away 2 (let me write this:) 43 00:02:47,557 --> 00:02:53,023 we took away 2, so we are going to be left with 12 blocks. 44 00:02:53,023 --> 00:02:56,423 And you see that there, the ones that I have not crossed out, there are 12 left, 45 00:02:56,423 --> 00:02:58,957 and here you have 3 of those X-blocks. 46 00:02:58,957 --> 00:03:02,361 Since we removed the same amount from both sides, 47 00:03:02,361 --> 00:03:08,357 our scale is still balanced. And our equation: 3x is now equal to 12. 48 00:03:08,357 --> 00:03:12,690 Now, this turns into a problem very similar to what we saw in the last video, 49 00:03:12,690 --> 00:03:17,083 so now I ask you: what can we do to isolate one x, 50 00:03:17,083 --> 00:03:21,428 to only have one 'X' on the left-hand side of the scale, 51 00:03:21,428 --> 00:03:24,552 while keeping the scale balanced? 52 00:03:28,163 --> 00:03:30,629 The easiest way to think about it is: 53 00:03:30,629 --> 00:03:35,154 If I want one X on this left-hand side, that is a third of the total X's here. 54 00:03:35,154 --> 00:03:38,963 So what if I were to multiply the left-hand side by one-third -- 55 00:03:38,963 --> 00:03:43,745 -- but if I want to keep the scale balanced, I have to multiply the right-hand side by one-third. 56 00:03:43,745 --> 00:03:46,229 If we can do that mathematically, 57 00:03:46,229 --> 00:03:49,563 Over here I can multiply the left-hand side by 1/3, 58 00:03:49,563 --> 00:03:55,163 and if I want to keep my scale balanced I also have to multiply the right-hand side by 1/3. 59 00:03:55,163 --> 00:04:00,563 Multiplying it physically literally means: just keeping a third of what we had originally 60 00:04:00,563 --> 00:04:03,896 We would get rid of two of these. 61 00:04:03,896 --> 00:04:06,029 If we want to keep a third of what we had here originally, 62 00:04:06,029 --> 00:04:08,896 -- there are 12 blocks left over after removing those first two -- 63 00:04:08,896 --> 00:04:15,429 so, 1/3 of 12: we are only going to have four of these little 1kg boxes left. 64 00:04:15,429 --> 00:04:21,363 Let me remove all but four. (so, remove those, and those...) 65 00:04:21,363 --> 00:04:24,659 And I have left [counts them] 4 here. 66 00:04:24,659 --> 00:04:27,563 And so, what you are left with, the only thing you have left, 67 00:04:27,563 --> 00:04:32,554 is this 'X' - I will shade it in to show this is the one we actually have left - 68 00:04:32,554 --> 00:04:37,058 and then we have these 1 kilogram boxes. 69 00:04:37,058 --> 00:04:40,866 You see it mathematically over here: 1/3 * 3x 70 00:04:40,866 --> 00:04:45,763 -- or you could have said 3x divided by 3 -- either way, that gives us 71 00:04:45,763 --> 00:04:49,736 -- these threes cancel out, so that would give you an 'X' 72 00:04:49,736 --> 00:04:56,633 and on the right-hand side: 12 * 1/3 - which is the same as 12/3, is equal to 4. 73 00:04:56,633 --> 00:05:02,600 And, since we did the same thing to both sides, the scale is still balanced. 74 00:05:02,600 --> 00:05:07,894 So you see that the mass of this thing must be the same as the mass of these 4 left-over blocks. 75 00:05:07,894 --> 00:05:10,829 x must be equal to 4 kilograms.