# SW4 EQUIVALENCE

This eighth grade mathematics lesson focuses on equations. It is a lesson in which the already introduced topic is practiced. The lesson is taught in Swiss French and is 45 minutes in duration. There are 22 students in the class.

Time | Caption |
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00:00:12 | [ Bell ] |

00:00:16 | Okay, then. |

00:00:24 | Okay, then, hello take a seat. So to start with, we are going to correct the problems that were due for today, that means exercise 948... |

00:00:36 | Or we can practice one of the theorems of equivalence. |

00:00:44 | Especially this one which, in the first place, is the one which tells us that one must add up an expression. Therefore, page 948. |

00:00:56 | We will start nicely, we will start with Gregoire... With A, X plus five equals eight. |

00:01:03 | Well, we must take away five. |

00:01:05 | We do minus five. |

00:01:06 | We do minus five, therefore X is equal to three. |

00:01:07 | Mmm. |

00:01:09 | X is equal to three. Okay? We will maybe do the most complicated ones on the board, the ones from the second line. For, er, the B. |

00:01:19 | Guillaume. |

00:01:20 | Er, X plus three is equal to minus five. We must take away er... minus three. |

00:01:25 | So we mustn't take away minus three. |

00:01:27 | It makes minus three. |

00:01:28 | So we must add minus three or do minus three. |

00:01:30 | It makes minus three. |

00:01:31 | It makes minus three. |

00:01:32 | And X is equal to eight, to minus eight |

00:01:35 | To minus eight, X equals minus eight. Questions up until now? C? Er... Florence. |

00:01:43 | Er, X minus six equals 30. Therefore we do plus six, which means X equals 36. |

00:01:48 | X equals 36, exactly... Letter D? Katia. |

00:01:56 | Er... well er, the solution is five. |

00:02:00 | Uh huh, the solution is five. What do you do? That's what interests us. Now that we are learning what to do, what do you do? |

00:02:07 | Well- |

00:02:08 | We know that in an equation, we must put a little vertical bar, and what must we do? |

00:02:15 | What do we do to make our equation equivalent. |

00:02:16 | I did three times five. The first is 15, then the second is two times five plus five. |

00:02:21 | So there you're groping about. |

00:02:23 | Yeah. |

00:02:24 | You're searching like this- The idea is to do what? It's to avoid all these stories of groping about. |

00:02:30 | When we have an equation of this sort... it's the... We have three X is equal to two X plus five. What does that mean? |

00:02:41 | What's the basic idea in all these problems there? Herve? |

00:02:44 | To take away the X's. |

00:02:46 | Take away the X. Here we must take away the X, but what are we trying to do? What are we trying to do as a final aim? Yes? |

00:02:53 | (inaudible) X is equal to something. |

00:02:55 | X is equal. So we must put on one side the X's, on the other side whatever doesn't have any X's. So we have the theorems of equivalence that we saw. |

00:03:03 | There's one which says I can add or take away a number... Okay? |

00:03:11 | There's one that says I can add or take away expressions, some literal part, okay? |

00:03:18 | And then there's a third one where we can multiply by a number that is not equal to zero. Are we okay? |

00:03:24 | So here, what do we do? Valentine? |

00:03:27 | We do minus two X. |

00:03:29 | We do minus two X. |

00:03:30 | Well there it stays one X to the three X. |

00:03:33 | On the left side it stays as one X. Three X minus two X makes one X. |

00:03:36 | (inaudible) the two X and then, er- |

00:03:38 | It suppresses the two X and what stays? |

00:03:39 | What stays is X is equal to plus five. |

00:03:42 | It stays as X is equal to plus five, and then directly one can say to oneself, well yes, my solution is well... five... Are we okay? |

00:03:46 | (inaudible) |

00:03:52 | Okay, Katia? Because if you try by doing it by groping about, I know that we did it, er, by successive tests- |

00:04:01 | We tried that, but we saw it was tedious and long. So now we have a new method. |

00:04:07 | Aurore, what can we say for E? |

00:04:09 | We do minus four X. |

00:04:11 | We have five X is equal to minus eight plus four X. Therefore we do minus four X. Which gives us X equal to? |

00:04:16 | Minus eight... |

00:04:17 | X equals minus eight... Questions up until now? For F? For F maybe I'll write it up... We have minus eight X is equal to minus nine X... plus seventeen. |

00:04:34 | We will come to this side. Yeah, Herve. |

00:04:36 | Well, er, since we want to eliminate the eight, well since we want to find X, we put, we do plus eight X. |

00:04:44 | So if we do plus eight X... What happens if we do plus eight X? |

00:04:49 | Ha, yeah... |

00:04:51 | We will have everything, we'll have nothing, and then we'll have the X's and the numbers on the right side. Have we won? |

00:04:58 | (inaudible) |

00:04:59 | It's not want we want? What do we want? |

00:05:00 | Well- |

00:05:01 | There should be one X. |

00:05:02 | So, Camille? |

00:05:03 | Nine X. |

00:05:04 | We do plus nine X. In doing plus nine X- in doing plus nine X what's interesting is that here we have X, because plus nine X minus eight X- |

00:05:13 | It makes X. |

00:05:14 | It makes X is equal to? |

00:05:16 | Seventeen. |

00:05:17 | Seventeen... and then directly one can have our solution which is, well, 17. So it's clear that sometimes we are going to try and get the X's. |

00:05:26 | There it's advantageous, we always have the X's on the left. |

00:05:30 | Maybe for G and H still, we'll put them up on the blackboard. So for G we have three X plus five... is, is equal to two X, Ludivine? |

00:05:39 | Minus (inaudible). |

00:05:40 | Minus? |

00:05:40 | Minus 30. |

00:05:41 | Minus 30? |

00:05:42 | Yeah. |

00:05:43 | Okay... so we can do it all in one stage. |

00:05:46 | No. |

00:05:48 | So one could- but one can do it in two... So what do we want to do first? |

00:05:53 | (inaudible) |

00:05:55 | Minus two. |

00:05:56 | So we want to do minus two X. And then it gives us? |

00:05:58 | (inaudible) |

00:05:59 | X plus five (inaudible). |

00:06:00 | X plus five is equal to? |

00:06:03 | Minus three. |

00:06:04 | Minus three... and what do we want to do now? What's bothering us? |

00:06:07 | Minus five. |

00:06:08 | It's the five which is bothering us, therefore we do... |

00:06:10 | Minus five. |

00:06:11 | Minus five. Therefore X is equal to? |

00:06:13 | Minus 35. |

00:06:14 | Minus 35. Okay... But we're going to try and take in the next one, we're going to try and do it in one stage directly. When |

00:06:23 | we write the next one down, we have 18 X. It's H... We have eighteen X, er, minus 24. |

00:06:30 | (inaudible) |

00:06:33 | Plus 17 X... There we're going to try and do it in one stage. I don't have enough space. It's going to be okay. Eighteen X minus 24 is equal to |

00:06:41 | 72 plus 17 X... What do you want to do? What must we choose first? |

00:06:50 | Minus 17. |

00:06:51 | So minus 17 X. It means we've chosen that on the right side we had more X's. We're okay? Okay, so if we decide to write minus |

00:06:58 | 17 X without doing the operation, we say- well then. I have already decided that on the right there would be more X's and then I'll put them on the left. |

00:07:06 | Are we okay? On the left the X's... So on the right we only have the- |

00:07:10 | The numbers. |

00:07:11 | Okay. So what is bothering us on the left now? It's- |

00:07:17 | Minus 24. |

00:07:18 | The minus 24. Therefore what must I do to take away this minus 24? |

00:07:21 | Plus 24. |

00:07:22 | Plus 24. Therefore I can directly write at the same time plus 24... and I do the two things at the same time. |

00:07:29 | What I'm going to be able to do, we shall see in the following exercises, it's to do multiplications and additions at the same time. But on the other hand... |

00:07:37 | Plus if I add up, instead of adding up monomials, if I add up polynomials, it's the same thing. Since I add up several |

00:07:44 | monomials. Okay? Therefore the 18 X that I'm going to subtract minus 17 X, there's going to stay? |

00:07:50 | X. |

00:07:51 | And then what am I going to do with my 24? Well, the minus 24? I'll be staying with the Y. The seven- the |

00:08:04 | minus 17 X, when it will arrive here, it will make the 17 X go away. And then the 24, when it arrives here it will |

00:08:11 | give me? Ninety-six. |

00:08:18 | And there we don't need a calculator to get to that S is equal to 96. Is there a question on this exercise 948? |

00:08:26 | So I'll let you continue with 949. |

00:08:31 | For those who finish through 950, there's 951, 952, 953, |

00:08:40 | 954. There's enough to amuse oneself until 957. |

00:09:30 | Sir? |

00:09:31 | Yes. |

00:09:32 | (inaudible) 949, I don't need to (inaudible). |

00:09:38 | But you've done it right here. |

00:09:40 | But yes, but A, we did with you. |

00:09:42 | Ah! So? |

00:09:46 | And, er, here B. |

00:09:47 | Yeah, you want to isolate- you want to isolate what? |

00:09:52 | Er, I- yeah, but I haven't understood. |

00:09:54 | But you want to isolate what, and you want to keep what? You decided on the left you put X's? It's what you've decided there. You tried to do |

00:10:01 | this. What's bothering you? It's that one there. |

00:10:03 | Yeah, therefore I do minus. |

00:10:06 | No. How do you take away seven? How do you get minus seven to become zero? |

00:10:12 | Er, I do plus- plus eight... er! Plus seven. |

00:10:18 | You do plus seven, hum. You take- so do plus seven. So now what happens on that side there? I do plus seven, |

00:10:25 | two X minus seven plus seven. The two X's will stay there. |

00:10:29 | Two X is equal to seven. |

00:10:30 | And then, no! Minus 19, when there's plus seven it makes minus 12. |

00:10:34 | Uhum! |

00:10:35 | It's correct. You've badly written the things, but your calculation is correct. |

00:10:38 | Yeah. |

00:10:39 | Two X is equal to minus 12. Now you want that only- that only X, therefore you- |

00:10:43 | Times a half. |

00:10:44 | Times a half. You find X equals minus six. |

00:10:46 | Yeah, I did it correctly. |

00:10:47 | It's correct. |

00:10:50 | So there, the same thing. You decide, you have X's only on the left. What's bothering you? |

00:10:55 | The five? |

00:10:56 | The five. Therefore? |

00:10:58 | I do plus minus five. |

00:11:01 | You do minus five. You write wrong things down, but you do the calculations correctly. You do minus five, therefore. You will find the four X here, and you |

00:11:07 | find 20 minus five, it equals, well? |

00:11:09 | Fifteen. |

00:11:10 | Fifteen. And then you must divide by- no, you must divide-there's not 15 which must- you must simply divide by? |

00:11:12 | Seven? |

00:11:16 | Twenty? |

00:11:17 | By? |

00:11:18 | By a quarter. |

00:11:19 | By quarter, therefore we need a quarter. You must find 15 quarters as a solution. |

00:11:22 | Sir? |

00:11:23 | Yes! |

00:11:24 | The thing (inaudible) must it always be on the left? |

00:11:27 | Not necessarily, you can also put it on the right. |

00:11:29 | Always X on the left? |

00:11:32 | Listen, generally people try and have it on the left because we've gotten used to saying X equals. But we'll see, maybe later, |

00:11:38 | er, there are some, later on here, where it's nicer to have it on the right because it's positive. Therefore we can directly find what X equals. Because sometimes we |

00:11:46 | will have minus one X, so we must multiply again by minus one. |

00:11:49 | Sir? |

00:11:50 | The purists always want X on the left. But is it really good? Yes! |

00:11:57 | It's always the same thing, 949? |

00:12:01 | Yeah, it's always the same thing but you can't do two things at the same time. You can't do- so sometimes you can't |

00:12:05 | do times a third then minus 12. To begin with, you must isolate- to start with- no- that one, you're not allowed to do it. The times a third. |

00:12:08 | To start with, you do minus 12. So what will you be staying with? |

00:12:10 | We inverse. |

00:12:16 | Bah! Three X. |

00:12:18 | Three X. |

00:12:19 | (inaudible) 48. |

00:12:20 | Three X, three X equals? |

00:12:21 | Forty-eight? |

00:12:22 | No... |

00:12:22 | No, er, 36. |

00:12:23 | If you do minus 12- |

00:12:24 | Thirty-six. Ah bother, I shouldn't have taken away the 36, it was there. So equals to 36. And then you can do times? |

00:12:32 | A third. |

00:12:33 | There now, is what I was saying before. We can't do several additions at the same- together. So we're going to do additions and multiplications together. |

00:12:38 | Ah! Yeah, okay. |

00:12:39 | Hum, hum? |

00:12:40 | Yeah, yeah, it's okay. |

00:12:40 | It's okay. |

00:12:41 | Thank you. |

00:12:41 | You're welcome. |

00:12:43 | Sir? |

00:12:44 | Yes! |

00:12:45 | There's a problem with C. |

00:12:47 | What's the problem with C. |

00:12:48 | I can't- I can't do it, C. |

00:12:49 | Take away five. |

00:12:50 | Yeah. |

00:12:51 | Okay. So what happens when you take away five? |

00:12:54 | Oh well, there also it goes away. |

00:12:56 | Well yeah. So write the thing again. The four X stays. |

00:12:58 | Yeah. |

00:12:59 | Equals? Fifteen. Then now. When four objects are worth 15 francs, what do you do to know the price of an object? |

00:13:08 | Well, I divide by, er, four? |

00:13:11 | Yeah, take a quarter. |

00:13:12 | Ah! Yeah. |

00:13:14 | It needs a quarter, and what happens? |

00:13:17 | (inaudible) |

00:13:19 | X equals fifteen over four. |

00:13:21 | Yeah. |

00:13:22 | The fraction is in numbers. |

00:13:26 | How? |

00:13:27 | No, X equals? You can't put the fraction between- under the fraction- under- the bar of fraction under the equality. X equals 15 over four. |

00:13:37 | (inaudible) |

00:13:38 | Yeah, if you like. |

00:13:42 | Yes. |

00:13:44 | There, if we divide by four, the plus five must also be divided by four? |

00:13:47 | Oh well, yeah. Therefore we don't divide by four? |

00:13:49 | No. |

00:13:50 | No? |

00:13:51 | If we take away the five, afterwards we have four X is equal to 15. |

00:13:53 | And then. |

00:13:54 | And then we can. |

00:13:55 | Then it's not divided by four fifteen? |

00:13:57 | Yes, but it does have decimal points. |

00:13:59 | Ho, poor little one, if it does have decimal points, are they so painful, decimal points? |

00:14:01 | No. |

00:14:02 | Ha, good. |

00:14:03 | But, Sir, (inaudible)? |

00:14:06 | So we leave the fraction, we will leave 15 quarters. Exactly. |

00:14:09 | (inaudible) |

00:14:11 | (inaudible) seventy-five (inaudible) the fraction? |

00:14:13 | But it's 15 quarters. You've divided 15 by four, that's 15 quarters. |

00:14:17 | Er, (inaudible). |

00:14:20 | Here you do minus 12, but when you did minus 12 here you touched only that 12 there. The three X you left alone. |

00:14:26 | (inaudible) |

00:14:27 | Leave it as three X. It stays three X here... And now you must still do a stage. So that's annoying because you... |

00:14:28 | (inaudible) do three. |

00:14:34 | You have already done the next one but... we do times one third, then we will find? |

00:14:37 | Third. |

00:14:39 | Uh... |

00:14:40 | X is equal to? |

00:14:42 | Twelve. |

00:14:43 | Twelve. |

00:14:44 | Humm. |

00:14:45 | Humm. But what do you do here? |

00:14:48 | Er, plus seven. |

00:14:49 | Plus seven. |

00:14:52 | What are you going to find? |

00:14:54 | Er, 12 X is equal to seven. |

00:14:55 | Well no... Minus 19 plus seven? Minus? |

00:15:00 | Well, uh... |

00:15:04 | Minus 19 plus seven? |

00:15:06 | Yeah, we must add that... yeah, well 26. |

00:15:10 | Tss-Tss. |

00:15:11 | Er, 25. |

00:15:12 | Tss-Tss. |

00:15:14 | Minus 19 plus seven? |

00:15:18 | You've got 19 objects missing, someone gives you back seven, how many do you have missing? |

00:15:22 | Well, er, 12. |

00:15:25 | So. |

00:15:27 | Then after that, what do we do? |

00:15:29 | Times one. |

00:15:34 | You want to take away the two? |

00:15:35 | Yeah, well times a second. |

00:15:36 | A half. |

00:15:37 | Yeah, a half. |

00:15:38 | Which will give, we take away the X. X equals. |

00:15:41 | Equals, is equal to... |

00:15:45 | Mental calculation. |

00:15:46 | Yes I know. |

00:15:47 | Twelve divided by two... the minus, it stays. |

00:15:49 | Minus six. |

00:15:50 | There now. |

00:15:51 | (inaudible) |

00:15:53 | Therefore here you must still put X is equal to 12 and here you must put X is equal to minus six. |

00:15:57 | Minus six. (inaudible) Thank you. |

00:16:00 | Sir, Sir! |

00:16:06 | What is it? |

00:16:08 | How do we go about for the two? Because I can do it, but in two stages. |

00:16:11 | Well yeah, and what? We never said one should do only one stage. |

00:16:15 | (inaudible) a half? |

00:16:17 | Well yeah... But yeah, one can maybe do 10 stages if it's necessary. We never said one should do it in only one stage. One mustn't be |

00:16:25 | a minimalist, either. |

00:16:29 | When we put there, er, we always first try to take away the part, er, on the right? |

00:16:38 | Yeah exactly. We try to take away the literal part on the right, if we decide to put it on the left. But, er, for F, for example, it can |

00:16:45 | be interesting, or for E, to put it on the right for example. To avoid the minuses, but one can do it in both cases. |

00:16:49 | Sir! |

00:16:50 | Yes. |

00:16:51 | There now I've found for half of X, I will multiply by two to find X. |

00:16:55 | Why? I'm not okay with what you've done there. You've done a multiplication and a subtraction at the same time. |

00:17:01 | Well, you said it was (inaudible). |

00:17:02 | No, don't do it at the same time. One stage after the other. Excuse me, Gregoire... I think that, let's stop for awhile, |

00:17:09 | please. We look at A, we did it together as an example. Do you remember? And B also? It astonishes me, |

00:17:20 | though, because a lot of you did it wrong. |

00:17:24 | So we'll maybe just do B again together. Two X minus seven is equal to 19. So it's clear that you have two worries. |

00:17:37 | Here. You have one X which is multiplied by two. You want X on it's own. |

00:17:41 | That's the least difficult. |

00:17:48 | Thank you. |

00:17:51 | Here you've got to realize something. You can't do everything at the same time. You mustn't think we have a problem, we put a |

00:17:59 | bar, then we have the solution underneath. We can maybe do one, two, three, four stages if it's needed... There's the stages also of literal |

00:18:07 | calculations. You'll see that when we calculate the data and there's brackets or something else, there will be some stages where we'll simply have |

00:18:11 | to work with the left part and the right part separately. Here, first thing, isolate everything unknown on the same side. |

00:18:22 | Therefore, what's bothering you here? |

00:18:25 | Minus seven. |

00:18:26 | It's the minus seven. Therefore we do one stage with the minus seven... We make it plus seven. |

00:18:32 | And then we find two X is equal to? |

00:18:35 | Minus 12. |

00:18:36 | Minus 12. And now we do one stage where we do times a- |

00:18:40 | Half. |

00:18:41 | Half... Okay, and then we find one X is equal to minus six so the solution is minus six... If we take C... You have it nearly |

00:18:50 | exactly C, you've all done it. Four X plus five is equal to 20. So it's very well, er... Herve, to say I divide by four, |

00:19:02 | and at the same time I do minus five... It's what you did, it's that one? |

00:19:08 | Yeah, at D. |

00:19:09 | Ah no, you did that in the next one, D. But look already here. It's the same thing here. We first try to isolate the X. There's only one place in X. |

00:19:18 | It's on the left, so what's bothering us? |

00:19:20 | The minus five. |

00:19:21 | The minus five... Therefore I find four X is equal to 15. And then even if there are decimal points it doesn't matter. Now I do times a quarter. |

00:19:32 | And I find X is equal to 15 quarters. And what does it mean that it's 15 quarters? There are those who would prefer to put three, decimal point, seventy-five. |

00:19:39 | I remind you it's better as a fraction, because if by any chance it wasn't a fraction which gives a rational number which |

00:19:47 | finishes itself, it's not interesting... Okay? A number with decimal points which finish. If we have periodical numbers it's never very interesting, we |

00:19:54 | can't use them on the calculator. On the other hand, 15 quarters we can put in... Therefore S is equal to 15 quarters. So that's true. |

00:20:03 | When one looks at D for example, where one has four X, it's what in D? Four X plus five again. |

00:20:10 | (inaudible) |

00:20:13 | Two X minus 16? |

00:20:14 | Thirteen. |

00:20:15 | Thirteen. Thank you. When we have this, it's true one would want to say: I've got multiples of X on both sides. Valentine? |

00:20:22 | So we put the bar and we do minus, er, minus two X. |

00:20:26 | Times a half. |

00:20:27 | So why do times a half? |

00:20:29 | To do a (inaudible). |

00:20:30 | So I tell you honestly, it's not good to do times a half. We're going to write what Herve said, then we're going to look at how we can |

00:20:35 | complicate a problem which wasn't too complicated to start with. |

00:20:38 | We can (inaudible). |

00:20:39 | How to complicate it when one can do it simply... that's Herve's proposition. Look, I do times a half, I am- |

00:20:46 | Okay, when doing times a half, let's look at what happens. I'll change colors on purpose for you to realize that it's not |

00:20:50 | the ideal. If I do times a half, how many X's do I have here? |

00:20:52 | Two. |

00:20:53 | Two X. |

00:20:54 | No, half of X. |

00:20:55 | No. |

00:20:56 | No. I've got an idea. Times half X. |

00:20:58 | You can't do times half X... You can't do times letters... It's forbidden, we've seen. Do you want me to show you again on the |

00:21:06 | transparency what happens? Do you remember? Suddenly it has curves. We have raised one solution. Therefore we only have three possibilities |

00:21:13 | to do something... All numbers, add letters, or multiply by a number which is not equal to zero... We can't |

00:21:21 | multiply by a literal part... We're not allowed to do that. Therefore when you say to me a half of X, but do times a half. It |

00:21:29 | doesn't bother me in that case there, the only difference is in my point of view, I absolutely haven't gained anything... because first of all I still have X's on |

00:21:37 | both sides... And what's more I've gained fractions. And since we all like working with fractions, we want to have them in |

00:21:45 | the solution, the fractions. We're okay? Gregoire, we want to have them in the solution? It's correct or not? But to have them now it's |

00:21:53 | not comfortable... After, you don't know anymore how to add fractions and then you get into trouble. Why get into trouble? |

00:22:02 | All right? On one's sheet of paper when one had something one could do easily. So do we continue doing times a half here- |

00:22:07 | Herve? Do you insist? |

00:22:09 | No. |

00:22:10 | No. Thank you. |

00:22:12 | So let's try to start like before, using times. To use times we need more polynomials. That one only has monomials on |

00:22:22 | the left or the right. Okay? We isolate. Therefore for the first we shall say what? |

00:22:27 | We can do (inaudible)? |

00:22:33 | So it's correct, it's correct, Camille. We can do minus two X. One can also do minus five. Because at the moment we say we'll do |

00:22:42 | minus two X, we've decided the X's will be on the left. |

00:22:44 | Nine. |

00:22:46 | We can do both at the same time. |

00:22:47 | We can put them one after the other. |

00:22:48 | So there one can put them one after the other because it's two additions. |

00:22:50 | All right. |

00:22:51 | Are we obliged to do them both at the same time? |

00:22:52 | No, you can do them separately. After there's a problem of production, of efficiency, when you do exercises and one tells you I would like you to |

00:22:59 | do 15 in 45 minutes, er, if you have two stages you will lose a little time. But that's training. |

00:23:07 | (inaudible) |

00:23:09 | It's better to get the correct answer than to finish quickly with all the results wrong, you're right... So here we get to two X is equal to... |

00:23:18 | because minus two X minus five. |

00:23:20 | Minus one. |

00:23:21 | Minus? |

00:23:22 | Minus (inaudible). |

00:23:23 | No, minus 13. |

00:23:25 | Minus 18. |

00:23:26 | But how much is it when one does minus 13 minus five? |

00:23:27 | Minus 18. |

00:23:28 | Well, minus 13 minus five, you've got 13 objects missing, someone asks you for five more, you've got 18 missing. |

00:23:32 | Ha, yeah. |

00:23:33 | (inaudible) divided by two. |

00:23:34 | It seems to me, I don't know... And then after? |

00:23:37 | Divided by a half. |

00:23:39 | No. |

00:23:40 | Not divided, times a half, yeah. |

00:23:41 | There now, either we do times a half or we divide by two, but we can't... times a half. X is equal to? |

00:23:48 | Minus nine. |

00:23:49 | Minus nine... Okay. The S is equal to minus nine. |

00:23:57 | Is there a question on this? Can we continue now? Let's advance. |

00:24:04 | Yes? |

00:24:06 | Er, there how do we do it? |

00:24:08 | So everywhere there are brackets, to start with we take away the brackets. |

00:24:11 | (inaudible) |

00:24:13 | No. |

00:24:14 | It's wrong? |

00:24:15 | It's wrong. |

00:24:17 | Ha, yeah, nine is wrong. |

00:24:21 | That is wrong. |

00:24:25 | The X cube is wrong. Ha, it was three X actually. |

00:24:27 | Yeah. |

00:24:28 | Ha, yeah. |

00:24:30 | It makes- it makes seven. |

00:24:32 | Me, I would like it better when you have something like that. So there do as Ludivine said. You write again the object when you've taken away the brackets. |

00:24:40 | Yeah, okay. |

00:24:41 | You do here the thing like this, you can say I'm doing literal calculation. |

00:24:45 | Yeah. |

00:24:47 | And then I find four X minus five is equal to six... 27... Then now I can work as I did before... |

00:24:51 | Six X over 27, |

00:24:56 | but now if we do minus six X to pass it on this side (inaudible) to minus two. |

00:24:59 | It's not interesting. We have minus, yeah. |

00:25:02 | Well, then we do minus four X and we pass (inaudible). |

00:25:03 | There then, so we decide to have the X's on the right. It doesn't matter. |

00:25:08 | Ah yeah, but- |

00:25:09 | If X is equal to three or three is equal to X? What changes for you? |

00:25:11 | Nothing. |

00:25:13 | Okay. So it's what I told you. There are purists who absolutely want to have the X on the left. But finally as far as efficiency, |

00:25:19 | in what we'll want to do later on to find problems, resolve problems, it doesn't take us very far. |

00:25:29 | And after we do it like that. |

00:25:31 | No you can't... After what must you do? |

00:25:36 | On that. |

00:25:38 | Minus 27. Okay. And what will we get? |

00:25:41 | Well it's minus... 32. |

00:25:43 | Minus 32 is equal to? |

00:25:46 | (inaudible) |

00:25:47 | Therefore- |

00:25:49 | Times a half. |

00:25:50 | Times a half. And you find X? |

00:25:53 | (inaudible) |

00:25:54 | Yeah, 32 over two. Thirty-two over two. |

00:25:59 | Therefore there it makes minus 16. |

00:26:00 | Minus 16 is equal to X. |

00:26:03 | Ha, yeah. |

00:26:04 | S equals minus five. Anyway, one writes S equals, so whether one has X equals minus 16 or minus 16 equals X. |

00:26:11 | Yeah, it doesn't matter. |

00:26:12 | The element on the left is equal to the one on the right. Whether X is on the left or the right. All right? |

00:26:17 | Yeah. |

00:26:18 | Okay? |

00:26:18 | Sir? Sir? |

00:26:19 | One can't have zero X. |

00:26:22 | No, it would be better not to have zero X, because then we're not well. |

00:26:24 | But because there it gives us zero X if we do minus 17 X. |

00:26:27 | Well, yeah. |

00:26:28 | It takes away each time, er. |

00:26:29 | Therefore you're going to find how much? Zero, yeah that's right. Well yeah, it's going to start to get interesting. Wait, let's look if you've already taken- |

00:26:31 | Ha yeah, it makes (inaudible). |

00:26:35 | the correct data. That one, yeah, that's going to become interesting. What do you find? You do minus 17 X. |

00:26:41 | I do minus 17 only. |

00:26:42 | No you must do minus 17 X, okay. You take away this 17 X to put it here. You've had the idea to do it like this. You do minus 17 X. |

00:26:47 | Therefore you find two here. |

00:26:49 | Yeah. |

00:26:50 | So write two. Two equals minus two... So now. |

00:26:55 | (inaudible) |

00:26:57 | You can't do. |

00:26:58 | But it's wrong, but it's not possible. |

00:27:00 | (inaudible) |

00:27:01 | So it's not wrong, it's that this equation doesn't have a solution. |

00:27:05 | Wah! (inaudible). |

00:27:06 | Is there some X which will permit you to say: whatever X you chose, you will have two equals minus two? There's something which |

00:27:12 | doesn't work, therefore you're going to say X equals the whole? |

00:27:14 | Empty. |

00:27:15 | To the empty set, right? |

00:27:17 | (inaudible) |

00:27:18 | Yes, but that is the set containing the empty set, therefore it's no longer empty. |

00:27:22 | Ha ha, what does that mean? |

00:27:24 | It means (inaudible). |

00:27:25 | That's all. The word there, that symbol means empty set. |

00:27:29 | So when there's... when there's, when it's impossible I must put only that? |

00:27:31 | Sir! |

00:27:32 | Sir! |

00:27:33 | Sir! |

00:27:34 | That's right, yeah. |

00:27:35 | E I understand very well. |

00:27:37 | No, why do you do minus five X? Is there anywhere here where there's a minus five X? |

00:27:43 | Well no, but to have X alone. |

00:27:45 | We didn't say one must have X alone there. We must first take away that X there. Is it X which is bothering you? |

00:27:50 | Yeah. |

00:27:51 | I'm coming, I'm here. If I take this X, what do I do to take away this X? I don't do minus five X? |

00:27:57 | I do minus X. |

00:27:58 | You do minus X. So now do only minus X. |

00:28:00 | E:00] |

00:28:02 | Yes? |

00:28:03 | Is it possible that equations don't work like that one? |

00:28:07 | Where are you there? |

00:28:08 | At (inaudible). It's not possible? If four X is equal to four X. |

00:28:13 | No, it's not possible. |

00:28:14 | So what do we put? |

00:28:16 | Well, look at what happens. Do something. |

00:28:18 | Well, yeah, but- |

00:28:19 | You must do minus four X. |

00:28:20 | Yeah. |

00:28:21 | So do minus four X, then look at what happens. I'll come back. |

00:28:23 | Sir! |

00:28:24 | Yes. |

00:28:25 | Er, here what do we do? |

00:28:28 | For E... Well minus four X plus 15, well what. You do minus 15. |

00:28:34 | Yeah (inaudible). |

00:28:35 | I did minus four X... Your 15 falls, then minus 15 plus two... That doesn't bother me. Eleven minus 15? |

00:28:43 | And for E? Is it correct here, if I put this? |

00:28:47 | (inaudible) the minus four, the over four it's also here? |

00:28:49 | But why do you divide by four when you have an addition? You mustn't do that when you have an addition, because |

00:28:56 | otherwise each element of the addition will divide up. There you're again in the case we just showed before, with the case that |

00:29:02 | Herve said where I divide first, then after, you have 15 quarters with 11 quarters. |

00:29:07 | Well, we must do minus 15. |

00:29:08 | For the first one you do minus 15. At the first isolate the X or multiply them by X. |

00:29:14 | There, four X, er, four X. |

00:29:16 | Well you take away four X. |

00:29:18 | But (inaudible). |

00:29:19 | Zero equals, no you find zero equals minus one. Possible, not possible? |

00:29:23 | Impossible. |

00:29:24 | Impossible. S equals? |

00:29:26 | Well, Q something. |

00:29:27 | (inaudible) |

00:29:36 | The slash in the middle. |

00:29:37 | Sir? |

00:29:38 | Sir! After, can one do times 15? |

00:29:39 | So after, when you have the four X alone, you can do times a quarter. |

00:29:43 | Aah! |

00:29:44 | Sir? |

00:29:45 | Yes. |

00:29:45 | But no, Sir! |

00:29:46 | I'm coming, I'm coming. |

00:29:48 | Is it possible that (inaudible)? |

00:29:53 | No. No, because if you do plus five X, you've got X here but five X here also. |

00:29:58 | Ha, yeah. |

00:29:59 | Sir! Is it right, that one? That one also. |

00:30:05 | Yes. That one's okay- and then that one... why is it- what's this six X? Where does it come from, this six X? Well, what I don't understand you've (inaudible) got a |

00:30:13 | solution in X there. You haven't eliminated your X's? You haven't eliminated your X's! |

00:30:17 | But no, but- if I eliminate that one, after that one has minus something- |

00:30:21 | But it doesn't matter... It's impossible to have minus. After, we'll do minus something. |

00:30:29 | Is this correct? |

00:30:33 | No. |

00:30:34 | Why? |

00:30:35 | No. If you take away three X here... So look at one thing. We'll look at this all together, please, |

00:30:42 | Ah! No, no, no. |

00:30:46 | Please, we'll look at this all together. |

00:30:48 | E:00] |

00:30:51 | Shh! Er, Camille get back to your seat please- we'll look at- I would just like to show you something- because there's- there's |

00:30:58 | the idea- it's- we- we will listen to- to what Valentine says, and then we'll try to see what- what doesn't match in her story. So |

00:31:07 | for the one, there's minus four X plus 15 is equal to 11. So Valentine said: I would like to have X on the left. We're okay, I'm going to have X on |

00:31:16 | the left, all right? So she says to herself but I want X. Therefore you go from minus four X to X, I do plus five X, right or wrong? |

00:31:24 | Yes. |

00:31:25 | Yeah, so she says to herself, I'm going to do plus five X, then at the same time you do minus 15, all right? |

00:31:28 | Yeah. |

00:31:29 | So then, I write this. |

00:31:33 | (inaudible) copycat. |

00:31:34 | Who said that, copycat? |

00:31:36 | Me! |

00:31:37 | You. You also want to do that. |

00:31:38 | Well, yeah. |

00:31:39 | Yeah, so, If I do plus five X. I remind you that everything I do here will be on both the left and the right side of the equality. Therefore if I |

00:31:49 | do plus five X, yes Miss, you'll have X here, the 15 will fall, wonderful, equal? But only on the right, the five X, it will |

00:32:01 | reappear. And then you'll have five X plus? |

00:32:06 | Minus four. |

00:32:07 | Minus four, it's all you've gained Miss. Now you have everything- X's. Before you had an advantage, |

00:32:13 | Valentine, you had an advantage, that here you had X's only on one side. Now, you've won so much that you've got X's on both sides... |

00:32:22 | So one shouldn't want to add something to raise (inaudible). One must say to oneself: do I have X's on both sides or not. That's not |

00:32:31 | at all okay, because now you've won nothing. I remind you everything you do, I repeat, I repeat once more, everything you |

00:32:37 | do here will apply itself on both sides of the equality. Therefore if you do plus five, because you only look at that because you've only |

00:32:46 | looked- you've got an idea- you've put blinders there, all right? You know those blinders, one puts on horses, all right? So if you've |

00:32:54 | put blinkers and then you only look at part of your equation; stand back and look at all of the equation. The equation here, when I |

00:33:02 | look at it entirely, as it is- I will erase the one above so we see it well, all right! I've even erased the 15 with that. When |

00:33:11 | you look at this, what happens? You say to yourself ah, yeah, great! You're happy. Why? You've got on only one side with |

00:33:20 | X's. They don't create a problem, they are already on the left. So I leave them on the left. I don't bother about the X's for the time being. What |

00:33:29 | is bothering me here? In a global way, what's bothering me? |

00:33:33 | The 15. |

00:33:34 | It's the 15, it's all. There's only the 15, which is bothering me to isolate the X's. So I do only the minus 15 and I find minus |

00:33:41 | four X is equal to? |

00:33:44 | Minus four. |

00:33:45 | Minus four. |

00:33:46 | Then we do times, times a quarter. |

00:33:47 | Haha. |

00:33:48 | Minus a quarter. |

00:33:49 | Times minus a quarter. |

00:33:50 | Times, so a quarter to take the four away, okay? But what's bothering me also? |

00:33:56 | It's the minus. |

00:33:57 | It's the minus. That one there doesn't bother me. |

00:34:00 | Times minus a quarter. |

00:34:01 | Times minus a quarter. The minus to take away the minus... And then the quarter to take away the four. Then after I have X, then I apply this minus- |

00:34:11 | A quarter times minus a quarter equals minus four. Therefore I find X equals? |

00:34:16 | One. |

00:34:17 | One... okay? One must be careful with this. One must work only with the elements you have in the equation. You mustn't start |

00:34:25 | looking for others which please you. There are already enough in here, you must not look for others. |

00:34:34 | (inaudible) 250. |

00:34:36 | To do calculations, you must take away the brackets. |

00:34:38 | And is that all? |

00:34:39 | Then after you do the same problem. |

00:34:41 | Sir? |

00:34:46 | Yes! |

00:34:47 | Can you come and see? |

00:34:48 | So we will start there, I go round like this. |

00:34:50 | Er, here there we do times 10, that like, after we have no more commas here? |

00:34:54 | Well yeah, and you've found X equals zero two. Exactly. |

00:34:59 | (inaudible) is it correct? |

00:35:02 | I don't know if it's correct but- |

00:35:04 | We do plus (inaudible). |

00:35:05 | If you do minus two X here. If you take away two X and then you've got two. |

00:35:10 | Yeah, but there aren't anymore at all. |

00:35:12 | You've got two pencils. Minus two pencils, how many have you got? |

00:35:15 | No more. |

00:35:16 | So. |

00:35:17 | It does... minus seven X. |

00:35:20 | Equals? |

00:35:21 | Twenty-one. |

00:35:22 | Twenty-one... Then now how are you going to take away this minus X? Now it's good. Now you have a multiple of X alone. So |

00:35:31 | what do you do? You're not going to add this 21 on the same side? |

00:35:36 | After, well, we do divided by seven X. |

00:35:40 | No, not seven X, we want to keep the X. |

00:35:42 | Times (inaudible) a seventh. |

00:35:44 | Times minus to take away this minus, then a seventh to take away this seven. |

00:35:49 | Ha, yeah. |

00:35:51 | Okay? |

00:35:52 | Hmm. |

00:35:53 | And then it will give us what? So here we're going to be able, we'll rub out already (inaudible) with the eraser. |

00:35:58 | Why is it minus two X at the top? |

00:36:01 | But because look at these two X, they are bothering us. These two X there, that's the one we want to take away. |

00:36:05 | Why isn't it the minus seven X? |

00:36:06 | Well because otherwise we must put the 21 on the other side, it's the same thing. As we see that there's only X's already here, we will |

00:36:12 | try and put everything this side... we choose a side where we want to put the X's. |

00:36:16 | Hmm. |

00:36:18 | We can put it the other way round if you wish. |

00:36:20 | But after that, this thing we must also do it at 21? |

00:36:23 | Ha, well of course. I said everything we do behind this vertical bar here, we must do it on both sides. |

00:36:31 | Why does it stay (inaudible), why (inaudible)? |

00:36:32 | So it makes X, but 21 divided by seven. |

00:36:36 | Er, three. |

00:36:37 | Three. Then there was a little minus, therefore it makes minus three. You can rub out what you had here. |

00:36:41 | Ha, therefore X becomes minus three. |

00:36:42 | But yeah, simply. |

00:36:44 | And then that one? |

00:36:47 | Ha. That one is more problematic. You mustn't divide by four. |

00:36:49 | But, Sir. |

00:36:50 | Yes. |

00:36:51 | (inaudible) the minus seven X there? |

00:36:53 | Well, because I did minus two X. |

00:36:55 | Minus five X or minus two X. |

00:36:56 | I- yes. I must still go over there behind. |

00:36:59 | The solution must be the solution which is the least, it must be more. |

00:37:02 | Must be X... Yeah. Yeah. That's why we do as you're doing here. You can do times minus one, but you could very well have done |

00:37:08 | directly minus one. You do times minus one... Then 42. |

00:37:12 | Quarter, you can put twenty-one and a half. |

00:37:14 | Yeah. |

00:37:16 | There was a question here on this side. Who was it? Ha, yeah, Aurore. |

00:37:21 | Er, here (inaudible) we do there we would put minus 21, that doesn't do anything above? |

00:37:25 | No, it would be minus five X minus 21. |

00:37:27 | (inaudible) minus three. |

00:37:30 | Well, yeah. |

00:37:31 | So it still does something? |

00:37:33 | Well, yeah, it does something. |

00:37:35 | (inaudible)? |

00:37:37 | No... Why do you want minus 21? |

00:37:38 | Sir. Is this correct? |

00:37:41 | I had. I had. |

00:37:42 | No you've done times, times a seventh, minus a seventh, so the 21 is affected by the times minus a seventh. |

00:37:46 | Is it correct? |

00:37:47 | Yeah. So I'll do F. I'll put F up. Yes, Herve? |

00:37:52 | (inaudible) |

00:37:53 | Well, precisely, I'll do it. It's wrong there. So I'll come- I'll do F, then after we'll look at G together, because G has something tricky. |

00:38:01 | We'll quickly look at F. So F, it's five X, er, minus five X is equal to two X plus 21. Is that correct? |

00:38:09 | Yeah. |

00:38:10 | Okay, so again here, I've already got on the left only X's... Therefore I'll pass- I'll eliminate the X's on the right. How do I eliminate the X's? |

00:38:22 | Minus two X. |

00:38:23 | I do minus two X. What does it give me? It gives me minus seven X is equal to 21. What's bothering me now? |

00:38:31 | The minus seven. |

00:38:32 | It's the minus seven, it is being multiplied... It is being multiplied. Therefore times minus one. |

00:38:39 | A seventh. |

00:38:40 | A seventh... Which gives us X equals? |

00:38:43 | Minus three. |

00:38:44 | Minus three, simply. So now it's true on the other hand for that part... So there's S equals minus three. |

00:38:57 | It's true for the G part. |

00:39:05 | The G part, what do we have that's tricky? We have four X is equal to four X... minus one. So how are we going to see that? |

00:39:12 | It's not possible! |

00:39:17 | It's impossible. Valentine? |

00:39:19 | Because it would need a place, for example four X is equal to three X minus one. Otherwise we can't take away the X's. We can't take away one. |

00:39:29 | Let's try to take it away, let's try to be as methodical as we can, then let's see what happens. Let's see, where do we find, so- |

00:39:36 | Times four X. |

00:39:38 | You're okay that this four X is bothering you. |

00:39:40 | Yeah. |

00:39:41 | We do minus four X. |

00:39:42 | So we do minus four X. |

00:39:43 | So it does zero equals zero. |

00:39:44 | We find zero X. Therefore zero is equal to minus one. Is that possible? |

00:39:50 | No! |

00:39:51 | No. Therefore when you get to a case like this... |

00:39:54 | B:00] |

00:39:55 | What does this mean graphically? Four X equals four X minus one. If we did it again graphically as we did it here. |

00:40:02 | It's parallel. |

00:40:04 | Yes Sir Jean-Pierre! It's parallel. Do you remember that? If I do the axis of the X's here, the axis of the function G of X and D of X, the |

00:40:14 | function on the left and the function on the right... four X it's what? (inaudible) the origin of four X, I start from where? Four X, the force- the- the- |

00:40:24 | the equation which says: G of R of R, X goes on four X. |

00:40:32 | It starts from zero. |

00:40:35 | It starts from zero. Right. We'll put it in one color. That one there. Here, it starts from zero and each time it advances it goes up how much? |

00:40:43 | Four. |

00:40:44 | Up four. Okay... it goes up like this. |

00:40:53 | And then it goes down also each time. Each time I go backwards, I go down four. Are we okay? |

00:40:57 | Yeah. |

00:40:58 | That's okay? That's the function of G, the left element. The right element... what nice little colors here would |

00:41:07 | distinguish this from blue? We don't have much... a little orange, a green. We'll see the difference but I don't have anymore orange... So the |

00:41:20 | function of the right side. I'll write it here... it says- it says from R to R, X goes on four X minus one. Therefore it means I go from where? |

00:41:33 | Minus one. |

00:41:34 | I go from minus one on- there and then, each time I advance one, I go up four. |

00:41:39 | Well it would do a (inaudible). |

00:41:41 | It is parallel. |

00:41:42 | It is, well, parallel. Therefore when do they meet each other, these two parallel lines? |

00:41:45 | Never. |

00:41:46 | Never. Therefore the- what we're looking for as a solution, it's the points that meet each other. Therefore there's none. Therefore one can say it's |

00:41:53 | it's- Do you visualize geometrically what's happening? Okay? Therefore that case, it's S equals- parallel. Okay? |

00:42:02 | Okay? That's the function of the right side. |

00:42:06 | Sir. |

00:42:08 | Yes. |

00:42:09 | Can I go to the bathroom? |

00:42:11 | You can wait two minutes. The bell will ring in two minutes. Yeah? |

00:42:17 | The first, I didn't understand why (inaudible). |

00:42:21 | What's bothering you? Well then, how do we take away a fraction? |

00:42:24 | Well, we (inaudible) another fraction. |

00:42:26 | Well yeah, which one? To take away the five, you did what? If you had only five X, what would you do? |

00:42:31 | Well, er, times a fifth. |

00:42:33 | A fifth. If you had only the- you would do times? Times two. |

00:42:41 | (inaudible) |

00:42:42 | No, times two. Times two, therefore times two fifths. Oh, I've put green, that's my print. |

00:43:05 | So I'll let you continue now, on this 948 and we'll correct next time the- the- the other ones, all right? We are |

00:43:13 | obliged. We'll correct H, I, K. Yeah. |

00:43:18 | Sir, (inaudible). |

00:43:20 | Hum um...(inaudible) we do. |

00:43:25 | Is it correct? |

00:43:52 | I've got 949. |

00:44:14 | Sir? |

00:44:15 | Yes. |

00:44:16 | Come and see. |

00:44:17 | Of course. |

00:44:21 | (inaudible) minus, er, minus, the X either negative or positive? |

00:44:26 | It had better be positive. It would avoid us- because every time there'll be minuses we'll make a mistake. What's bothering you? |

00:44:32 | Well there, it's to leave it like that. |

00:44:33 | No. Here you've got six X, here you've got five X- you've got X. |

00:44:36 | Yeah. |

00:44:37 | Well, which is the one bothering you? Where are there the most X's? On the left or on the right? So you stay on the left and then you take away. |

00:44:43 | Ah yeah, so, I do minus one. |

00:44:44 | Minus X, yeah, you do minus X. |

00:44:56 | Well, it does (inaudible)... and then after- to do- divide by, er. |

00:44:59 | Not before doing division. As long as you have a polynomial, you don't do division. It's a principle you've got to keep to. |

00:45:07 | Therefore now? |

00:45:08 | We do plus 10. |

00:45:09 | You do plus 10. |

00:45:12 | [ Bell ] |

00:45:14 | Okay then... Okay then, we'll see each other after the break. |