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.

TimeCaption
00:00:12[ Bell ]
00:00:16Okay, then.
00:00:24Okay, 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...
00:00:36Or we can practice one of the theorems of equivalence.
00:00:44Especially this one which, in the first place, is the one which tells us that one must add up an expression. Therefore, page 948.
00:00:56We will start nicely, we will start with Gregoire... With A, X plus five equals eight.
00:01:03Well, we must take away five.
00:01:05We do minus five.
00:01:06We do minus five, therefore X is equal to three.
00:01:07Mmm.
00:01:09X 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.
00:01:19Guillaume.
00:01:20Er, X plus three is equal to minus five. We must take away er... minus three.
00:01:25So we mustn't take away minus three.
00:01:27It makes minus three.
00:01:28So we must add minus three or do minus three.
00:01:30It makes minus three.
00:01:31It makes minus three.
00:01:32And X is equal to eight, to minus eight
00:01:35To minus eight, X equals minus eight. Questions up until now? C? Er... Florence.
00:01:43Er, X minus six equals 30. Therefore we do plus six, which means X equals 36.
00:01:48X equals 36, exactly... Letter D? Katia.
00:01:56Er... well er, the solution is five.
00:02:00Uh 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?
00:02:07Well-
00:02:08We know that in an equation, we must put a little vertical bar, and what must we do?
00:02:15What do we do to make our equation equivalent.
00:02:16I did three times five. The first is 15, then the second is two times five plus five.
00:02:21So there you're groping about.
00:02:23Yeah.
00:02:24You're searching like this- The idea is to do what? It's to avoid all these stories of groping about.
00:02:30When 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?
00:02:41What's the basic idea in all these problems there? Herve?
00:02:44To take away the X's.
00:02:46Take 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.
00:02:55X 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.
00:03:03There's one which says I can add or take away a number... Okay?
00:03:11There's one that says I can add or take away expressions, some literal part, okay?
00:03:18And then there's a third one where we can multiply by a number that is not equal to zero. Are we okay?
00:03:24So here, what do we do? Valentine?
00:03:27We do minus two X.
00:03:29We do minus two X.
00:03:30Well there it stays one X to the three X.
00:03:33On 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-
00:03:38It suppresses the two X and what stays?
00:03:39What stays is X is equal to plus five.
00:03:42It 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)
00:03:52Okay, Katia? Because if you try by doing it by groping about, I know that we did it, er, by successive tests-
00:04:01We tried that, but we saw it was tedious and long. So now we have a new method.
00:04:07Aurore, what can we say for E?
00:04:09We do minus four X.
00:04:11We have five X is equal to minus eight plus four X. Therefore we do minus four X. Which gives us X equal to?
00:04:16Minus eight...
00:04:17X 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.
00:04:34We will come to this side. Yeah, Herve.
00:04:36Well, er, since we want to eliminate the eight, well since we want to find X, we put, we do plus eight X.
00:04:44So if we do plus eight X... What happens if we do plus eight X?
00:04:49Ha, yeah...
00:04:51We 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)
00:04:59It's not want we want? What do we want?
00:05:00Well-
00:05:01There should be one X.
00:05:02So, Camille?
00:05:03Nine X.
00:05:04We 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-
00:05:13It makes X.
00:05:14It makes X is equal to?
00:05:16Seventeen.
00:05:17Seventeen... 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.
00:05:26There it's advantageous, we always have the X's on the left.
00:05:30Maybe 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?
00:05:39Minus (inaudible).
00:05:40Minus?
00:05:40Minus 30.
00:05:41Minus 30?
00:05:42Yeah.
00:05:43Okay... so we can do it all in one stage.
00:05:46No.
00:05:48So one could- but one can do it in two... So what do we want to do first?
00:05:53(inaudible)
00:05:55Minus two.
00:05:56So we want to do minus two X. And then it gives us?
00:05:58(inaudible)
00:05:59X plus five (inaudible).
00:06:00X plus five is equal to?
00:06:03Minus three.
00:06:04Minus three... and what do we want to do now? What's bothering us?
00:06:07Minus five.
00:06:08It's the five which is bothering us, therefore we do...
00:06:10Minus five.
00:06:11Minus five. Therefore X is equal to?
00:06:13Minus 35.
00:06:14Minus 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
00:06:23we write the next one down, we have 18 X. It's H... We have eighteen X, er, minus 24.
00:06:30(inaudible)
00:06:33Plus 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
00:06:4172 plus 17 X... What do you want to do? What must we choose first?
00:06:50Minus 17.
00:06:51So 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
00:06:5817 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.
00:07:06Are we okay? On the left the X's... So on the right we only have the-
00:07:10The numbers.
00:07:11Okay. So what is bothering us on the left now? It's-
00:07:17Minus 24.
00:07:18The minus 24. Therefore what must I do to take away this minus 24?
00:07:21Plus 24.
00:07:22Plus 24. Therefore I can directly write at the same time plus 24... and I do the two things at the same time.
00:07:29What 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...
00:07:37Plus if I add up, instead of adding up monomials, if I add up polynomials, it's the same thing. Since I add up several
00:07:44monomials. Okay? Therefore the 18 X that I'm going to subtract minus 17 X, there's going to stay?
00:07:50X.
00:07:51And then what am I going to do with my 24? Well, the minus 24? I'll be staying with the Y. The seven- the
00:08:04minus 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
00:08:11give me? Ninety-six.
00:08:18And there we don't need a calculator to get to that S is equal to 96. Is there a question on this exercise 948?
00:08:26So I'll let you continue with 949.
00:08:31For those who finish through 950, there's 951, 952, 953,
00:08:40954. There's enough to amuse oneself until 957.
00:09:30Sir?
00:09:31Yes.
00:09:32(inaudible) 949, I don't need to (inaudible).
00:09:38But you've done it right here.
00:09:40But yes, but A, we did with you.
00:09:42Ah! So?
00:09:46And, er, here B.
00:09:47Yeah, you want to isolate- you want to isolate what?
00:09:52Er, I- yeah, but I haven't understood.
00:09:54But 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
00:10:01this. What's bothering you? It's that one there.
00:10:03Yeah, therefore I do minus.
00:10:06No. How do you take away seven? How do you get minus seven to become zero?
00:10:12Er, I do plus- plus eight... er! Plus seven.
00:10:18You do plus seven, hum. You take- so do plus seven. So now what happens on that side there? I do plus seven,
00:10:25two X minus seven plus seven. The two X's will stay there.
00:10:29Two X is equal to seven.
00:10:30And then, no! Minus 19, when there's plus seven it makes minus 12.
00:10:34Uhum!
00:10:35It's correct. You've badly written the things, but your calculation is correct.
00:10:38Yeah.
00:10:39Two X is equal to minus 12. Now you want that only- that only X, therefore you-
00:10:43Times a half.
00:10:44Times a half. You find X equals minus six.
00:10:46Yeah, I did it correctly.
00:10:47It's correct.
00:10:50So there, the same thing. You decide, you have X's only on the left. What's bothering you?
00:10:55The five?
00:10:56The five. Therefore?
00:10:58I do plus minus five.
00:11:01You 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
00:11:07find 20 minus five, it equals, well?
00:11:09Fifteen.
00:11:10Fifteen. And then you must divide by- no, you must divide-there's not 15 which must- you must simply divide by?
00:11:12Seven?
00:11:16Twenty?
00:11:17By?
00:11:18By a quarter.
00:11:19By quarter, therefore we need a quarter. You must find 15 quarters as a solution.
00:11:22Sir?
00:11:23Yes!
00:11:24The thing (inaudible) must it always be on the left?
00:11:27Not necessarily, you can also put it on the right.
00:11:29Always X on the left?
00:11:32Listen, generally people try and have it on the left because we've gotten used to saying X equals. But we'll see, maybe later,
00:11:38er, 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
00:11:46will have minus one X, so we must multiply again by minus one.
00:11:49Sir?
00:11:50The purists always want X on the left. But is it really good? Yes!
00:11:57It's always the same thing, 949?
00:12:01Yeah, 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
00:12:05do 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.
00:12:08To start with, you do minus 12. So what will you be staying with?
00:12:10We inverse.
00:12:16Bah! Three X.
00:12:18Three X.
00:12:19(inaudible) 48.
00:12:20Three X, three X equals?
00:12:21Forty-eight?
00:12:22No...
00:12:22No, er, 36.
00:12:23If you do minus 12-
00:12:24Thirty-six. Ah bother, I shouldn't have taken away the 36, it was there. So equals to 36. And then you can do times?
00:12:32A third.
00:12:33There 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.
00:12:38Ah! Yeah, okay.
00:12:39Hum, hum?
00:12:40Yeah, yeah, it's okay.
00:12:40It's okay.
00:12:41Thank you.
00:12:41You're welcome.
00:12:43Sir?
00:12:44Yes!
00:12:45There's a problem with C.
00:12:47What's the problem with C.
00:12:48I can't- I can't do it, C.
00:12:49Take away five.
00:12:50Yeah.
00:12:51Okay. So what happens when you take away five?
00:12:54Oh well, there also it goes away.
00:12:56Well yeah. So write the thing again. The four X stays.
00:12:58Yeah.
00:12:59Equals? Fifteen. Then now. When four objects are worth 15 francs, what do you do to know the price of an object?
00:13:08Well, I divide by, er, four?
00:13:11Yeah, take a quarter.
00:13:12Ah! Yeah.
00:13:14It needs a quarter, and what happens?
00:13:17(inaudible)
00:13:19X equals fifteen over four.
00:13:21Yeah.
00:13:22The fraction is in numbers.
00:13:26How?
00:13:27No, 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)
00:13:38Yeah, if you like.
00:13:42Yes.
00:13:44There, if we divide by four, the plus five must also be divided by four?
00:13:47Oh well, yeah. Therefore we don't divide by four?
00:13:49No.
00:13:50No?
00:13:51If we take away the five, afterwards we have four X is equal to 15.
00:13:53And then.
00:13:54And then we can.
00:13:55Then it's not divided by four fifteen?
00:13:57Yes, but it does have decimal points.
00:13:59Ho, poor little one, if it does have decimal points, are they so painful, decimal points?
00:14:01No.
00:14:02Ha, good.
00:14:03But, Sir, (inaudible)?
00:14:06So we leave the fraction, we will leave 15 quarters. Exactly.
00:14:09(inaudible)
00:14:11(inaudible) seventy-five (inaudible) the fraction?
00:14:13But it's 15 quarters. You've divided 15 by four, that's 15 quarters.
00:14:17Er, (inaudible).
00:14:20Here 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)
00:14:27Leave 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.
00:14:34You have already done the next one but... we do times one third, then we will find?
00:14:37Third.
00:14:39Uh...
00:14:40X is equal to?
00:14:42Twelve.
00:14:43Twelve.
00:14:44Humm.
00:14:45Humm. But what do you do here?
00:14:48Er, plus seven.
00:14:49Plus seven.
00:14:52What are you going to find?
00:14:54Er, 12 X is equal to seven.
00:14:55Well no... Minus 19 plus seven? Minus?
00:15:00Well, uh...
00:15:04Minus 19 plus seven?
00:15:06Yeah, we must add that... yeah, well 26.
00:15:10Tss-Tss.
00:15:11Er, 25.
00:15:12Tss-Tss.
00:15:14Minus 19 plus seven?
00:15:18You've got 19 objects missing, someone gives you back seven, how many do you have missing?
00:15:22Well, er, 12.
00:15:25So.
00:15:27Then after that, what do we do?
00:15:29Times one.
00:15:34You want to take away the two?
00:15:35Yeah, well times a second.
00:15:36A half.
00:15:37Yeah, a half.
00:15:38Which will give, we take away the X. X equals.
00:15:41Equals, is equal to...
00:15:45Mental calculation.
00:15:46Yes I know.
00:15:47Twelve divided by two... the minus, it stays.
00:15:49Minus six.
00:15:50There now.
00:15:51(inaudible)
00:15:53Therefore here you must still put X is equal to 12 and here you must put X is equal to minus six.
00:15:57Minus six. (inaudible) Thank you.
00:16:00Sir, Sir!
00:16:06What is it?
00:16:08How do we go about for the two? Because I can do it, but in two stages.
00:16:11Well yeah, and what? We never said one should do only one stage.
00:16:15(inaudible) a half?
00:16:17Well 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
00:16:25a minimalist, either.
00:16:29When we put there, er, we always first try to take away the part, er, on the right?
00:16:38Yeah 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
00:16:45be interesting, or for E, to put it on the right for example. To avoid the minuses, but one can do it in both cases.
00:16:49Sir!
00:16:50Yes.
00:16:51There now I've found for half of X, I will multiply by two to find X.
00:16:55Why? I'm not okay with what you've done there. You've done a multiplication and a subtraction at the same time.
00:17:01Well, you said it was (inaudible).
00:17:02No, don't do it at the same time. One stage after the other. Excuse me, Gregoire... I think that, let's stop for awhile,
00:17:09please. We look at A, we did it together as an example. Do you remember? And B also? It astonishes me,
00:17:20though, because a lot of you did it wrong.
00:17:24So 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.
00:17:37Here. You have one X which is multiplied by two. You want X on it's own.
00:17:41That's the least difficult.
00:17:48Thank you.
00:17:51Here 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
00:17:59bar, 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
00:18:07calculations. 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
00:18:11to work with the left part and the right part separately. Here, first thing, isolate everything unknown on the same side.
00:18:22Therefore, what's bothering you here?
00:18:25Minus seven.
00:18:26It's the minus seven. Therefore we do one stage with the minus seven... We make it plus seven.
00:18:32And then we find two X is equal to?
00:18:35Minus 12.
00:18:36Minus 12. And now we do one stage where we do times a-
00:18:40Half.
00:18:41Half... 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
00:18:50exactly 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,
00:19:02and at the same time I do minus five... It's what you did, it's that one?
00:19:08Yeah, at D.
00:19:09Ah 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.
00:19:18It's on the left, so what's bothering us?
00:19:20The minus five.
00:19:21The 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.
00:19:32And 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.
00:19:39I remind you it's better as a fraction, because if by any chance it wasn't a fraction which gives a rational number which
00:19:47finishes itself, it's not interesting... Okay? A number with decimal points which finish. If we have periodical numbers it's never very interesting, we
00:19:54can'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.
00:20:03When 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)
00:20:13Two X minus 16?
00:20:14Thirteen.
00:20:15Thirteen. Thank you. When we have this, it's true one would want to say: I've got multiples of X on both sides. Valentine?
00:20:22So we put the bar and we do minus, er, minus two X.
00:20:26Times a half.
00:20:27So why do times a half?
00:20:29To do a (inaudible).
00:20:30So 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
00:20:35complicate a problem which wasn't too complicated to start with.
00:20:38We can (inaudible).
00:20:39How to complicate it when one can do it simply... that's Herve's proposition. Look, I do times a half, I am-
00:20:46Okay, 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
00:20:50the ideal. If I do times a half, how many X's do I have here?
00:20:52Two.
00:20:53Two X.
00:20:54No, half of X.
00:20:55No.
00:20:56No. I've got an idea. Times half X.
00:20:58You 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
00:21:06transparency what happens? Do you remember? Suddenly it has curves. We have raised one solution. Therefore we only have three possibilities
00:21:13to do something... All numbers, add letters, or multiply by a number which is not equal to zero... We can't
00:21:21multiply 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
00:21:29doesn'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
00:21:37both sides... And what's more I've gained fractions. And since we all like working with fractions, we want to have them in
00:21:45the 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
00:21:53not comfortable... After, you don't know anymore how to add fractions and then you get into trouble. Why get into trouble?
00:22:02All right? On one's sheet of paper when one had something one could do easily. So do we continue doing times a half here-
00:22:07Herve? Do you insist?
00:22:09No.
00:22:10No. Thank you.
00:22:12So let's try to start like before, using times. To use times we need more polynomials. That one only has monomials on
00:22:22the left or the right. Okay? We isolate. Therefore for the first we shall say what?
00:22:27We can do (inaudible)?
00:22:33So 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
00:22:42minus two X, we've decided the X's will be on the left.
00:22:44Nine.
00:22:46We can do both at the same time.
00:22:47We can put them one after the other.
00:22:48So there one can put them one after the other because it's two additions.
00:22:50All right.
00:22:51Are we obliged to do them both at the same time?
00:22:52No, 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
00:22:59do 15 in 45 minutes, er, if you have two stages you will lose a little time. But that's training.
00:23:07(inaudible)
00:23:09It'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...
00:23:18because minus two X minus five.
00:23:20Minus one.
00:23:21Minus?
00:23:22Minus (inaudible).
00:23:23No, minus 13.
00:23:25Minus 18.
00:23:26But how much is it when one does minus 13 minus five?
00:23:27Minus 18.
00:23:28Well, minus 13 minus five, you've got 13 objects missing, someone asks you for five more, you've got 18 missing.
00:23:32Ha, yeah.
00:23:33(inaudible) divided by two.
00:23:34It seems to me, I don't know... And then after?
00:23:37Divided by a half.
00:23:39No.
00:23:40Not divided, times a half, yeah.
00:23:41There now, either we do times a half or we divide by two, but we can't... times a half. X is equal to?
00:23:48Minus nine.
00:23:49Minus nine... Okay. The S is equal to minus nine.
00:23:57Is there a question on this? Can we continue now? Let's advance.
00:24:04Yes?
00:24:06Er, there how do we do it?
00:24:08So everywhere there are brackets, to start with we take away the brackets.
00:24:11(inaudible)
00:24:13No.
00:24:14It's wrong?
00:24:15It's wrong.
00:24:17Ha, yeah, nine is wrong.
00:24:21That is wrong.
00:24:25The X cube is wrong. Ha, it was three X actually.
00:24:27Yeah.
00:24:28Ha, yeah.
00:24:30It makes- it makes seven.
00:24:32Me, 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.
00:24:40Yeah, okay.
00:24:41You do here the thing like this, you can say I'm doing literal calculation.
00:24:45Yeah.
00:24:47And then I find four X minus five is equal to six... 27... Then now I can work as I did before...
00:24:51Six X over 27,
00:24:56but now if we do minus six X to pass it on this side (inaudible) to minus two.
00:24:59It's not interesting. We have minus, yeah.
00:25:02Well, then we do minus four X and we pass (inaudible).
00:25:03There then, so we decide to have the X's on the right. It doesn't matter.
00:25:08Ah yeah, but-
00:25:09If X is equal to three or three is equal to X? What changes for you?
00:25:11Nothing.
00:25:13Okay. 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,
00:25:19in what we'll want to do later on to find problems, resolve problems, it doesn't take us very far.
00:25:29And after we do it like that.
00:25:31No you can't... After what must you do?
00:25:36On that.
00:25:38Minus 27. Okay. And what will we get?
00:25:41Well it's minus... 32.
00:25:43Minus 32 is equal to?
00:25:46(inaudible)
00:25:47Therefore-
00:25:49Times a half.
00:25:50Times a half. And you find X?
00:25:53(inaudible)
00:25:54Yeah, 32 over two. Thirty-two over two.
00:25:59Therefore there it makes minus 16.
00:26:00Minus 16 is equal to X.
00:26:03Ha, yeah.
00:26:04S equals minus five. Anyway, one writes S equals, so whether one has X equals minus 16 or minus 16 equals X.
00:26:11Yeah, it doesn't matter.
00:26:12The element on the left is equal to the one on the right. Whether X is on the left or the right. All right?
00:26:17Yeah.
00:26:18Okay?
00:26:18Sir? Sir?
00:26:19One can't have zero X.
00:26:22No, it would be better not to have zero X, because then we're not well.
00:26:24But because there it gives us zero X if we do minus 17 X.
00:26:27Well, yeah.
00:26:28It takes away each time, er.
00:26:29Therefore 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-
00:26:31Ha yeah, it makes (inaudible).
00:26:35the correct data. That one, yeah, that's going to become interesting. What do you find? You do minus 17 X.
00:26:41I do minus 17 only.
00:26:42No 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.
00:26:47Therefore you find two here.
00:26:49Yeah.
00:26:50So write two. Two equals minus two... So now.
00:26:55(inaudible)
00:26:57You can't do.
00:26:58But it's wrong, but it's not possible.
00:27:00(inaudible)
00:27:01So it's not wrong, it's that this equation doesn't have a solution.
00:27:05Wah! (inaudible).
00:27:06Is there some X which will permit you to say: whatever X you chose, you will have two equals minus two? There's something which
00:27:12doesn't work, therefore you're going to say X equals the whole?
00:27:14Empty.
00:27:15To the empty set, right?
00:27:17(inaudible)
00:27:18Yes, but that is the set containing the empty set, therefore it's no longer empty.
00:27:22Ha ha, what does that mean?
00:27:24It means (inaudible).
00:27:25That's all. The word there, that symbol means empty set.
00:27:29So when there's... when there's, when it's impossible I must put only that?
00:27:31Sir!
00:27:32Sir!
00:27:33Sir!
00:27:34That's right, yeah.
00:27:35E I understand very well.
00:27:37No, why do you do minus five X? Is there anywhere here where there's a minus five X?
00:27:43Well no, but to have X alone.
00:27:45We didn't say one must have X alone there. We must first take away that X there. Is it X which is bothering you?
00:27:50Yeah.
00:27:51I'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?
00:27:57I do minus X.
00:27:58You do minus X. So now do only minus X.
00:28:00E:00]
00:28:02Yes?
00:28:03Is it possible that equations don't work like that one?
00:28:07Where are you there?
00:28:08At (inaudible). It's not possible? If four X is equal to four X.
00:28:13No, it's not possible.
00:28:14So what do we put?
00:28:16Well, look at what happens. Do something.
00:28:18Well, yeah, but-
00:28:19You must do minus four X.
00:28:20Yeah.
00:28:21So do minus four X, then look at what happens. I'll come back.
00:28:23Sir!
00:28:24Yes.
00:28:25Er, here what do we do?
00:28:28For E... Well minus four X plus 15, well what. You do minus 15.
00:28:34Yeah (inaudible).
00:28:35I did minus four X... Your 15 falls, then minus 15 plus two... That doesn't bother me. Eleven minus 15?
00:28:43And for E? Is it correct here, if I put this?
00:28:47(inaudible) the minus four, the over four it's also here?
00:28:49But why do you divide by four when you have an addition? You mustn't do that when you have an addition, because
00:28:56otherwise each element of the addition will divide up. There you're again in the case we just showed before, with the case that
00:29:02Herve said where I divide first, then after, you have 15 quarters with 11 quarters.
00:29:07Well, we must do minus 15.
00:29:08For the first one you do minus 15. At the first isolate the X or multiply them by X.
00:29:14There, four X, er, four X.
00:29:16Well you take away four X.
00:29:18But (inaudible).
00:29:19Zero equals, no you find zero equals minus one. Possible, not possible?
00:29:23Impossible.
00:29:24Impossible. S equals?
00:29:26Well, Q something.
00:29:27(inaudible)
00:29:36The slash in the middle.
00:29:37Sir?
00:29:38Sir! After, can one do times 15?
00:29:39So after, when you have the four X alone, you can do times a quarter.
00:29:43Aah!
00:29:44Sir?
00:29:45Yes.
00:29:45But no, Sir!
00:29:46I'm coming, I'm coming.
00:29:48Is it possible that (inaudible)?
00:29:53No. No, because if you do plus five X, you've got X here but five X here also.
00:29:58Ha, yeah.
00:29:59Sir! Is it right, that one? That one also.
00:30:05Yes. 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
00:30:13solution in X there. You haven't eliminated your X's? You haven't eliminated your X's!
00:30:17But no, but- if I eliminate that one, after that one has minus something-
00:30:21But it doesn't matter... It's impossible to have minus. After, we'll do minus something.
00:30:29Is this correct?
00:30:33No.
00:30:34Why?
00:30:35No. If you take away three X here... So look at one thing. We'll look at this all together, please,
00:30:42Ah! No, no, no.
00:30:46Please, we'll look at this all together.
00:30:48E:00]
00:30:51Shh! 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
00:30:58the 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
00:31:07for 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
00:31:16the 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?
00:31:24Yes.
00:31:25Yeah, so she says to herself, I'm going to do plus five X, then at the same time you do minus 15, all right?
00:31:28Yeah.
00:31:29So then, I write this.
00:31:33(inaudible) copycat.
00:31:34Who said that, copycat?
00:31:36Me!
00:31:37You. You also want to do that.
00:31:38Well, yeah.
00:31:39Yeah, 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
00:31:49do 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
00:32:01reappear. And then you'll have five X plus?
00:32:06Minus four.
00:32:07Minus four, it's all you've gained Miss. Now you have everything- X's. Before you had an advantage,
00:32:13Valentine, 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...
00:32:22So 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
00:32:31at all okay, because now you've won nothing. I remind you everything you do, I repeat, I repeat once more, everything you
00:32:37do 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
00:32:46looked- 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
00:32:54put 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
00:33:02look 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
00:33:11you look at this, what happens? You say to yourself ah, yeah, great! You're happy. Why? You've got on only one side with
00:33:20X'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
00:33:29is bothering me here? In a global way, what's bothering me?
00:33:33The 15.
00:33:34It'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
00:33:41four X is equal to?
00:33:44Minus four.
00:33:45Minus four.
00:33:46Then we do times, times a quarter.
00:33:47Haha.
00:33:48Minus a quarter.
00:33:49Times minus a quarter.
00:33:50Times, so a quarter to take the four away, okay? But what's bothering me also?
00:33:56It's the minus.
00:33:57It's the minus. That one there doesn't bother me.
00:34:00Times minus a quarter.
00:34:01Times 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-
00:34:11A quarter times minus a quarter equals minus four. Therefore I find X equals?
00:34:16One.
00:34:17One... okay? One must be careful with this. One must work only with the elements you have in the equation. You mustn't start
00:34:25looking for others which please you. There are already enough in here, you must not look for others.
00:34:34(inaudible) 250.
00:34:36To do calculations, you must take away the brackets.
00:34:38And is that all?
00:34:39Then after you do the same problem.
00:34:41Sir?
00:34:46Yes!
00:34:47Can you come and see?
00:34:48So we will start there, I go round like this.
00:34:50Er, here there we do times 10, that like, after we have no more commas here?
00:34:54Well yeah, and you've found X equals zero two. Exactly.
00:34:59(inaudible) is it correct?
00:35:02I don't know if it's correct but-
00:35:04We do plus (inaudible).
00:35:05If you do minus two X here. If you take away two X and then you've got two.
00:35:10Yeah, but there aren't anymore at all.
00:35:12You've got two pencils. Minus two pencils, how many have you got?
00:35:15No more.
00:35:16So.
00:35:17It does... minus seven X.
00:35:20Equals?
00:35:21Twenty-one.
00:35:22Twenty-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
00:35:31what do you do? You're not going to add this 21 on the same side?
00:35:36After, well, we do divided by seven X.
00:35:40No, not seven X, we want to keep the X.
00:35:42Times (inaudible) a seventh.
00:35:44Times minus to take away this minus, then a seventh to take away this seven.
00:35:49Ha, yeah.
00:35:51Okay?
00:35:52Hmm.
00:35:53And then it will give us what? So here we're going to be able, we'll rub out already (inaudible) with the eraser.
00:35:58Why is it minus two X at the top?
00:36:01But because look at these two X, they are bothering us. These two X there, that's the one we want to take away.
00:36:05Why isn't it the minus seven X?
00:36:06Well 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
00:36:12try and put everything this side... we choose a side where we want to put the X's.
00:36:16Hmm.
00:36:18We can put it the other way round if you wish.
00:36:20But after that, this thing we must also do it at 21?
00:36:23Ha, well of course. I said everything we do behind this vertical bar here, we must do it on both sides.
00:36:31Why does it stay (inaudible), why (inaudible)?
00:36:32So it makes X, but 21 divided by seven.
00:36:36Er, three.
00:36:37Three. Then there was a little minus, therefore it makes minus three. You can rub out what you had here.
00:36:41Ha, therefore X becomes minus three.
00:36:42But yeah, simply.
00:36:44And then that one?
00:36:47Ha. That one is more problematic. You mustn't divide by four.
00:36:49But, Sir.
00:36:50Yes.
00:36:51(inaudible) the minus seven X there?
00:36:53Well, because I did minus two X.
00:36:55Minus five X or minus two X.
00:36:56I- yes. I must still go over there behind.
00:36:59The solution must be the solution which is the least, it must be more.
00:37:02Must 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
00:37:08directly minus one. You do times minus one... Then 42.
00:37:12Quarter, you can put twenty-one and a half.
00:37:14Yeah.
00:37:16There was a question here on this side. Who was it? Ha, yeah, Aurore.
00:37:21Er, here (inaudible) we do there we would put minus 21, that doesn't do anything above?
00:37:25No, it would be minus five X minus 21.
00:37:27(inaudible) minus three.
00:37:30Well, yeah.
00:37:31So it still does something?
00:37:33Well, yeah, it does something.
00:37:35(inaudible)?
00:37:37No... Why do you want minus 21?
00:37:38Sir. Is this correct?
00:37:42No you've done times, times a seventh, minus a seventh, so the 21 is affected by the times minus a seventh.
00:37:46Is it correct?
00:37:47Yeah. So I'll do F. I'll put F up. Yes, Herve?
00:37:52(inaudible)
00:37:53Well, 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.
00:38:01We'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?
00:38:09Yeah.
00:38:10Okay, 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?
00:38:22Minus two X.
00:38:23I do minus two X. What does it give me? It gives me minus seven X is equal to 21. What's bothering me now?
00:38:31The minus seven.
00:38:32It's the minus seven, it is being multiplied... It is being multiplied. Therefore times minus one.
00:38:39A seventh.
00:38:40A seventh... Which gives us X equals?
00:38:43Minus three.
00:38:44Minus three, simply. So now it's true on the other hand for that part... So there's S equals minus three.
00:38:57It's true for the G part.
00:39:05The 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?
00:39:12It's not possible!
00:39:17It's impossible. Valentine?
00:39:19Because 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.
00:39:29Let'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-
00:39:36Times four X.
00:39:38You're okay that this four X is bothering you.
00:39:40Yeah.
00:39:41We do minus four X.
00:39:42So we do minus four X.
00:39:43So it does zero equals zero.
00:39:44We find zero X. Therefore zero is equal to minus one. Is that possible?
00:39:50No!
00:39:51No. Therefore when you get to a case like this...
00:39:54B:00]
00:39:55What does this mean graphically? Four X equals four X minus one. If we did it again graphically as we did it here.
00:40:02It's parallel.
00:40:04Yes 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
00:40:14function 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-
00:40:24the equation which says: G of R of R, X goes on four X.
00:40:32It starts from zero.
00:40:35It 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?
00:40:43Four.
00:40:44Up four. Okay... it goes up like this.
00:40:53And then it goes down also each time. Each time I go backwards, I go down four. Are we okay?
00:40:57Yeah.
00:40:58That's okay? That's the function of G, the left element. The right element... what nice little colors here would
00:41:07distinguish 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
00:41:20function 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?
00:41:33Minus one.
00:41:34I go from minus one on- there and then, each time I advance one, I go up four.
00:41:39Well it would do a (inaudible).
00:41:41It is parallel.
00:41:42It is, well, parallel. Therefore when do they meet each other, these two parallel lines?
00:41:45Never.
00:41:46Never. 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
00:41:53it's- Do you visualize geometrically what's happening? Okay? Therefore that case, it's S equals- parallel. Okay?
00:42:02Okay? That's the function of the right side.
00:42:06Sir.
00:42:08Yes.
00:42:09Can I go to the bathroom?
00:42:11You can wait two minutes. The bell will ring in two minutes. Yeah?
00:42:17The first, I didn't understand why (inaudible).
00:42:21What's bothering you? Well then, how do we take away a fraction?
00:42:24Well, we (inaudible) another fraction.
00:42:26Well yeah, which one? To take away the five, you did what? If you had only five X, what would you do?
00:42:31Well, er, times a fifth.
00:42:33A fifth. If you had only the- you would do times? Times two.
00:42:41(inaudible)
00:42:42No, times two. Times two, therefore times two fifths. Oh, I've put green, that's my print.
00:43:05So I'll let you continue now, on this 948 and we'll correct next time the- the- the other ones, all right? We are
00:43:13obliged. We'll correct H, I, K. Yeah.
00:43:18Sir, (inaudible).
00:43:20Hum um...(inaudible) we do.
00:43:25Is it correct?
00:43:52I've got 949.
00:44:14Sir?
00:44:15Yes.
00:44:16Come and see.
00:44:17Of course.
00:44:21(inaudible) minus, er, minus, the X either negative or positive?
00:44:26It had better be positive. It would avoid us- because every time there'll be minuses we'll make a mistake. What's bothering you?
00:44:32Well there, it's to leave it like that.
00:44:33No. Here you've got six X, here you've got five X- you've got X.
00:44:36Yeah.
00:44:37Well, 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.
00:44:43Ah yeah, so, I do minus one.
00:44:44Minus X, yeah, you do minus X.
00:44:56Well, it does (inaudible)... and then after- to do- divide by, er.
00:44:59Not 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.
00:45:07Therefore now?
00:45:08We do plus 10.
00:45:09You do plus 10.
00:45:12[ Bell ]
00:45:14Okay then... Okay then, we'll see each other after the break.