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Intro

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La notion d'équation

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La résolution d'équations

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[Music]

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hello in this video I suggest you

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watch the entire course on the chapter

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of equations the subject of this sequence

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and to explain and remind you of the

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most important elements of this

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chapter more precisely I

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will present the concept to you of equations the

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principles of solving an equation

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and we will see the particular case of

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the equation produced then to prepare for

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a test or even an exam you

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will also need training on

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exercises this is especially important for this chapter

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for the resolution of equations I

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therefore advise you to click on the link

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which will take you to other videos

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offering numerous exercises on the

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theme of equations we can begin

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so let's start by addressing the notion

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of equations and already the question that we

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can ask ourselves ask c but what is an

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equation

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well you hri answer we should already

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introduce a little bit of vocabulary

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and in particular the notion of unknowns

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when do we need one of an

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unknown what is an

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unknown what is an

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unknown number in an equation to

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understand it we will start from a small

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problem of a small

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geometry problem very simple problem which starts

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from a rectangle and in my rectangle I

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know one of the dimensions this one which

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is 5 then 5 5 cm 5 km if it is a

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land under five meters finally it doesn't

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matter but I don't know this

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dimension I don't know it not and we're

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going to see it I'd like to determine it 7 7 lengths

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so what we're going to do is we're going to

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give it a name and seven lengths a

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good I'm going to call it x why not

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j 'could have called it her or

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something else, it doesn't matter, it's a choice, this

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length is an unknown length, we

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agree,

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the question comes up, the question is

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what is the length x of this rectangle

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so that its perimeter is equal to 30

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but then if the perimeter is equal to 30

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I therefore have a way of determining the

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length x but how to do it well the

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technique consists of putting the problem

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into an equation but I still haven't said

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what it is was just an equation but we

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'll see right away the perimeter equals 30 then 30

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is the perimeter will I have a

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way of expressing the perimeter in

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another way than saying it's 30

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yes I can I can using the

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data that I noted at the start on my

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rectangle which are x and 5

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but if that makes x and that makes five

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games can therefore calculate the perimeter

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according to x what will

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the perimeter be c 'is therefore the length of the

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circumference of the figure x + 5 + 6 + 5

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I will write it x + 5 + 6 + 5

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that is the perimeter of my rectangle

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no I said earlier that the

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perimeter c30 also in other words the

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perimeter is indeed equal to the

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logical perimeter but here I am

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writing an equality where I have the

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left member which is different from the right member

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on one side I say that the perimeter is

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equal to x + 5 + 6 + 5

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because I know a technique to

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calculate the perimeter and on the other hand

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I say that the perimeter is equal to 30

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because that is

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the one stated which tells us in this

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way I created an equality with

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2 in an unknown number

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and well here we have just defined what

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an unknown is an unknown not

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it is quite simply a number whose

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value we do not know but we are trying to

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determine it if it is possible

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sometimes it's not even possible but then

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what is an equation is

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an equation we have one in front of our

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eyes here is an equation equation

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what is equal equality

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it is an equality which contains an unknown number

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so I say an unknown number but we

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will see later finally when I say

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later it is in the years which will

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follow that an equation can also

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contain several unknowns it is not

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necessarily one here it is a single

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unknown well we have there our first finally

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perhaps not for you

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our first equation then a

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salah equation

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particularity of being able to be written in

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different ways for example this one

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x + 5 + 6 + 5 equal to 30

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it is not well described like that we

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could reduce all that because here

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I can group 5 and 5 which make ten

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games by the c5 +57 idiot and x + 6 I

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was talking about cxp 6 no longer x + 6 made 2 x

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x so from x in other words our equation

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can be writing 2x plus 10 equal to 30

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we say that writing x + 5 + 6 + 5 equal to

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37 equivalent to aytré 2x plus 10 equal to

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30 so here is our equation written in two

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different ways

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so be careful an equation is

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necessarily an equality that for example 2x

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plus simply say it's not an

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equation it's an expression which

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contains an unknown number but it's

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not an equality it's not an equation

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an equation there is necessarily an

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equal sign in its writing let's continue we

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therefore know what an unknown is

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we know what an equation is

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so what is it to solve

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an equation the resolution is when

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we manage to find the solution of our

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problem is to solve an equation

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found the value of x found the

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value of x so that 2 x + 10 is equal

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to 30 or so that x + 5 + 6 + 5 is equal

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to 30 and that's all work that

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needs to be done in this chapter

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and that is why it is very

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important to train and do

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exercises because it is the most difficult part

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when it comes to solving equations so

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this equation here I I'm not going to show you

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now how we solve it, it's

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not at all the purpose of this first

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paragraph

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but maybe you see the solution,

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it's quite easy to guess but in

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any case it will allow us

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to move on to what is

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the solution of an equation

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so I start from my equation 2x plus

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10 equal to 30 and I said we are not going to

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solve the equation

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so I am not going to show you here the

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technique to find x but on the other hand

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here it is

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easy to guess the value of x

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the value that works and that of the

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value of the x we want to think that

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these

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when I replace

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its by ten in my equation I get

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something true let's check

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twice more replace 10 + 10 2 times 10 22

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+10 30

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indeed I find the value

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of 30 so we can say that x equals 10

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and solutions of our equation well the

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solution of the equation

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is quite simply the number hidden

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under the known one that is to say the value

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of the unknown and that's it I repeat

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all the work of solving

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the the equation was found here

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the solution of my equation we will see

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a second example a little more complex

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where we are asked to check if a

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number and solution of an equation is

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therefore we would like to know if 5

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therefore if x equal to 1 .5 and solution of

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the equation 3x minus equals 4x plus three

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then we see there that we again have an

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equation an equality with what with

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inside a number which is unknown according to

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this unknown number there appears twice

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in this equation

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which fact that it is much

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harder to guess the solution to this

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equation unlike earlier we

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were able to guess that these ten here

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to solve this equation

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you would really have to have a real

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technique for solving equations

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that we don't have it yet but in any case we

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could still check 6.5 and

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solutions say that there we are given

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the solution that fell from the sky

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and we are told yes or no these solutions

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or not and well for that we just have to

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replace five in these two expressions

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and see if we actually have

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legality well let's go that's what we're going to

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do I first take the expression 3x

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-1 and I replace I therefore replace x by

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5 that's three times 5 - 1 then three

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times 5 15 - 14 in the left member

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gets 14 and we are going to do the same with the

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right member so I also replace

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by 5 to check if these solutions

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therefore give us 4 x 5 + 3g replace x

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with 5 4 times 5 20 and 3.23 finally

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I obtain 14 on the left side and

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23 on the right side we can clearly see

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that we do not have the equality we wanted

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to have is therefore there no we n 'has not

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the equality conclusion is good 5 is

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not solution of our equation 5 is

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not solution of the equation they said that 5

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does not verify the equation

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we must find a number which verifies

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this equation it is to say that when

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I replace it works well

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I obtain equality here this is not the

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case it is not a solution so for

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these on the other hand I am not going to give you the

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solution will perhaps only at the end of

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this sequence which will arrive on its own and

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I will still give the solution at the

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end of the sequence

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so continue with the resolution

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of equations so what I am going to

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explain here is more of a concept but

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I am not going to fully develop the

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technique of solving equations for

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those really I invite you to

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watch the videos which are in

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the red playlist explains in detail

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all the solving techniques but

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here we are just going to stay on the

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notion of resolution to understand what

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it is solve an equation

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then for that but we are going to start from an

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equation it is what is simpler and

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I am talking about the equation 2x minus three

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equal to 5

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we therefore have there an equality equation with

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in it therefore two expressions and we see

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that in the left hand side I find

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my unknown

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say that the final goal

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is to have x equal to a number at the end

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so I don't know what this number is if

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I knew it well I would have solved my

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equation of course but that's

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the objective at fav illegal 2.5 yes you

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know really hungry is to find the

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value of x and that you must always

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have in mind when you want to

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solve an equation and the whole

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principle of solving equations is

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found here it is this passage which is

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complex

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the passage from my initial equation to

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the solution so when we look

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more closely at the solution here we see that we

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have x alone the equal symbol and the

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number it is important to understand

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how the solution x

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equals a number there are more x and the

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gx alone in other words network of an

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equation this will amount to isolating x in

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the equation

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modified when writing this equation we

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saw it earlier we can modify

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the writing of an equation mellal has

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modified in a very strong structural way

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so that at the end gx alone and

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when we solve an equation this is what we

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must always have in mind

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managed to isolate x well sure to

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get there we have resolution techniques

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if these techniques that I

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said I will not explain parents details

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but still I will recall the

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fundamentals on this equation is therefore

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I do not develop much the

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resolution technique if that -this

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is a problem for you, I strongly invite you

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to watch all the videos that

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are in the playlist, from the

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simplest it goes to the most complicated

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so let's tackle this equation 2x

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minus three equal to 5

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so we remember that the objective is

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to arrive at x equal to a number therefore isolated

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x

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the problem is that in the state we

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see that x is really not alone

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it is accompanied here by a small 2

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there is one minus there is 1 3 if we want to

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isolate

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get rid

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of the 2 right away the answer is no

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so I'm going a little quickly not so on that

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quite simply because 2 x x and a

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multiplication is a priority we ca

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n't get rid of it like that so what we

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could start by doing here it is

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to get rid of this 3 which is there

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so how to get rid of this 3

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well the technique will consist of

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adding 3 I note it here to

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remind me on the left and right of my

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equation why because if I have an

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equality and I add three to the left

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and 3 to the right and well we keep the equality

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imagine you are the same height as

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someone you climb on a very small

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very small promontory which is 3 cm

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well everyone is on the

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promontory 2.3 the other also what's

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happening you stay with the same

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low waist is exactly what I

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'm doing here

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I add +3 to the left and +3 to the right

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what does that do then that makes 2 x - 3 + 3 and

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I do the same on the right equal to 5 + 3

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but why I did that well I

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remember it because I want to

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get rid of those - 3 and to

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get rid of a -3 it just

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add are opposites that is to say + 3

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we can see it clearly here the minus 3 and the

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plus 3 merge that makes zero I

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simply have x left on the

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left side what does it merge for me stay on

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the right hand side bat there I do them

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q5 +38 we still have so our equation

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is still the same equation

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only we wrote it

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differently

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writing that is equivalent to writing that

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I just added three on each side

00:14:41

is equivalent to writing that juju

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simplify the expressions and there we

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slowly notice I am in the process

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of isolating x we continue how to

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now get rid of the

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little 2 which is there and well it is a

00:14:54

multiplication by 2 to get rid

00:14:57

of a multiplication by 2 what

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do we do well we divide by two

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so I'm going to divide by two on both sides

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that gives credence so dividing by two is

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like adding a denominator of 2.1

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we agree so that makes

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x / 2 equal

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8 / 2

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we agree that if we have two

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objects of the same size

00:15:30

if I cut them in two each half to

00:15:32

the same size well that's what I

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did here g2x equal to 8

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I divide 2x by 2 e division by 2 beat

00:15:38

I keep the equality but what is

00:15:41

the point of doing that well we said

00:15:42

it it's that when we multiply by two

00:15:44

then divide by 2 behind

00:15:45

what happens and well all that is

00:15:47

eliminated we are simply left with x x

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equals 8 over 2 in other words x equal to 4

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and well there I I managed to isolate

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equation which is x equal

00:16:10

to 4 and if we really want to be

00:16:12

convinced of this we can check that these

00:16:15

are just for that you just have to

00:16:18

take this solution value and

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inject it into the starting equation and

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look that this solution verifies

00:16:29

the initial equation then normally

00:16:30

yes unless we are wrong we are not going to

00:16:32

write it we will just do it like that

00:16:34

and sega the card I therefore replace its

00:16:36

with four that makes me twice 4 -

00:16:39

3 then twice 4 8 8 - 3 5

00:16:42

it works well indeed x equal to 4

00:16:46

checks the starting equation x equal to 4

00:16:49

and solutions then it remains to treat the

00:16:53

case of the equation produced already we must

00:16:55

recognize what is a

00:16:56

product equation is a product equation it is

00:16:59

an equation formed of a product but

00:17:02

strong but structurally of a product

00:17:05

so the expression on the left we will

00:17:08

see is a product for example x -5 2 x

00:17:15

+ 3 equal to zero is a

00:17:19

products equation as we recognize it

00:17:23

then we do not recognize it what I

00:17:25

say them on the left

00:17:26

I therefore have factors necessarily a

00:17:28

product that is to say that that is not

00:17:32

a products equation that is a sum

00:17:35

of 6 months 5 + 2 x + 3 it's a sum

00:17:38

no I want to

00:17:40

product so on the left I really have two

00:17:44

factors it can be two big factors

00:17:45

but it's necessarily factors on the

00:17:48

right I necessarily have equal to zero

00:17:52

we will see why not something else

00:17:54

if I have here ips - five factors of 2 x +

00:17:59

3 equal to 1 then indeed here I have

00:18:01

a product but it is not called an

00:18:03

equation product necessarily equal to

00:18:05

zero but why do we stop on this

00:18:08

type of equations and by the others for

00:18:09

example it is legal at 1 what is

00:18:10

it less good and good quite

00:18:12

simply because we will benefit from a

00:18:14

property on the zero product which will

00:18:16

allow us to solve

00:18:18

this type of equations quite easily and the

00:18:20

following property tells us if a product of

00:18:23

zero factors then at least one of the

00:18:30

zero factors we can write it in a

00:18:33

more algebraic way 6 at x b equal to zero

00:18:37

then at x b equal to zero then equal to

00:18:42

zero him or b equal to zero I hope that

00:18:46

you understand the whole principle of

00:18:49

solving this type of equation, that is to

00:18:50

say that we will start from a single

00:18:53

equation to arrive at two

00:18:57

separate equations which means that what I said

00:19:00

earlier about solving

00:19:01

equations is a little different here we

00:19:03

see another technique which consists

00:19:06

of not wanting to isolate x immediately

00:19:09

in a first step in any case

00:19:10

we can clearly see here that isolated

00:19:21

xkrs stop that is to

00:19:23

say a second degree equation and

00:19:25

we don't know how to do that we will have to wait a

00:19:27

few more years

00:19:28

in any case if we rely on this

00:19:31

property then we know how to do it

00:19:33

because the property tells us so if

00:19:35

a product of zero factors then

00:19:37

at least one of the zero factors, one of which

00:19:39

means either this one is zero x - 5

00:19:42

is equal to zero or then 2 x + 3 and

00:19:49

equalize

00:19:54

either him or him it is to say that in

00:19:58

the case where it is this one which is equal to

00:20:00

zero we agree that all of this will

00:20:02

necessarily be zero since I will have 0 times

00:20:04

something which is equal to zero

00:20:05

in the case where it is the one this which is

00:20:07

equal to zero no problem either if

00:20:09

g10 here I will have something x 0 passed

00:20:12

also equal to zero so it works and if

00:20:14

it is both if it is

00:20:16

both it is not for the no longer 0 x

00:20:18

0 that makes zero so at least one

00:20:20

must be at least one of the two which is

00:20:21

equal to zero and so it is enough

00:20:24

to solve these two small

00:20:25

equations separately to find

00:20:29

this time the solutions of my

00:20:31

equation because yes there will be two so

00:20:33

we are still going to solve these two

00:20:35

little equations so the first one is

00:20:38

quite simple it is enough here so to

00:20:39

add plus 5 on both sides as

00:20:42

we did everything at the hour that makes x - 5

00:20:44

+ 5 equal to zero + 5

00:20:48

so there I have minus 5 + 5 which

00:20:50

goes away I have left x equal to 5

00:20:53

we therefore have the first solution which

00:20:56

is x equal to 5

00:20:58

on the other side g2x +3 equal to zero so

00:21:01

I'm going to start getting rid of the

00:21:03

plus three like we did

00:21:04

earlier so for that well I

00:21:08

subtract by 3 from both sides so

00:21:13

I let's go quite quickly for the resolution

00:21:14

here once again I repeat it to

00:21:16

see the exercises in detail and in a

00:21:19

more precise way

00:21:20

you need to click on the link or so

00:21:24

the plus 3 the minus 300 goes there here goes so

00:21:27

I had 0 - 3 left that is to say minus 3

00:21:29

in other words there we find 2 x equal to

00:21:32

minus 3 to finish I just need to

00:21:35

divide by 2 on both sides in

00:21:37

this way the two directions go there

00:21:39

remains x equal to -3 2 me x equal to 5 x

00:21:46

equal to -3 2 me are the two solutions

00:21:49

of my equation is yes if I replace

00:21:51

by five in there

00:21:52

well it will work the equation will be

00:21:56

verified but if I replace x by -3 2

00:21:59

me in there l the equation will also be

00:22:01

verified I'll let you

00:22:03

do it if you want to see it here and

00:22:06

I said that at the end of the video I

00:22:08

will give the solution to the

00:22:10

introductory equation and here it is

00:22:12

I'll let you pause the video if

00:22:14

you want to read this in detail or check

00:22:17

if you actually found the right

00:22:18

solution which was minus 4 in the meantime

00:22:20

this sequence is finished but I invite

00:22:22

you to do lots of exercises

00:22:23

to train yourself

Description:

Dans cette vidéo, je te propose de revoir tout le cours sur le chapitre des équations. L’objet de cette séquence est de te rappeler et de t’expliquer les éléments les plus importants du chapitre : 0:00 Intro 0:45 La notion d'équation 9:52 La résolution d'équations 16:51 L'équation-produit 👍 Site officiel : https://www.maths-et-tiques.fr/ Twitter : https://twitter.com/mtiques Facebook : https://www.facebook.com/unsupportedbrowser Instagram : https://www.facebook.com/unsupportedbrowser

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