 Hi, this is Chichu. Welcome to my channel. Now what we're going to do in this video is We're going to talk about the equal sign. We're going to take Look at how we go about solving equations And if you want to know more about this material Then we've covered some of the stuff in the language of mathematics Actually a lot of the stuff in the language of mathematics in series 3a and 3b And we're using some of the concept in series 4 where we're doing unit conversion and we've also talked a little bit about this in ASMR math for when we took a look at the power of zero Which is basically giving us the power to be able to solve equations Okay, so if you want to go a little bit further into what we're going to talk about right now because we're going to take a look at Just some of the basic concepts of how to move around an equal sign But if you want more information dig down a little bit deeper then you can start off from series 3a Go all the way to series 4 and Take a look at the one video we did at least one video we did for ASMR math We took a look at the power of zero Okay now what we're gonna do is Let's do I want to bought some new supplies fresh colors Because the previous pens were getting a little faint All right, so I got a whole bunch of colors to play with right now, but we're gonna start off with black basically at first And what we're gonna do is we're gonna start off With exactly the same place that we started off with in the language of mathematics in series 3a and that's basically Giving us an appreciation of we're just learning mathematics that there are two main types of Problems questions were asked to do one of them is to simplify Expressions and the other one is to solve equations or to graph functions, right and the difference between them is this So the first type of questions is that they ask us to simplify things? Expressions and the next type of questions they ask us to solve equations, right? So Now this is Solving equations is what we're going to talk about in this video and the way we reach that point is this when We're asked to simplify expressions and mathematics. We're basically asked To use some of the operations that we know a multiplication division Edition and subtraction basically those mean for operations and you got exponents and radicals some other stuff Basically use some of the symbols some of the powers that we have in math to simplify things, right? so for example if we're given a Fraction 4 over 8 we always try to simplify this and We know that 4 goes into 4 1s 4 goes into 2 4 goes into 8 twice So 4 over 8 reduces down to 1 over 2, right? And we always want to express things in their simplest form because well Mathematics is about simplifying everything, right? trying to analyze the world and The best way to do it is just to simplify things, right? Just for example You never go into the store and if some things, you know Priced at $2. They don't ask you to Pay 8 over $4, right? They don't ask you to pay the square root of $4 to them, right or 2 to the power of $1, right? They just ask you straight out to pay $2, right? That's what it means to simplify expressions and the way we've Done when we're starting out learning math What we first learn how to do is to deal with addition subtraction multiplication division, right? We learn how to use the basic operations, right? Which is addition subtraction multiplication and division and Once we learn these things then we learn how to deal with exponents, right? Things to a certain power and we talked about the stuff in series 2 of the language of mathematics And we've covered some of the stuff for ASMR math too as well, right? So we deal with exponents and then Usually we deal with radicals, which are basically the opposite of exponents, right? Because one of the things that We end up seeing in math, which is brilliant, which isn't really always mimicked in the real world is in mathematics Whenever for whatever you can do you can usually undo, right? So Subtraction is really the opposite of addition division is really the opposite of multiplication and radicals are the opposite of Powers right or exponents, I guess into the integer, right? Because this thing can also be expressed as That's what a radical is. It's the denominator in the exponent, right? So in math For our purposes everything that we do we can undo, right? Again, not mimicked in the real world, right? If I take this glass or this cup, right? And if I break it, I Can't put it back together the way it was, right? I can Glue it together as best as I can, but it's gonna be little things missing and you'll be able to see the craft So I won't be able to undo the breaking of this but in mathematics It's a language you've come up with to quantify the world for everything we do we can undo, right? If we give something a quantity, we can take that quantity away, right? That's what simplifying means. You're taking expressions and you're simplifying them. I shouldn't erase this I should have kept it on there, right? But I was gonna show you an example. So Whenever we're asked to simplify something What they're asking us to do is to take an expression and crunch it down Represented as in represented in its simplest form, right? We did 4 over 8 this reduces down to 1 over 2, right? 1 over 2. You can also combine like terms. You can go 2x plus 5x this simplifies down to 7x, right? And there's a whole bunch of other things you can do. You can go 2 to the power of 3 divided by 4 squared, right? Well 2 to the power of 3 is 8 4 squared is 16 and this simplifies again down to 1 over 2, right? So that's what it means when they ask you to simplify something and you're basically using the abilities we've acquired when it comes to adding things subtracting things, multiplying things, dividing things Taking the roots of things and Kicking it up to a certain power, right? The other type of questions, problems that were given when we're learning mathematics at the beginning stages of learning math anyway is Not to simplify expressions, but to solve equations and the only addition to What we've learned here, right? Because the beauty of mathematics is math is Built on five axioms, right? There's only five rules that the rest of mathematics is built on, right? Incredible, right? So what we learn in math at the beginning stage of mathematics Also applies to the later stages of mathematics, right? The more complex we go So if we learn how to add and subtract multiply and divide and take radicals and powers of things That still applies as a power required into the next step And the next step from simplifying is solving and solving all it does is introduces a new symbol Which is the equal sign, right? So whenever we're solving for equation, the only Additional thing we need to learn is How to deal with the equal sign, what the rules are To solve an equation, what the rules are to move around the equal sign, right? It's a new card in play, a new power, a new tool The rules, what we learn how to deal with all these things still apply, right? So what we're going to do in this video is Learn how to move around an equal sign while still having the ability to do all this, okay? And the way math works is Is once we start combining things the meaning for one symbol Has to take into account the meanings of the other symbols, right? So what we're going to do is we're going to learn what the equal sign means And we're going to learn how the powers that we acquired of additional subtraction multiplication division and the radicals and exponents How that plays in with these things, with the equal sign, okay? As far as what the equal sign represents, okay? Now the equal sign is basically What it is You can think about it as a teeter-totter, right? A balance So when I put down an equal sign, right? And I put something on this side Of the equation and I put something on this side of the equation, right? Then what I'm saying is This side of the equation has to equal this side of the equation, right? We talked a little bit about, well, a fair bit about this in Series four of the language of mathematics, right? But basically what we're talking about is Whatever I put here and whatever I put here, if I'm saying, if I'm putting an equal sign between them I'm saying they both represent the same trait, the same quantity, the same anything I want them to represent, okay? So for us to be able to deal with an equal sign, which is basically For us to be able to solve equations, we're gonna have to know how to add and subtract, multiply and divide Take things to a power and to be able to take Radicals, to understand what radicals represent, right? And that's stuff we talked about in series one and series two of the language of mathematics, okay? So let's take a look at some simple equations, and I'm gonna assume you already know how to Do all those things, right? Add and subtract, multiply and divide, and how to deal with exponents and radicals So let's assume we're gonna question some simple stuff, something like, you know, if they asked us to solve Right? For x, given that x plus 2 is equal to 5 So the way you can think about this is this Think about the equal sign as a sort of a scale, as a teeter-totter to a certain degree, right? Right now what I'm saying is x plus 2 Whatever my x is, it's you can put anything you want in there, right? My unknown plus 2 is equal to 5, right? I'm saying that this side of the equation equals this side of the equation, right? So if you think about that as a teeter-totter or a scale, I always like using a scale What that means is if I take something away from this side For this scale to stay exactly level, I have to take the same amount from that side, right? If I'm going to add something to this side, I have to add the same amount to this side So to keep the scale level An equal sign says this Whatever you do on one side of the equal sign, you have to do to the other side of the equal sign That's it. That's what the equal sign means That's the introduction of the equal sign to the powers that we already know, that we already have, right? Multiplication is multiplication, right? We already know how to multiply, we already know how to divide We already know how to add, we already know how to subtract We already know how to take things to the exponent We already know how to Take the roots of things, right? With the equal sign The additional information that we're learning is this Is that whatever you do to one side of the equation, you have to do to the other side of the equation Okay The second part is that to get rid of something you have to do the opposite to it, right? So that's sort of kicking in some of the information that we already know from just basic operations Adding, subtracting, multiplying, dividing. If you have five, if you want to make it zero, you've got to subtract five, right? For every Thing you do, you can undo. If you're adding something, you can subtract. If you're multiplying something, you can divide it out Right? If you're taking something to a certain power, you can Take it back down again by taking its root, right? Equal sign says this, use that power, right? Use that power to balance this Equal sign, right? So this is a sentence I've already said Now I want to solve an equation. All I need to do is Get x by itself. So what I need to do is is to get rid of the two Right? If I want to get rid of the two If I get rid of the two that gets x by itself, right? If I get rid of the two on this side, then I have to Get rid of a two on this side because it's a scale, right? So for me to get x by itself here The two is adding to the x. So the only way for me to get Rid of this two is to subtract out of two, right? The two is adding to the x. So the way I'm going to get rid of it is I'm going to go minus two on this side of the equation The equal sign says if I do something on this side, I got to do it to the other side So I got to subtract two on this side My equal sign whenever I'm solving equations. I like to line up my equal sign. I line up my equal sign I went x plus two minus two. This is all in the same line, right? So what I have in this Level I guess I never include this, but I'm just gonna show it so you see it Right now on this line. I have x plus two minus two is equal to five minus two right So if we're combining like terms Which is really just all we're doing is simplifying this side and we're simplifying that side, right? So we're just simplifying right now, but we're incorporating that with What the equal sign? Has told us we have to do It's a scale So if we're going to simplify this side of the equal sign Then two minus two is zero. So on this side, we just have x left Five minus two is three right We've got x by itself That means we've solved for x. That's what it means if you're solving for something you're trying to get it by itself The question would be solve for x I got x by itself. I solve for x. What is x equal to x equals three? Okay, good enough Let's do something more complicated and I'm going to kick this up Most likely exponentially every every example we do, right? I'm going to add one more rule I don't know if it's exponential, but it'll be fast Let's say we had another type of question. Let's say we had this two x plus x minus two. Why she has to make it plus two equals five Let's make it a prettier five five plus seven Right and the question would be like the previous one, but it's not just solve they always ask you to solve for x Or whatever variable they want you to solve for right there could be w could be u it could be something else right So what we end up doing is We use the powers that we acquired when we Learn how to simplify things. So whatever we do whenever we're trying to solve an equation We try to simplify each side first, right? so What we're going to do is we're going to combine like terms as much as we can anyway So over here, we've got two x and x can combine and five plus seven We can add right Before we start moving things around From one side to the other and equal sign simplify each side first, right? Line up your equal sign Two x plus x is three x plus two Five plus seven is 12 right Now we got no more simplifying to do on either side Now we can Use what the equal sign allows us Right, we can do what the equal sign allows us to do which is Move things around the equal sign from one side to the other But the only way we can move it from one side to the other Is to undo what's being done to it on this side, right? And because the equal sign is a scale Whatever we do on one side, we must also do the on the other side, right? So there's you know two rules sort of playing with each other. It's the same thing Right, but you know, it's really one rule So right now what we want to do is Get x by itself. That's what it means. So for x Get x by itself What are we going to do? Well, we've got to undo what's being done to the x, right? What are they doing to the x? They're multiplying x by three. That's what three x means And they're adding two now if I try to get rid of the three first Then that means I'm going to have to divide This whole thing by three because whatever you do on one side You have to do it to the other side But you got to do it to that side in its entirety, right? You can't just do it to one of the terms Right, this side is three x plus two So if I want to divide out three some people make this mistake they go Oh, I want to divide three. I divide this by three and I divide this by three That's a mistake because You can't just divide this term by three you got to divide the whole term by three So you would have to divide this whole thing by three, right? You can think about each side as being within brackets. That's an entire thing Okay, now if we divide this side by three then This simplifies down to Basically, this becomes three x over three plus two over three because what we're doing is We're dividing each term by three, right? So we're going divide by three divide by three And if you do this you end up having x plus Two over three is equal to four and you can simplify from there and we will do stuff like this but There's an easier way to go about these things the easy way to go about solving equations Is to do the opposite of bed mass, right? in When we're simplifying things, right? And You know when we learn how to add and subtract multiply and divide and exponents and radicals There's another thing in there called brackets, right? There's something they've told us to do which is always use bed mass, right? bed mass and bed mass means brackets exponents Division and multiplication these two have the same weight Right and addition subtraction these two have the same weight, right? So they always tell us to do brackets first and then exponents and then multiply and divide and then add and subtract When you're solving equations when you're trying to undo things not when you're trying to simplify when you're trying to simplify You're doing bed mass When you're trying to solve when you're trying to get an x by itself The way you really want to do it is to a certain degree Do it backwards Get rid of your addition of subtractions first the separate terms Do your divisions and multiplications if you can right and then deal with exponents and brackets Okay, so let's take a look at how this works So for us what we're going to do first is we're going to get rid of the two Okay, because what that's going to do is we're going to get rid of the two That means there's only one term on this side. So when I divide I'm not going to get a fraction on the term where my Variable is where my x is right? So there's an easy way to solve equations and it's a hard way to solve equations So what I'm going to do is I'm going to subtract out two from this side I'm going to subtract out two on this side as well, right? So minus two minus two now one thing I Do early on when I when I teach My students how to solve equations. I don't write it out like this I like to see things as visual So what I do when I'm trying to move things for addition of subtraction specifically when I'm Getting rid of a positive or negative What I do is I grab whatever it is that I'm moving to the other side and I move it over circle and with an arrow and I always remember I've Mention this I don't know gazillion times. I guess the sign in front of the number always goes with the number, right? So I always circle the sign in front of the numbers or the terms, right or the variables, right? So if I was going to move the x over it would have been a positive x I would grab in the lower And whenever I move over In equal sign whenever I jump over an equal sign, right the sign changes. So this becomes minus two Okay So This is gone gone to the other side and its sign has changed become negative I line up my equal sign on this side all I got left is three x and on this side is 12 minus two is 10 cool now Again, according to our bed mass. We took care of Do we do purple or do we do blue bed mass? bed mass So I took it took care of the addition and subtraction that was going on Now I got a multiplication, right? So to get rid of a multiplication, we do the opposite to it, which is division, right? The opposite of multiplication is division So what I'm going to do is I'm going to divide this side by three So if we divide this side of equation by three, we're going to divide the other side of equation by three That's what an equal sign means So this side becomes All we got left is an x And on this side, we got 10 over 3 and 10 over 3 3 goes into 10 three times you got 1 over 3 left. So 10 over 3 is really We have room to simplify this sure we do x well not simplify but express it as different That's a improper fraction. That's a mixed number, right? So x is equal to 1 and 1 third, right? So we solved four x Let's do another one. Let's solve for another x. Let's give us a little bit more space Let's do something a little bit more complicated, right? So Solve do we should we change the variable? Let's change the variable. So We're not always just solving for x, I guess Which I really like to do actually solve for w right Which means get w by itself. Okay So let's say we had this 5 w plus 7 w minus 4 is equal to 6 w plus 7 okay So the question says solve for w which means get w by itself. Okay So what we're going to do in general what I do is I combine life terms on either side first, right? I simplify each side first and once you learn how to do this I'll show you the next step you could do it one thing and You know multiple things in one line, but for now, let's just simplify each side first, right? So first thing we do is we line up the equal sign And we combine our life terms So all we got here is 5 w and 7 w Nothing else combines here. Nothing combines there So right now we got 5 plus 7 is 12. So we got 12 w minus 4 equals 6 w plus 7, right? If we're trying to solve for w that means we have to get w by itself, right? Well, I have a w on this side and I have a w on that side That means I need to bring all my w's to one side so I can combine them, right? So what I'm going to do is I'm going to bring this w over to this side So this is right now when there is no sign in front of a number of variable means it's positive This is positive 6 w when it comes over When I get rid of 6 w on the side to get rid of a positive 6 w I have to subtract 6 w So what I have to do if I'm going to subtract 6 w from there I'm going to subtract 6 w from here, which is basically grabbing and moving it over in the sign changes Okay Now at the same time I'm doing this Because if I end up doing this I'm going to have 6 w left But I'm going to have a negative 4 here. So what I'm going to do in the same staff is I'm going to grab this guy And bring it over to the other side That's a negative 4 negative 4 moves over it comes positive 4 So what I have now is I line up my equal signs 12 w minus 6 w is 6 w Nothing else left here. I took the negative 4 over 6 w's come this way. It's not there anymore 7 plus 4 is 11 And then what I do What's being done to the w the w is being multiplied by 6 So I need to get rid of this multiplication right What do I do? 6 times w the opposite of multiplication is division. So I divide The w by 6 Or 6 w by 6 I divide this side by 6 to get rid of the 6 So I divide this side by 6. So right now I got 11 over 6 And 11 over 6 well That's a mixed Improper fraction 6 goes into w once And we've got 5 left over 5 over 6 right you need to know how to deal with fractions extremely important extremely important Let's do more complicated. Let's introduce division in the first statement in the first problem Right right the beginning Let's assume we had this We had 5 w over 4 plus 1 over 2 is equal to 6 now Whenever you get fractions and equations What I you know, there's a simple way to do things and there's a harder way to do things the hard way to do things This one is actually not that much harder The other way but once you start getting more complicated Expressions might more complicated equations. It becomes harder doing it This way which is basically grab this guy bring it over and then multiply by four or cross multiply stuff Whenever I'm given an equation where there's fractions in the equation The first thing I do is I multiply the equation by the common denominator right And the common denominator for the whole equation not for one side right the common denominator for this equation Because that's six. That's just six over one the common denominator for the entire equation is four Right and I'm assuming you know how to add and subtract fractions Right and if you know how to add and subtract fractions, then you know about common denominators And that's one thing You know, I try to emphasize As many times as I can is Because a lot of people use their calculators to do things That only gets you so far if you're relying on the calculator to deal with fractions Because the higher level of math minus you go the less Calculator you can end up using right you can have variables and their expressions and You know rational functions rational equations. You can't deal with those with a calculator. You need to know how to deal with Fractions right and the common denominator here Is four so I'm going to multiply the whole equation by four right? What that means is every term gets multiplied by four The whole side of this equation this side if we're going to make you know, write this more complicated We're going to multiply this side equation by four I'm going to multiply that side of the equation by four because the equal sign says If I do something on one side of the equation, I have to do it to the other side of the equation Now the reason I'm going to multiply by the common denominator Is because when we multiply by the common denominator, the denominators disappear Right, I'll just do a little recap of this For multiplying fractions, but I'm going to assume This is going to be obvious right so we're going to have Four multiplying this and this and we're going to have four multiplying this right So five w over four times four Now if you're that's four over one if you're multiplying fractions remember Extremely important the most important thing about multiplying fractions is reduced before you multiply Makes life a lot easier in general 99.99999% of the time makes life easier right Reduced before you multiply with fractions and with fractions anything from the top can kill anything from the bottom As long as there's no plus and minus between them and four kills four So all we got left here is five w right? That's the reason we're multiplying by the common denominator We're getting rid of the denominators. We're getting rid of the fractions and that's what Most people want to do anyway right off the bat and right off the bat. That's exactly what we're doing So this side four kills four. So we got five w Minus now remember This four killing this four didn't mean it kills the Four that's multiplying everything. It's just killed the four that's multiplying it All right, there's another four coming here multiplying this guy Well one over two two goes into four twice. So this is two Is equal to well, there's nothing to reduce here six times four is 24 Now what we can do is I should have actually done this one in red. So you see it. We'll do another one more complicated as well So what we do now is we grab the two bring it over that's plus two right So what we have is five w Five w is equal to 26 And then we divide this side by five Right and we divide we have to divide the other side by five. So Divide by five divide by five. So on this side, we've got w is equal to 26 over five Right, we solved four w because we got w by itself Okay Let's do one where we got the variable in the denominator So if we have variable in the denominator, let's say we have five over w minus four over three is equal to seven We'll do a squared one on the next one so What we need to do now is Get rid of the denominators right off the bat and the way we're going to do it is we're going to multiply this whole expression by The common denominator, which is three w right? So we're going to take this whole equation not expression this whole equation and multiply it by Three w because we need a three and we need a w right adding the subtracting fractions So when three w let's write this clean three w Multiplied by Three w so three w multiplies this this and this right Five over w times three w. W kills w three. It's only three left. So three multiplies the 15 Becomes uh multiplies the five becomes 15 minus Four over three times three w three kills the three so only w multiplies this so become minus four w Is equal to and over here. There's nothing in the denominator Right, so it's just three w times seven which is 21 w All right. Now what I need to do is I need to isolate w. So I'm going to grab this two again Bring it over on this side, right because nothing says you always have to have w on this side You could have w on the other side, right? I could have easily brought this w over here and brought the 15 over and combined the w is on this side But I like combining I like making my w's positive or my x's whatever i'm solving for positive right off the bat So I don't have to divide by a negative number, right? So if I bring it over to that side, that's my that's plus 4 w So now I have 15 equals 25 w right so 15 equals 21 plus 4 is 25 w And now what do I do? I divide both sides by 25 divide by 25 Divide by 25 All right And I can simplify this right while 25 kills 25. I'm doing the opposite of what's being done to the w which is 25 is multiplying w so I'm dividing the 25 out So I got to do it on this side because that's what the equal sign tells me to do So on this side, I just have w now. I'm not going to write w on this side All right, well, maybe we have enough room to readjust it afterwards and on this side I can simplify this fraction right because remember you're still simplified. Whenever you can you try to simplify Crunch as you go, right? So five goes into this three times five goes into that five times. So on this side, I got three over w Oh three over five, right? This side I got three over five And since I like my w's on the left side on this side of the equation I'm just going to rewrite this as w is equal to three over five right You can just Flip it right the whole thing Let's do more complicated. Let's do an equation where we have a more complicated expression in the denominator What if we had five over x plus one minus One over three one over two. What was it four over two four over three? four over three Is equal to seven now the common denominator Here is three times x plus one, right That's what the common denominator is for adding subtracting right fractions So we're going to multiply this whole thing by three and x plus one So that's going to multiply this this and this So when we do it on this side x plus one kills x plus one We'll talk a lot more about this stuff later And we've talked a lot about this in series 3b how to deal with rational Expressions in a large part anyway, we will talk about a lot more about them as well, right? But this is basically you can think about this as one thing. That's one thing, right? We need it So this kills that so just three multiplies that which is 15 minus Three kills three. So it's just x plus one multiplying the four four times x plus one is equal to three Bracket x plus one multiplies the seven because there's nothing in the denominator to reduce it, right? So seven times three is 21 times x plus one Okay, now what we do we go back To what we knew before for simplifying expression. So we try to simplify this thing. So negative four comes in multiplies these Line up your equal sign. So that's 15 minus four x minus four 21 comes in and multiplies this and this, right? This is what we should know how to do, right? Based on Multiplying dividing and subtracting simplifying expressions, right? Think of each side as separate So this becomes 21 plus 21. Oh, sorry 21x So this becomes So this becomes 21 x plus 21 And then what we try to do is We try to bring the axis to one side and the numbers to the other side, right? So actually we can combine these two guys first, right? So simplify each side first line up your equal sign You don't always have to but at the beginning is a good idea to do this first because You're sort of getting used to the rules of moving around the thing, right? So 15 minus four Is 11. So right now the sign in front of the number always goes in numbers. So this is We're gonna put the 11 first makes it look cleaner 11 minus four x and this one is 21x plus 21 And then what I'm going to do is I'm going to move The axis to one side and the numbers to the other side So again, I'm going to take the negative four bring it over It becomes plus four x I'm going to take this 21 bring it over here becomes minus 21 right So on this side, I'm going to have 25x and on this side I'm going to have subtracting 21 from 11 I get negative 10 Right And then what we do we divide both sides equation by 25 right divide by 25 divide by 25 This part was all just Simplified So right now I can just put the x on this side because it's by itself x is equal to Let's write this a little bit bigger x is equal to Five goes into 10 twice five goes into 25 five times. So it's equal to negative two over five That's solving something a little bit more complicated And if we want to go step further than this what we can do is Introduce exponents and radicals All right, okay, so let's assume we had the following two x squared Minus eight is equal to negative seven x squared plus 12 right So if they want us to solve for x right, which means get x by itself Um What we're going to do first is we're going to simplify each side of the equation first So we take a look at this one We can't simplify anything there. Everything's already simplified as much as it can go same with that side There's nothing to simplify. So what we're going to do is we're going to bring The x's to one side of the equation and bring the numbers to the other side Now this sort of changes if we have x's to different powers, right? And we'll deal with that next one and we've already talked about in ASMR math And we talked about the power of zero, right? But for now because we only have x squares, we're going to bring this guy over That's a negative seven x squared. This becomes plus seven x squared This side I'm going to bring over becomes plus eight All right, I get rid of a negative eight on one side I'm moving to the other side. It comes positive the only way to get rid of a negative eight is to add eight All right, I'm going to line up my equal sign negative two x squared plus seven x squared is five x squared 12 plus eight is 20 And then what I'm going to do is I'm going to divide by five, right? Divide by five And five kills five. So I have x squared left on this side, right? x squared is equal to 20 divided by five is four And Now we have exponents, right? Well, the opposite of exponent An integer to a power is to put it over a denominator or take the root of it, right? We've talked a lot about this in series two of the language of mathematics, right? We've got a lot of videos on exponents and radicals. So To get rid of a square, I have to do the opposite of square, which is taking the square root So what I do is I take the square root on this side and I take the square root on this side Whatever you do on one side, you're going to do it to the other side Square root of x squared is just x and this side is just two. So x is equal to two And we solve for x More complicated than this Let's say we had this And again, you can think about it as us doing The backwards of what bet bass is, right? Let's assume we had something like this 4 over x plus 1 squared minus 1 over 2 is equal to 11 over 4 So what we're going to do is we're going to combine Like terms first simplify as far as we can and we can't really simplify anything here And this is already simplified as much as it can go So what we're going to do is Because these are fractions Right, we can multiply the whole equation by the common denominator, right? And if we multiply the whole equation by the common denominator This is what we end up getting and What's that going to do is going to make it a little bit more complicated, right? So what we're going to what we can do is multiply this whole thing by I'll do this But just write it out But I'm going to show you the other way to do it, right because there's certain cases where To simplify, combine like terms first, it becomes easier, right? So what we can do is multiply this whole thing as far as the numbers go the common denominator is 4 And as far as the expressions go it's x plus 1 squared x plus 1 squared, right? So we're going to multiply each term by this now That's going to make it A little bit harder Instead of doing the following Okay, this is what we're going to do now. This is the other method Where I mentioned in the simpler one, but I didn't do it that way because It would have been Taking more steps to do it That way, but this one's going to be easier steps. So what I'm going to do is I'm going to Move this guy over to this side first. So that's plus 1 over 2 To get rid of a negative half I have to add a half here So I add a half there or grab it move it sign changes Line up my equal sign I add up these guys as a common denominator, right? And we know how to add fractions common denominator here is 4 so This needs to become 4 I multiply this by 2 to make it 4 So I multiply this by 2 so that becomes 2, right? So the whole thing's over 4 and 11 plus 2 is 13. So this is 13 over 4 right On this side, I have 4 over x plus 1 squared, right? Okay. So this is 4 over x plus 1 squared Now we've talked about this which is Sort of the concept of cross multiplication and again, we talked about this in series 3a where We talked about cross multiplication, right? How we can solve equations where we have one fraction Equal to another fraction, right? All we have to do is Cross multiply and this is something we're using for unit conversion as series Series 4 of the language of mathematics, right? So what we're going to do is we're going to cross multiply. We're going to grab this duet key Kick it up here and we're going to grab that duet key and kick it up there On this side I'm going to line up the equal sign on this side. We've got 4 times 4 is 16 On this side, we're going to have 13 times x plus 1 squared Now we still need to get x by itself, right? We're still trying to reach x. It's like peeling peeling crunch crunch warm around jump All right Do whatever you need to do to get x by itself within the rules of mathematics. Okay So what we're going to do is Because this is 13 times this duet key. We're doing bed mass backwards, right? So we're doing bed mass bed mass We did our subtraction addition We're into multiplication and division because we've got exponents and brackets, right? So what we need to do is deal with multiplication and divisions first So this is 13 times this duet key, right? I'm not going to go into the brackets and exponents yet because I haven't dealt with all my multiplication and division And then we can get into brackets and exponents, right? So what we're going to do is Because this is 13 times this expression that has the x within it. I'm going to divide this side by 13 And I'm going to divide this side by 13 Right because 13 kills 13 So on this side of the equation, I'm just going to write it over here because we can All right. So this side I got x plus 1 squared is equal to 16 over 13 right Now I need to get to the x I have exponents here and I have brackets here Bed mass let's write it out again. Bed mass is I dealt with the addition of subtraction I dealt with the multiplication and division As much as I could Now I got exponents and then I got brackets. So I got to deal with the exponents first Okay, so I got to get rid of that squared To get rid of the squared on one side of the equation You have to take the square root of it And if I do something on one side of the equation, I got to do it to the other side of the equation So I'm going to take the square root. Let's do this in red I should have done this one in red too, right? 13 And 13 Gross it out So what I'm going to do is I'm going to take the square root on this side I'm going to take the square root on this side So right now the square root of x plus 1 the whole thing squared the square root kills the squared So on this side, I got x plus 1 Is equal to square root of 16 is 4 Square root of 13 is just the square root of 13 All right And then what do I got to do? I got to move The one Over there, right because I hit the brackets bet mass and whatever is happening inside the bracket The rules of mass still apply bet mass still applies, right? So Bet mass is sort of layered, right? You do your bet mass Until you've gone through the rounds and then if you still have Things you need to simplify move around the equation You still got to do that But you still have to follow the rules of bet mass, right? So I need to get rid of the one here So I grab this one and bring it over this becomes mine as well All right Should we erase this give us a little bit more up here. Yeah, let's do it here So right now I got Brought it over I got x is equal to 4 over Square root of 13 Minus 1 Now there's multiple ways you can deal with this Because we have a radical and a denominator We got a radical and a denominator But I'm going to just deal with it by itself. I'm not going to rationalize the denominator yet Right. I could rationalize the denominator. Let's erase the rest of these Right So right now We got x by itself you want you can punch this in your calculator, but Generally don't right you want to get exact values. We have a radical we have you That doesn't simplify. So that's an irrational number. Okay, so what we want to do is Simplify this side because we've already solved for x technically. We've already solved for x solving for x means Get x by itself. I've already got x by itself and x is equal to 4 divided by the square root of 13 minus 1 But I'm going to clean that up because I can All right using the rules of mathematics that we already knew of how to simplify expressions, right? So all I'm going to do is just simplify that side Well, the way I'm going to simplify that side is Common denominator here is square root of 13 Just adding fractions. This side is just 4 that becomes minus the square root of 13, right? Because that's over 1 I have to multiply top and bottom by square root of 13 So this becomes 4 minus the square root of 13 over square root of 13 Okay, that's what x is and whenever I have x by itself on one side or a function by itself on one side I don't need to always write x because this side is Haven't changed. I'm keeping it at x. So if I don't write anything here, it means that's x that's x That's x You know until I move something over then I have to rewrite it again, right? Now what I'm going to do I'm just going to rationalize the denominator here because I want to Right because I can because it cleans it up a little bit So square root of 13 multiplies this and this Square root of 13 times that it becomes 4 square root of 13 Square root of 13 times square root of 13 is just 13. So minus 13 over 13 and that's what my x is and whenever you get to the last Location last step where you have your x Equaling something then usually put down your x write it down complete the sentence because that's what you know An equation really is x is equal to This do a key Okay, and from there it goes on to more complicated Questions that some of the stuff we talked about When we looked at the power of zero right where we had Things that we could factor and the only reason why we could solve those equations was it was basically Because the only way you can multiply things we do a little review of that Let's do a little review of that. But it's really important that concept is ridiculously important. So, you know, if we had Let's say we had An equation where we needed to solve for the x, right? Let's say we had this 2 x squared plus 3 x squared minus 8 x Minus 7 is equal to 2 x minus 8 Okay Let's assume we had this equation and they want us to solve this equation so what we end up doing again is Simplified like terms on either side that we can right so we can combine this guy and this guy And we can bring this guy over here, right? That becomes minus 2 x and instead of Bringing the 7 to the other side. Okay, and actually what I'm going to do is I'm going to change this to a plus So this thing works out nicely. Okay, I'm going to change this guy to a plus Okay, now instead of bringing the negative 7 over to that side With practice you'll notice this but what I have here is is x squared terms and an x and I can't combine those So what I'm going to end up having is a quadratic function If we've talked a fair bit about right, so what I'm going to do right now is I'm going to bring this guy over here Which becomes minus 8 So if we're combining like terms what I end up having is this is 5 x squared 8 x minus 2 x is going to be negative 10 x negative 7 minus 8 is going to be negative 15 is equal to 0 right and this is something that We should recognize why some of the previous videos we've done, right? This is a quadratic function and it looks like a complex complex quadratic, right complex trinomial, right? But it's not because since this side is equal to zero because I brought everything over from that side, right? If I subtract it out everything and everything counts of each other out on this side of the equation, I have zero, right? So what I can do right now is I can Simplify my work, right? I can make Do this in a way where I make life easier for myself. I'm just going to move this up a little bit So it's going to be 5 x squared So we have enough from 5 x squared minus 10 x minus 15 is equal to 0 Now one of the rules we have with the equal sign is Whatever you do on one side, you got to do it to the other side, right? And if I'm trying to simplify life for myself, I can notice that there is a GCF in these expressions, which is 5 So what I can do is I can divide this side of the equation by 5 Right get rid of a 5 on this side and I can get rid of it on this side as well Okay Now be careful doing this because you should only do it for numbers never do this with variables. We've talked about this in the past where You know We did examples where we're eliminating solutions. We don't want to eliminate any solutions, right? Because when they're asking us to solve for something, that's something doesn't necessarily have to have one answer. It could have multiple answers, right? And usually the highest number of answers we can get are the power for polynomials anyway for polynomial functions equations It's the highest power in the equation, right? So the maximum number of solutions for this is 2, right? Because that's squared But we'll talk a lot more about this as well. Okay, so what I can do is divide out of 5 because I'm doing one thing on one side I can do it to the other side Well, what that means is I line up my equal sign 0 divided by 5 is 0 This side didn't get any more complicated, but this side became easier Dividing this side by 5 means I'm dividing every term by 5 5 goes into 5 You get this minus 2x minus 3 Right You can also think about it as you took out a gcf of 5, right? There's a 5 out here All of those are 1 negative 2 and negative 3 and then 5 kills 5, right? Now what I'm going to recognize is this is a simple trinomial and I can factor it, right? So I factor this Again, series 3 and 3a and 3b is where you want to be I look for two numbers and multiply to give me negative 3 out to give me negative 2, which is negative 3 and plus 1 Now this kicks into What I mentioned, which is the power of 0 in an equation Which is the only way for us to multiply two things To give us 0 Is if at least one of them is 0 So it gives us the ability to solve this equation by splitting these these terms And writing as x minus 3 is equal to 0 And x plus 1 is equal to 0 And then I grab the negative 3 and bring it over We'll do it in red It becomes plus 3 and I grab the 1 and bring it over becomes minus 1 so This thing when they're solving for x becomes x is equal to 3 and this one is x is equal to negative 1 and those are our solutions. Those are those are what x can be That and that I don't want to go any deeper in different types of equations that we can solve because We've done a fair bit of this and we're going to do a lot more in the future Um, but this should give you an idea of what's required to solve an equation, which is basically You know a couple of extra things a couple of extra rules that come with an equal sign One of them being if you do something on one side you go ahead and do the other side And to get rid of something on one side you do the opposite of what's being done, right? Do bet mass backwards, I guess if you want to think about it that way, okay? Um, I thought this was a good opportunity to sort of review some of the rules of How to solve equations how to move around an equal sign? For asmr math anyway, because we haven't we haven't done it for asmr math We've done a little bit, but we haven't done some of the simpler stuffs Stuff but we did do it for a language of mathematics So I thought it'd be nice to do a review of the previous stuff and overlay that Concept for asmr math as well because it's really important I hope you enjoyed and I'll see you guys in the next video. Bye for now