Algebra Introduction - the basics - By tecmath
Transcript
| 00:0-1 | Could I welcome to the Tech Math channel . What | |
| 00:02 | will be having a look at this video is we're | |
| 00:04 | going to be starting to look at algebra . This | |
| 00:06 | is an introduction sort of video looking at algebra where | |
| 00:09 | we are starting to explore this really , really great | |
| 00:12 | branch of mathematics . So anyway , if it's your | |
| 00:15 | first time doing algebra review , you need a bit | |
| 00:17 | of brushing up . I think this is a really | |
| 00:19 | good video for you to see . So anyway , | |
| 00:21 | algebra is basically a branch of mathematics where we use | |
| 00:25 | letters or symbols as well as numbers and basically this | |
| 00:29 | letter or symbol , they take place when unknown value | |
| 00:32 | or a value which might change . So you might | |
| 00:34 | occasionally see this thing written as follows . You might | |
| 00:37 | say something like uh , you know , occasionally you | |
| 00:40 | might see this written as an X . X plus | |
| 00:42 | two , the course , an unknown number , like | |
| 00:45 | why or something like this . But we're using letters | |
| 00:48 | instead of numbers here and this tends to throw a | |
| 00:50 | few people so don't get too scared when you first | |
| 00:53 | see them because we much treat these letters the same | |
| 00:55 | as we treat numbers and normal mathematics . So I'll | |
| 00:59 | give you an example . They say we have something | |
| 01:01 | like a rectangle and I'll show you how we could | |
| 01:03 | do a eligible for example here . So we had | |
| 01:07 | a rectangle here and we we knew that the side | |
| 01:09 | length here was seven centimetres and this side length here | |
| 01:12 | was three centimeters . Okay , We'll say I said | |
| 01:17 | I wanted to work out the perimeter of this . | |
| 01:19 | Well the way we do it is as follows , | |
| 01:22 | okay , we'll get the This perimeter pay a equals | |
| 01:26 | would be equal to seven this side here As well | |
| 01:30 | as this side here , so plus three Plus this | |
| 01:33 | side here , plus seven plus his side . He'll | |
| 01:35 | just like , Okay , so all together , that | |
| 01:38 | would be 20 cm and that's all well and good | |
| 01:42 | and it's not too bad . Okay that's that's fairly | |
| 01:45 | easy . And what we're doing is we're adding the | |
| 01:47 | sides together . But what I'll do is I'll actually | |
| 01:49 | get rid of these numbers now and I'll write it | |
| 01:53 | slightly differently . So say we actually don't give you | |
| 01:56 | the actual sidelines here instead of actually give them as | |
| 02:01 | I'll give them something that can change . I get | |
| 02:02 | the length here is an L . And I give | |
| 02:04 | the width here . There's a W . And then | |
| 02:07 | let's say I want you to work out the perimeter | |
| 02:09 | . Okay . So we actually haven't got the length | |
| 02:11 | and the width stated here . We can still actually | |
| 02:14 | show a relationship to work out the perimeter as as | |
| 02:17 | follows . So the perimeter well it would be equal | |
| 02:21 | to like we did before . It's we've got a | |
| 02:22 | length and the wit . Okay so length plus the | |
| 02:25 | width plus this length plus the width . Okay , | |
| 02:32 | so we could actually take that a step further and | |
| 02:34 | we can say we've got two lengths . Okay , | |
| 02:36 | I'll put a little L there . I think maybe | |
| 02:38 | that will make it a bit easier to see that | |
| 02:39 | . Okay , so we got two lengths and we | |
| 02:42 | have to wits . So all of a sudden , | |
| 02:44 | what we've done is we've written this as an algebraic | |
| 02:46 | expression . Okay , instead of actually putting numbers there | |
| 02:51 | and a bit later on , what we might do | |
| 02:53 | is we can actually substitute values in here where we | |
| 02:55 | said say this Length here is seven . Okay , | |
| 02:59 | so two times seven and this one here was three | |
| 03:02 | , so two times three . And again , if | |
| 03:03 | we do that two times seven is 14 , 2 | |
| 03:06 | times three or six and we end up with 20 | |
| 03:08 | centimeters . But this way here is written algebraic lee | |
| 03:12 | . So this algebraic expression has two terms here has | |
| 03:15 | this Alan this w Okay . Uh and They also | |
| 03:20 | have this thing here which is called a coefficient in | |
| 03:21 | front of both . Have this coefficient of two . | |
| 03:23 | The number in front the of the algebra expression what | |
| 03:27 | we might call variable is called a coefficient . So | |
| 03:30 | that's that's an example of , say , something else | |
| 03:33 | to break . So from here , what we're going | |
| 03:35 | to be doing is we're gonna start actually having a | |
| 03:37 | look about how we can actually write certain algebraic expressions | |
| 03:41 | and then how we can simplify them . Okay , | |
| 03:43 | so it's a really , really good thing for you | |
| 03:45 | to be able to get the hang off before you | |
| 03:47 | start then launching into later algebra . So , say | |
| 03:50 | for instance what I wanted to do because I wanted | |
| 03:53 | to actually write four times the number . Okay , | |
| 03:56 | so that could actually write four times any given number | |
| 04:00 | here . Using algebra . So what we could do | |
| 04:03 | here , Yes . So this stuff what we're gonna | |
| 04:06 | do is we replace the word number four times the | |
| 04:09 | number will replace that word number with xo four times | |
| 04:13 | X . That have changed it to a letter here | |
| 04:16 | . Okay , so four times X . We can | |
| 04:18 | actually write this . We don't bother put the times | |
| 04:21 | in here a lot of the time . What we | |
| 04:22 | start to actually do is we write this is four | |
| 04:25 | X . So if you start seeing algebraic expressions on | |
| 04:28 | this four X , what it actually means is four | |
| 04:31 | times a given number , say X was for example | |
| 04:34 | three . Okay , four times three would be 12 | |
| 04:38 | or say it was 74 times seven would be 28 | |
| 04:42 | . Okay , so X can can change here . | |
| 04:45 | Um So I'll give you another example , say I | |
| 04:47 | wanted to write five times a certain number . How | |
| 04:49 | would I write that ? I'd rather five times . | |
| 04:53 | And you can choose any letter to , you do | |
| 04:54 | have to use X . No . What used a | |
| 04:57 | here ? But you might have used X . Okay | |
| 05:00 | . So five times 8 and we get five idea | |
| 05:04 | . Doesn't actually matter what actual better we use with | |
| 05:07 | this . Okay . Um often will use excess and | |
| 05:10 | we use wires and we use a and we use | |
| 05:12 | bees . But it's just mainly fairly a good thing | |
| 05:16 | to actually do is is to actually try to be | |
| 05:19 | a bit consistent with these . And I also try | |
| 05:21 | to use letters that often don't get confused with numbers | |
| 05:24 | . Like I think using our law something like oh | |
| 05:27 | , can be very confusing because it could look like | |
| 05:29 | a zero . Okay , so another thing we're doing | |
| 05:34 | algebra is we can also add certain algebraic expressions together | |
| 05:39 | , ones with life terms . And these are ones | |
| 05:42 | that contain exactly the same Pro numerous . I'll show | |
| 05:45 | you what I mean by this . Okay . So | |
| 05:48 | I would say something like um four X plus five | |
| 05:55 | Y plus two eggs . Okay . You can see | |
| 06:01 | here that we actually have a couple of these are | |
| 06:04 | pro numerals , here are the same . We have | |
| 06:05 | this four X . Here and we have to xia | |
| 06:08 | . We can actually group these together . Okay , | |
| 06:11 | so we can what we can do with these is | |
| 06:12 | four X plus two X . What we can actually | |
| 06:15 | do is we can group these guys together . So | |
| 06:18 | four X plus two X . Is six X . | |
| 06:20 | Because we're adding them together . Okay , so four | |
| 06:22 | X plus two X . Six X . And five | |
| 06:25 | . Why are you going to see is different ? | |
| 06:26 | We can't actually add this to six exit . It's | |
| 06:29 | a different prime minimal . So we can't actually group | |
| 06:31 | that in with that . So we actually put this | |
| 06:34 | over here is just plus five . Y . Okay | |
| 06:36 | , this is a way that you can actually simplify | |
| 06:40 | algebraic expression . So number one , what we can | |
| 06:42 | do this means six times X . And we can | |
| 06:44 | put those together . We can also add ones with | |
| 06:47 | like terms together , ones that are not like terms | |
| 06:49 | we can't add together . And I'll give you another | |
| 06:51 | example of unlike term , which is a fairly group | |
| 06:55 | want to be able to understand because it's another good | |
| 06:58 | thing that you sort of get is that the actual | |
| 07:01 | the powers of the term also matter . I'll show | |
| 07:04 | you what I mean by this say I would say | |
| 07:05 | something like X square . This means X times by | |
| 07:09 | itself . And I add this to eight X . | |
| 07:14 | And I take five x . And I add to | |
| 07:19 | And I say I want to simplify this expression . | |
| 07:21 | So what I want to do is group any of | |
| 07:23 | the like terms together you're gonna see here we've got | |
| 07:26 | X squared , we've got eight X . Negative five | |
| 07:29 | X . With it too . So first off we | |
| 07:32 | could actually say this negative five X . And eight | |
| 07:34 | X . Here can be grouped together but what about | |
| 07:36 | this X squared here ? And unfortunately can't group these | |
| 07:40 | guys together because this is a different power to exhale | |
| 07:43 | . Okay , so you can't group these guys together | |
| 07:45 | so they have to be in the same power so | |
| 07:47 | it's X . To the to you can't add it | |
| 07:49 | . If you can text the three , you couldn't | |
| 07:51 | add it . So we end up with the following | |
| 07:53 | been up with this X squared , we're going to | |
| 07:55 | add adx take away five X . Eight X . | |
| 07:59 | Take away five X . His three X . And | |
| 08:03 | then there's two here plus two also currently group . | |
| 08:06 | So we end up with this sort of expression , | |
| 08:09 | okay , I'll give you an example which I'll get | |
| 08:11 | you to answer by yourself . So what about a | |
| 08:14 | couple of these are going to get you to a | |
| 08:17 | group , these sorts of questions here . So I | |
| 08:20 | want you to group up uh three Y plus two | |
| 08:25 | X plus 41 How would you go about ? Group | |
| 08:31 | you guys ? Okay . So hopefully what you're going | |
| 08:34 | to do is going to realize that we have the | |
| 08:36 | same uh pro numeral here , the same letter here | |
| 08:39 | in Y . Three Y . Four Y . So | |
| 08:42 | you can group these guys together . Three Y plus | |
| 08:45 | four . Wife is seven more . And here we | |
| 08:49 | have this to actually can't be group any further . | |
| 08:50 | It's a completely different letter . So we end up | |
| 08:53 | with just place to X . Okay . And it | |
| 08:56 | seems a little bit strange at the moment . Okay | |
| 08:58 | . You might actually still get , this means seven | |
| 09:02 | times why ? And it's plus two times X . | |
| 09:07 | Okay . It's an important thing to understand also . | |
| 09:09 | And what we'll actually do a bit later on , | |
| 09:12 | we might give why a value will say when I | |
| 09:15 | say we said why equaled three and X equals to | |
| 09:20 | . What would our expression equal ? We might do | |
| 09:22 | that . Okay . That sort of thing does happen | |
| 09:24 | in that . So if I was to equal 37 | |
| 09:26 | threes the 21 Plus x equals to four equals 25 | |
| 09:31 | . Okay , so that's that's the sort of thing | |
| 09:34 | you might be doing in algebra . Um Okay , | |
| 09:39 | what are we gonna head on to next ? Um | |
| 09:42 | So we could also multiply and divide occasionally when we're | |
| 09:47 | talking about unlike terms . Okay , so we can | |
| 09:50 | add and subtract life terms . We can actually multiply | |
| 09:54 | unlike terms . And I'll show you what I mean | |
| 09:56 | by this say we would actually say what is six | |
| 10:00 | X times three ? Why ? Okay . We're asked | |
| 10:05 | to simplify this now with this what you might realize | |
| 10:10 | is okay , there's that they are different . They're | |
| 10:13 | unlike terms . We can actually um we can actually | |
| 10:17 | multiply this . I'll show you how to do this | |
| 10:19 | six times 3 . So we actually multiply these coefficients | |
| 10:23 | as normal . Also on top of this because he | |
| 10:26 | was actually right this this is six times x . | |
| 10:29 | Times three times why ? Okay , so six times | |
| 10:34 | three his I . D . We also times he | |
| 10:37 | by X . Okay . And we're also times in | |
| 10:40 | my wife . So we end up with 18 X | |
| 10:42 | . Y . This means 18 times X . Times | |
| 10:45 | Y . Okay so we end up multiplying these expressions | |
| 10:49 | together . So when you first doing these it's probably | |
| 10:52 | not a bad idea to actually write them out the | |
| 10:54 | long lines to get used to actually doing this seat | |
| 10:57 | . But you can actually do this . So I'll | |
| 10:59 | give you another example for you to go and say | |
| 11:01 | , I asked you to multiply five X . Times | |
| 11:06 | three white . Okay . So you might write this | |
| 11:12 | is five times X times three tom's boy . Okay | |
| 11:20 | . There's $2 here to five in the three . | |
| 11:22 | We can multiply each other . We get 15 times | |
| 11:26 | X . So we just put the X . Times | |
| 11:28 | X . Times Y . Okay because this You see | |
| 11:31 | here they're sitting next to each other . It just | |
| 11:32 | means that times $1 . So 15 X . Y | |
| 11:36 | . Another type of uh multiplication we can do and | |
| 11:40 | we do algebra , jesus . So many of them | |
| 11:42 | isn't there ? But they're not too bad . You | |
| 11:44 | do get used to the is this sort of thing | |
| 11:46 | ? So we actually had four X . And I | |
| 11:49 | wanted to multiply this by X . Okay . How | |
| 11:54 | would you go about doing this ? Well with this | |
| 11:58 | it's kind of a funny thing . But what you | |
| 12:00 | might realize that this is equal to again , at | |
| 12:03 | the start , we get first doing that you might | |
| 12:05 | like This is for times X times X . Dave | |
| 12:10 | was to put brackets around here . This exodus . | |
| 12:14 | Actually you're going to realize possibly that this is equal | |
| 12:17 | to X squared . A number times itself is said | |
| 12:20 | to be squared . Ok . And then what we | |
| 12:22 | have is we have this X squared . And we | |
| 12:24 | multiply this by four . So we ended up with | |
| 12:26 | four X squared . We have to give you one | |
| 12:30 | of these . What about we do three X times | |
| 12:41 | What about ? I'll do it won't do this , | |
| 12:42 | exile . Do I believe we've got a step further | |
| 12:44 | ? I'll do it times four x . So first | |
| 12:48 | off write it all out . So we have three | |
| 12:52 | times X . Times four times X . Okay , | |
| 13:00 | so first off we can do three times four . | |
| 13:03 | Just 12 times X times X . X times X | |
| 13:09 | . Is X . Word . Okay . So we | |
| 13:12 | might even write the start X times X . And | |
| 13:14 | your first working things out . And then you realize | |
| 13:16 | that this here is X squared equals 12 X squared | |
| 13:23 | . And when you first do we need you might | |
| 13:25 | get a few of them incorrect and it might be | |
| 13:27 | a bit of a bit hard when you're first starting | |
| 13:29 | , but don't get too despondent with it . You | |
| 13:31 | do get used to the what about just one more | |
| 13:34 | like this ? Um What about we do a bit | |
| 13:36 | of a more difficult woman ? What about I do | |
| 13:39 | say something like bonus too X . Y . And | |
| 13:45 | I'm going to multiply this by three Y . Okay | |
| 13:50 | . It's a pretty difficult one . Yeah . Again | |
| 13:53 | , so what we start off doing is we all | |
| 13:55 | right , equals and equals -2 tom objects times Why | |
| 14:03 | time story times Y . Okay . So we've written | |
| 14:07 | it all out this long way . Okay , so | |
| 14:10 | first off let's multiply the coefficients . So we've got | |
| 14:13 | minus two times three times three minus two times three | |
| 14:17 | . It's more than six . Okay ? We have | |
| 14:21 | an excess . I'll put that down . We have | |
| 14:23 | X . So at times it by X . And | |
| 14:26 | we also have two wives . So Y times Y | |
| 14:28 | . So this is why times why which is why | |
| 14:31 | squared okay . Of I'll tell you what I'll do | |
| 14:38 | with this . I'll give you a bunch of examples | |
| 14:40 | here right now to go through and we'll see how | |
| 14:42 | you go with these . Okay . And then we'll | |
| 14:44 | answer this . That was a whole big lesson on | |
| 14:47 | that , and I think it's a really really difficult | |
| 14:49 | one when you're first starting out , but don't feel | |
| 14:51 | too bad with this . Okay , because if you | |
| 14:53 | can get those you're pretty much we'll work out a | |
| 14:54 | lot of the algebra without much sweat . So what | |
| 14:57 | about first off , who started very simply ? Um | |
| 15:02 | can you simplify these sort of things ? So I'll | |
| 15:04 | put up five questions one , two , three far | |
| 15:13 | . Okay so the first one I'll put up Is | |
| 15:15 | as follows . What about I get you ? Simplify | |
| 15:18 | three X plus two . Why Mourners two X . | |
| 15:27 | Look at this one here , we'll go X . | |
| 15:30 | Where ? Plus three X Plus two x squared minus | |
| 15:40 | why ? Okay , we got this one here where | |
| 15:45 | we will go , what is three X . Times | |
| 15:51 | for ? Why ? What about we go ? What's | |
| 15:55 | three x . Times two X . And last of | |
| 16:01 | all ? What's uh -2 ? Why ? Times three | |
| 16:09 | X . Y . Yeah . Leave it there . | |
| 16:13 | Okay so we'll give that a guy , so pause | |
| 16:17 | it , give it a go see here you go | |
| 16:18 | . Will work through this particular city . Okay . | |
| 16:22 | So how did you go ? Did you cause that | |
| 16:24 | you just didn't cause it all with the guards . | |
| 16:26 | Let them work it out for me anyway . Hopefully | |
| 16:28 | poison . We'll see how you do anyway . Let's | |
| 16:30 | first off we'll get to this first question . Three | |
| 16:32 | X plus two Y minus two X . Their first | |
| 16:35 | off we had to recognize these life terms and these | |
| 16:38 | are the ones are the same . Pro knew all | |
| 16:40 | the same letter three X . Take away two x | |
| 16:44 | . His ex . Okay . And we're left with | |
| 16:48 | plus two Y . Okay . So hopefully you got | |
| 16:54 | that answer . What about this next one more again | |
| 16:58 | ? I'll underline this one's with the same coefficients and | |
| 17:01 | into the same power . So we have X squared | |
| 17:03 | plus three X plus two X squared minus Y . | |
| 17:06 | So x squared plus two X squared . We have | |
| 17:10 | three X squared . Uh then we have plus three | |
| 17:15 | X . Which is By himself , plus three x | |
| 17:19 | . And we also have minus Y . Which is | |
| 17:21 | also by itself . So we're going to not be | |
| 17:22 | able to do much more with those . So hopefully | |
| 17:25 | you went well with that . What about number three | |
| 17:27 | ? Three X times four ? Y . Okay . | |
| 17:30 | So this is going to be equal to three times | |
| 17:35 | X times full times . Why ? Okay , so | |
| 17:40 | three times before we multiply these coefficients and we get | |
| 17:43 | 12 and then we get the X there . And | |
| 17:47 | then we're also times into why they're so three times | |
| 17:50 | four is 12 times x times y . Okay , | |
| 17:56 | What about this next 1 ? three x times two | |
| 17:58 | x . So , first of all multiply these coefficients | |
| 18:03 | actually . So we're gonna end up with three times | |
| 18:05 | X . First off . All right , the whole | |
| 18:06 | lot of three times X times two times X coefficient | |
| 18:12 | three times two or six . And then we have | |
| 18:14 | X times X for X times X . His ex | |
| 18:18 | . Where did you come to that ? And last | |
| 18:22 | of all we have this minus two Y times three | |
| 18:26 | X . Y . So again , first off , | |
| 18:29 | I'll write them all out minus two times Wire times | |
| 18:33 | three times X times Why first we multiply these co | |
| 18:39 | efficiencies numbers so minus two times three is minus six | |
| 18:43 | . Hopefully got the minus 60 . Yeah , he | |
| 18:46 | has to be the truth there . And we've got | |
| 18:50 | one X . So I'm gonna put that down and | |
| 18:52 | we have Y times Y . So we have boy | |
| 18:55 | squared minus six X , y squared . You know | |
| 18:58 | , it actually doesn't matter . Also if you're ever | |
| 19:00 | in a different way of interpreting at minus six y | |
| 19:04 | squared X . It's exactly the same number . Okay | |
| 19:08 | , it still means minus six times X . Times | |
| 19:10 | y squared . And this one also has minus six | |
| 19:13 | times X times y squared . Just written in a | |
| 19:16 | different order . Okay . It's like this one up | |
| 19:18 | here . If you had read this is 12 YX | |
| 19:21 | . It's exactly the same number . I just tend | |
| 19:23 | to put the excess before I put the wise . | |
| 19:25 | Okay , it's just a bit of a habit I | |
| 19:26 | I've got into anyway . Hopefully that was some use | |
| 19:30 | to you in future videos . What we're gonna be | |
| 19:33 | having to look at is we're going to start actually | |
| 19:35 | taking this a little bit further where we're going to | |
| 19:37 | start expanding and working at different factors and things like | |
| 19:41 | this with algebra . So we're gonna make this algebra | |
| 19:43 | a little bit more complex anyway . I hope to | |
| 19:46 | see you then . Bye . |
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