Simultaneous Equations Math Lesson - By tecmath
Transcript
| 00:0-1 | Good day and welcome to the Tech mouth channel . | |
| 00:02 | What are we having a look at in this video | |
| 00:04 | ? Is these things called simultaneous equations . These are | |
| 00:07 | equations which looks like this , They're type of algebraic | |
| 00:09 | equation . This in response to a question that somebody's | |
| 00:12 | asked me on help on solving these . So hopefully | |
| 00:15 | it will be of some help . We're going to | |
| 00:17 | make a couple of videos to do with this . | |
| 00:18 | The first one we're going to be having to look | |
| 00:20 | at is just some parts of the simultaneous equations . | |
| 00:23 | Just a couple of the terms used , I won't | |
| 00:24 | bug you too much down with those . They're going | |
| 00:27 | to look at ways of solving them . And just | |
| 00:28 | a couple of the methods used . The second video | |
| 00:31 | I'm going to make , I want to do a | |
| 00:32 | bit of a shortcut method on how to solve these | |
| 00:34 | fairly quickly . Okay , so it's worth uh if | |
| 00:37 | you get this part of it , following the links | |
| 00:39 | that I'll put up to have a look at this | |
| 00:41 | next particular bit , because it does save a bit | |
| 00:43 | of work , but I think it's really good first | |
| 00:45 | off to understand how they work . Now . First | |
| 00:49 | off about to algebraic equations here and you have noticed | |
| 00:53 | that first off , a couple of things about them | |
| 00:56 | , algebraic equations are , there's a couple of things | |
| 00:58 | . First off we have these letters in it , | |
| 01:00 | these are known as variables , so that X is | |
| 01:02 | here and the wives are known as variables . This | |
| 01:05 | goes for any algebraic equation , whether no matter what | |
| 01:07 | the X , Y , Z as bees ends or | |
| 01:09 | whatever . Those particular letters are called variables . And | |
| 01:12 | we have these particular ones , these numbers in front | |
| 01:15 | of the variables called coefficients . Where we actually don't | |
| 01:18 | write a number in front of the variable . That's | |
| 01:21 | assumed to be a one . Okay , I think | |
| 01:23 | that's the basics . I think that should be able | |
| 01:25 | to start you off . So we have variables coefficients | |
| 01:29 | . Okay , some variables here and some coefficients . | |
| 01:32 | Okay , now the way that we solve simultaneous equations | |
| 01:36 | is this we pretty much just have to stuff around | |
| 01:40 | with entire equations to make the coefficient of a particular | |
| 01:44 | variable say X or why ? The same example here | |
| 01:48 | . Say we were trying to make the coefficient here | |
| 01:50 | the same . We might actually multiply an entire equation | |
| 01:54 | by two or three or four to make it the | |
| 01:56 | same as this one here . In this case we | |
| 01:58 | would multiply this Equation by two . The whole lot | |
| 02:02 | of it . Well , this is what we get | |
| 02:05 | . We'd end up with the following and I'm gonna | |
| 02:08 | just number these equations so we don't get thrown . | |
| 02:11 | So what we do is we do this . So | |
| 02:14 | multiplying equation too by two . Okay , so I'm | |
| 02:20 | going to end up with this particular set of equations | |
| 02:25 | . Okay , we're gonna end up with X times | |
| 02:29 | two . It's two X . Plus Yeah , three | |
| 02:36 | y times two is six Y . Then 19 times | |
| 02:44 | two is 38 . Okay , you're good with that | |
| 02:49 | . Hopefully you are . Then what I'm gonna do | |
| 02:51 | is I'm going to our we're now going to do | |
| 02:55 | this thing called elimination . So you got to notice | |
| 02:57 | I've actually got the coefficients in front of X . | |
| 02:59 | The same for equation one and equation too . Do | |
| 03:03 | you notice that ? Yeah , I'm not actually even | |
| 03:06 | uh just rewrite equation one here , just to start | |
| 03:10 | off with just I'm not gonna do this necessarily every | |
| 03:13 | single time , but just to try and let you | |
| 03:16 | work out what's going on . So again , equation | |
| 03:18 | one , I'm just rewriting out is two X Plus | |
| 03:21 | five y equals 33 . Okay , No , now | |
| 03:28 | what we do now is we have these coefficients at | |
| 03:31 | the same in front of here , we can literally | |
| 03:33 | start taking the coefficients off one another . Now it's | |
| 03:36 | really worth at this stage having a look at the | |
| 03:39 | variables off from one another and it's the way we | |
| 03:41 | do this is really worth having a look at the | |
| 03:42 | moment , you've got these two , the same look | |
| 03:44 | at these variables over here . Now you're going to | |
| 03:46 | see this one here is bigger than this one here | |
| 03:50 | . So usually what I do is with this , | |
| 03:52 | this is just something you got used to get used | |
| 03:54 | to with it . The one has a smaller coefficient | |
| 03:57 | here , you change all of these By minus enter | |
| 04:02 | , minus as I show you . I mean it's | |
| 04:03 | the same as times this by -1 . If you | |
| 04:06 | were to do this times this entire equation by -1 | |
| 04:10 | , this should become -2 acts . This here would | |
| 04:13 | become Get rid of that and become -5y . This | |
| 04:17 | year would become -33 . Do you see that ? | |
| 04:21 | Okay . Hopefully I haven't confused you too much by | |
| 04:25 | now . But what I'm gonna do is I'm gonna | |
| 04:26 | take equation one now from equation to by putting them | |
| 04:30 | together . So this is what we're going to end | |
| 04:33 | up with two X . Take away two X . | |
| 04:37 | Is zero . See that two X take away to | |
| 04:39 | exile , putting these two equations together . Now then | |
| 04:43 | what we do is we go six Y . Take | |
| 04:45 | away five Y . six , White Take Away five | |
| 04:48 | Y . We are left with Y . Okay positive | |
| 04:51 | . Why Equals 38 take away 33 . We end | |
| 04:56 | up with five . So that's the first part of | |
| 04:59 | our answer . Okay , I'll write it up over | |
| 05:02 | here . The first thing we have worked out is | |
| 05:05 | that why Equals five . That's the first part of | |
| 05:10 | our equation solved . Okay so I'm going to get | |
| 05:14 | rid of all this working out that we've done so | |
| 05:15 | far . We've ended up with the first bit of | |
| 05:20 | information that we want that y equals five . Then | |
| 05:23 | what we do is we go in and we start | |
| 05:24 | substituting values . Now substituting values means that where we | |
| 05:28 | see this , why we can pick one of these | |
| 05:30 | equations and we can instead of having why there we | |
| 05:32 | can actually make it into we can we can put | |
| 05:35 | five there instead . So I'm going to do it | |
| 05:36 | for this number down . I'm gonna do it for | |
| 05:37 | this equation down here . Okay for equation too this | |
| 05:41 | is what we're going to end up with . So | |
| 05:44 | X . Is still X . But instead now writing | |
| 05:46 | three , why we can think of this as being | |
| 05:49 | three Times 5 ? Yeah . So x plus 15 | |
| 05:54 | equals 19 . Okay , what number do you add | |
| 06:00 | to 15 to get 19 ? It's four . Okay | |
| 06:03 | so X Equals four and that's the second part of | |
| 06:08 | our answer . Why it was five X equals four | |
| 06:13 | . Yeah hopefully you get that . Hopefully it's not | |
| 06:15 | too bad . Uh There is an again an easier | |
| 06:18 | way of doing these . Uh I tell you now | |
| 06:20 | but I think it's really really worth going through and | |
| 06:23 | getting these really nailed down because there are also some | |
| 06:26 | variants on these and it's just a good thing to | |
| 06:28 | have nailed down . So what about I give you | |
| 06:31 | another example ? Alright this might take a few examples | |
| 06:35 | to get used to if you're you haven't done these | |
| 06:38 | before or if you struggle with these . Okay so | |
| 06:40 | the next example I'm going to give you , what | |
| 06:42 | about this one ? We have three X plus three | |
| 06:47 | Y . And that equals 18 and we'll have a | |
| 06:53 | second equation which is X plus seven Y . And | |
| 07:00 | that is going to equal 30 . Okay , there | |
| 07:05 | will be a fair few examples with this but I'm | |
| 07:06 | going to start getting faster and faster at working these | |
| 07:09 | out . Okay . And this will be an example | |
| 07:11 | where I start doing that we have to make again | |
| 07:13 | I'm gonna stuff around with this , this is the | |
| 07:15 | easiest variable stuff around with this . These are a | |
| 07:18 | bit yucky . You know , we can't we have | |
| 07:19 | to multiply it by some horrible number . We're going | |
| 07:21 | to get to this sort of example . We would | |
| 07:22 | do this a bit later but it's just as easy | |
| 07:24 | here to multiply as you might work out this equation | |
| 07:27 | by three . So if we are to do that | |
| 07:30 | what we end up with this equation , one this | |
| 07:32 | equation to on equation to what we end up with | |
| 07:35 | . And we multiply that all by three is we | |
| 07:37 | get the following we get three X . Okay because | |
| 07:41 | your time is up by three plus seven times three | |
| 07:43 | which is 21 Y equals . And you probably work | |
| 07:48 | this part out already three times 30 . It's naughty | |
| 07:53 | . Okay , equation one by the way is going | |
| 07:56 | to stay very very similar because it's three X plus | |
| 08:00 | three Y Equals eight . And we're happy with that | |
| 08:03 | because again the coefficients here in front of this variable | |
| 08:06 | is now the same . So we don't need to | |
| 08:08 | actually change it . Now . The next thing I | |
| 08:10 | said is you look at the look at the variables | |
| 08:12 | over here coefficients in front of him , this one's | |
| 08:15 | a smaller one . So these ones we're going to | |
| 08:17 | multiply this equation by -1 . Okay . Times that | |
| 08:21 | by -1 . So we're gonna end up with this | |
| 08:24 | is a minus . We're going to end up with | |
| 08:27 | this is a modest , so we're going to end | |
| 08:28 | up with this is a modest . Okay so let's | |
| 08:32 | see what we end up with . Okay , three | |
| 08:34 | X . Take away three X . We end up | |
| 08:36 | with zero 21 , Why take away three Y . | |
| 08:41 | We end up with 18 , why ? And then | |
| 08:46 | 90 takeaway 18 went up with the answer of 72 | |
| 08:52 | . Okay , what number of times by 18 To | |
| 08:56 | get 72 ? Well hopefully you can go 72 divided | |
| 09:00 | by 18 and you find out the following that why | |
| 09:04 | is indeed equal ? I'm just going to ride up | |
| 09:06 | over here , why equals four . Okay so let's | |
| 09:12 | rob these ones out and let's now substitute in these | |
| 09:21 | values into one of the equation I'm going to put | |
| 09:23 | it into I think this equation here , so for | |
| 09:27 | equation one substituted more values three X is going to | |
| 09:31 | stay the same plus three y three y three times | |
| 09:37 | four it's 12 and vehicles 18 . Now get I | |
| 09:42 | am assuming a bit of knowledge already with this but | |
| 09:46 | what we can do now is we can get rid | |
| 09:48 | of this 12 by taking it away , we have | |
| 09:50 | to take it away from this side as well . | |
| 09:52 | So we end up with the equation where three x | |
| 09:56 | We're taking 12 off . Both sides would take 12 | |
| 09:59 | . We're going to add equal six . Okay and | |
| 10:03 | then what we get Is the three x equals six | |
| 10:06 | x equals two . So why ? It was four | |
| 10:10 | X equals two . Okay , how did you go | |
| 10:15 | with that ? What about to go through another example | |
| 10:17 | . Now I want to recommend with these if you're | |
| 10:20 | somewhat struggling or even if you're going okay what you | |
| 10:22 | might do want to write them up , give them | |
| 10:24 | a go answering yourself and then see how you went | |
| 10:27 | . Okay , So what about this one ? This | |
| 10:29 | one is going to be a slightly different one because | |
| 10:33 | then I might have to give you a bit of | |
| 10:34 | a heads up on how you might do this , | |
| 10:35 | but You might be okay with this two , X | |
| 10:39 | plus two Y . And that equals 16 . We're | |
| 10:45 | gonna have X take three Y And that equals four | |
| 10:52 | . Yeah . You notice here the biggest difference we've | |
| 10:54 | got is we have a negative sign here . We | |
| 10:57 | will deal with that a little bit later . Okay | |
| 10:59 | , again , when you get these coefficients , just | |
| 11:01 | make sure that one of them is a negative and | |
| 11:03 | one positive , by the way , it probably actually | |
| 11:05 | might stuff around with on this one too . I'd | |
| 11:08 | go for and give it a go . So hopefully | |
| 11:11 | if you've pause the video and you're giving it a | |
| 11:13 | bit of a go and you're giving it a go | |
| 11:15 | . Let's solve this . So you might have got | |
| 11:18 | I'm going to stuff around with this particular one here | |
| 11:20 | because , you know , this is easy just to | |
| 11:22 | multiply this entire equation by two . So for equation | |
| 11:26 | to what we end up with is following , multiplying | |
| 11:30 | by two , we end up with two X . | |
| 11:32 | Take away six Y equals eight . All right , | |
| 11:36 | And this equation one stays the same . So we | |
| 11:40 | end up with two X plus two Y equals 16 | |
| 11:46 | . Okay , now we're going to notice here that | |
| 11:48 | we are again , we're gonna change the sign here | |
| 11:51 | . You notice this ? Okay . Um Let's do | |
| 11:55 | that . I think the easiest way to do this | |
| 11:58 | is as follows . What about we are because you | |
| 12:01 | might be thinking well , which one do we change | |
| 12:03 | ? Okay , I'll tell you what we'll do is | |
| 12:05 | not going to change the negative sign in front of | |
| 12:07 | the sea again . I think it's a bit of | |
| 12:10 | an easy one . This is something that comes out | |
| 12:12 | of the experience . Um so let's just multiply this | |
| 12:15 | entire equation by -1 . This has become minus , | |
| 12:18 | this becomes a plus and this becomes a minus . | |
| 12:22 | Okay , two X . Take away two X zero | |
| 12:28 | plus two Y plus six Y . We're now adding | |
| 12:32 | them because it's a positive and a positive . We | |
| 12:34 | end up with eight Y . How did you get | |
| 12:37 | that one ? Would you you fall for it 16 | |
| 12:40 | takeaway ? Eight is eight . So eight y equals | |
| 12:45 | eight . Maybe you could probably guess what why is | |
| 12:48 | going to be , why is going to be equal | |
| 12:52 | to one ? Did you get that ? So what | |
| 12:55 | we'll do is I'll rub these out and I'll do | |
| 12:56 | a bit of substitution . Okay , let's substitute in | |
| 13:02 | . Um Let's substitute into let's choose one of them | |
| 13:05 | . What about we choose equation uh Two might be | |
| 13:08 | the easiest one here . X . Take away three | |
| 13:13 | Y . Okay . Three times Y . Is three | |
| 13:16 | Equals four . What number do you if you would | |
| 13:21 | actually what nobody X . Take away ? Three calls | |
| 13:23 | for ? What number Can you subtract three from to | |
| 13:26 | get four X equals seven . So hopefully you're going | |
| 13:32 | pretty good with these . Okay , so the equation | |
| 13:35 | I'm going to put up now , is this 12 | |
| 13:37 | X . Take away three Y equals zero . That's | |
| 13:44 | gonna be equation one and then three X . Take | |
| 13:48 | away plus why equals 22 . And that will be | |
| 13:54 | So you're trying to give it a go . All | |
| 13:57 | right . So what are we going to do now | |
| 14:00 | to solve this equation ? I do the following . | |
| 14:02 | I'd actually this time I'd stuff around with this particular | |
| 14:06 | the wire here . Okay . And what you might | |
| 14:09 | do actually going to do it slightly differently ? You | |
| 14:12 | quit times just by three here , but I'm gonna | |
| 14:13 | do it slightly differently . You get chimes this equation | |
| 14:16 | to get these the same by three and this equation | |
| 14:20 | by two , because then we're gonna end up with | |
| 14:22 | this is a six and this is a six , | |
| 14:23 | so let's do that . Okay , so what we're | |
| 14:26 | going to end up with is six X . Okay | |
| 14:29 | , This first equation take away three times three is | |
| 14:32 | nine Y . And it equals zero because nothing times | |
| 14:35 | three is nothing and then three times three X times | |
| 14:40 | two is six X . Plus two . Why ? | |
| 14:45 | And it equals 44 . Okay , So now what | |
| 14:49 | do we do ? We then take one off the | |
| 14:51 | other ? So the one I'm going to stuff around | |
| 14:53 | with is I'm going to make this one here negative | |
| 14:57 | . All these negative rather than actually changing around too | |
| 15:00 | much I want to go that becomes minus that becomes | |
| 15:02 | A plus . This doesn't change because zero is zero | |
| 15:04 | . So six X . Take away six X . | |
| 15:07 | Is nothing . Uh to Y nine Y . We | |
| 15:12 | get 11 while we're getting those together and it equals | |
| 15:15 | 44 . So if 11 wires 40 for you can | |
| 15:19 | probably guess and hopefully you've got the answer of why | |
| 15:23 | being four . Okay , well substitute the values in | |
| 15:29 | . Try this a little bit quicker . I just | |
| 15:31 | uh family , family coming home . Just what you | |
| 15:35 | want to hear . Okay so what we are get | |
| 15:37 | is this So we substitute these values in uh we're | |
| 15:43 | going to do it and do the equation two years | |
| 15:45 | . So we end up with three X . Plus | |
| 15:49 | Why ? So why is four and equals 22 ? | |
| 15:53 | So taking four off both sides , we take four | |
| 15:56 | of here , we can take four of here . | |
| 15:58 | We end up with three x equals 18 . So | |
| 16:04 | X equals six . Okay one last uh one last | |
| 16:13 | one of these . Let's have a good so one | |
| 16:20 | last one of these and that will be it . | |
| 16:24 | I would say so . And then I'll make the | |
| 16:26 | other video . Probably not straight away anyway but it | |
| 16:29 | would be pretty quickly . So the last thing we | |
| 16:31 | do is probably the hardest one . And it's uh | |
| 16:34 | we're gonna be stuffing around with . You don't have | |
| 16:36 | to change too . Lots of them seven X . | |
| 16:39 | Plus three Y . And it equals 33 . And | |
| 16:45 | then we end up with three X . Plus five | |
| 16:50 | Y . And that equals 29 . So equation one | |
| 16:55 | equation to , we're going to multiply this equation here | |
| 16:59 | . Okay . They're looking at to give it a | |
| 17:01 | go . Yeah . Hopefully what you did is I'm | |
| 17:06 | going to do this and what I'm gonna do is | |
| 17:08 | I'm gonna change around for X . I'm gonna multiply | |
| 17:10 | this equation by three . Because I'm going to end | |
| 17:12 | up with 21 , I'm going to multiply this equation | |
| 17:15 | by seven . Okay so what I end up with | |
| 17:17 | this equation one equation too . Okay 21 x plus | |
| 17:23 | nine why equals 99 And 21 x Plus 35 Y | |
| 17:32 | Equals What's 29 times seven . Hopefully you can work | |
| 17:35 | that one out . Hopefully if you work that out | |
| 17:37 | you get the answer of 203 . So I'm going | |
| 17:43 | to change the signs here . Um gonna change it | |
| 17:45 | on this one again , it's a smaller one . | |
| 17:47 | So this will become minus , this will become minus | |
| 17:50 | and this will become minus . So 21 take away | |
| 17:53 | 21 is nothing . Okay 35 Y . Take away | |
| 17:58 | nine y . It's 28 why Equals 203 take away | |
| 18:06 | 99 is 104 . So 104 divided by 28 will | |
| 18:13 | give us our answer . Okay ? So hopefully the | |
| 18:16 | answer you're going to get when you do that is | |
| 18:19 | you will get why being equal before the substitute values | |
| 18:26 | in and this is what we'll get okay ? Um | |
| 18:32 | So I'm gonna substituting in down here in the equation | |
| 18:34 | to this time . Um What will end up with | |
| 18:37 | is three X plus five . I say five times | |
| 18:41 | four is 20 and equals 29 . So taking 20 | |
| 18:45 | off both sides , we're gonna end up with three | |
| 18:47 | X equals nine because we're taking this 20 off both | |
| 18:51 | sides . Therefore X Equals three . Now the link | |
| 18:56 | I'm going to put up and I've been putting up | |
| 18:57 | throughout this video is going to show you a different | |
| 18:59 | way of working these hours . Okay . It's a | |
| 19:01 | way that you can actually are , not stuff around | |
| 19:03 | with this so much of having to multiply this and | |
| 19:06 | multiply that uh , as much as we do here | |
| 19:08 | , it's a way of really , really quickly working | |
| 19:10 | these out anyway . I hope that was some help | |
| 19:13 | anyway , See you next time . |
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