06 - Solve Quadratic Systems of Equations by Substitution - Part 1 (Simultaneous Equations) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra . The title of | |
| 00:02 | this lesson is we're gonna solve quadratic systems of equations | |
| 00:05 | by substitution . This is part one and we have | |
| 00:07 | several parts with more complex problems . Now in the | |
| 00:10 | last lesson we introduce the concept of quadratic systems . | |
| 00:12 | It's a system of equations that have these comic sections | |
| 00:15 | involved which are all quadratic in nature . So we | |
| 00:17 | can have a circle and ellipse or we can have | |
| 00:20 | a parabola and hyperbole to . We can also have | |
| 00:22 | lines even though lines aren't quadratic . We can have | |
| 00:24 | those running around our system as well and we can | |
| 00:27 | have either no intersection points , no solutions or one | |
| 00:30 | or two or three or even up to four intersection | |
| 00:33 | points of the real solutions here . So when we | |
| 00:35 | solve these problems we're going to get sometimes no solution | |
| 00:38 | at all and sometimes we'll get one intersection point or | |
| 00:40 | two or three or four . And the reason is | |
| 00:42 | because of the geometry of how well we graph the | |
| 00:44 | things how many crossing points it has . And we've | |
| 00:46 | talked about that in the last lesson . So to | |
| 00:48 | solve these things mathematically , there's two techniques . The | |
| 00:51 | first one really is called substitution . That's we're gonna | |
| 00:53 | learn now later on , we're going to talk about | |
| 00:55 | addition . So for substitution , what we have to | |
| 00:57 | do is take one of the equations solve for variable | |
| 01:01 | and put it into the other equation . It's the | |
| 01:03 | same thing we debate lines . Let's take an example | |
| 01:05 | . It'll be much , much easier to understand . | |
| 01:07 | Let's say our quadratic system is X squared minus Y | |
| 01:12 | . Is equal to five and two eggs plus why | |
| 01:18 | Is equal to three . Now , before you solve | |
| 01:20 | anything , sometimes it's nice to know what you're looking | |
| 01:22 | at what you have . What is this equation this | |
| 01:25 | bottom one look like ? Well there's no square on | |
| 01:27 | the X . And there's no square on the Y | |
| 01:28 | . So , you know , it has to be | |
| 01:29 | a line . If there's no squares anywhere , it | |
| 01:31 | has to be a line . So that's some kind | |
| 01:33 | of line . I don't know how it shifted yet | |
| 01:34 | because I haven't plotted it , but I know that | |
| 01:36 | it's a line , I don't know what what the | |
| 01:38 | slope is or anything until I do more work on | |
| 01:40 | it . But I know it's online . What does | |
| 01:42 | this one look like ? It's an X . Term | |
| 01:44 | that squared , but the white term is not squared | |
| 01:46 | . And so we've done enough of these to know | |
| 01:48 | that that has to be a parabola right ? Like | |
| 01:50 | Y equals X squared . The Y terms not squared | |
| 01:52 | , but the X term is so that's a parabola | |
| 01:54 | . I don't know if it's upside down or sideways | |
| 01:57 | or anything else because I haven't done any work on | |
| 01:59 | it . But I know it's a Parabola plus a | |
| 02:00 | line . And when you think back for Parabola plus | |
| 02:03 | a line , you can have no solutions . If | |
| 02:04 | they don't intersect , you can have two solutions , | |
| 02:08 | you can have one solution . I don't think you | |
| 02:09 | can really have three solutions of a problem plus the | |
| 02:12 | line . So in other words , that's the geometry | |
| 02:15 | of what we have now . How do we figure | |
| 02:16 | out what the solutions are ? We have to solve | |
| 02:19 | something in one of these equations and put it into | |
| 02:22 | the other one . Now we have choices here . | |
| 02:23 | I mean there's no one way to do this . | |
| 02:26 | There's actually lots of ways to do it . But | |
| 02:28 | I see that I have a why here and why | |
| 02:30 | here ? So I want to solve for y and | |
| 02:32 | put it into the other equation . Now I have | |
| 02:34 | a choice , I can solve this equation for why | |
| 02:36 | and then once I have that I can stick it | |
| 02:38 | into this location and then proceed , or I can | |
| 02:41 | solve this equation for why . And I can plug | |
| 02:44 | it in up to here and solve . Now of | |
| 02:46 | course I could solve for X and put it in | |
| 02:47 | here . There's nothing wrong with that . It's just | |
| 02:49 | that because X is squared . If I solve this | |
| 02:51 | thing for X , I'm gonna have to stick it | |
| 02:53 | in , I'm gonna have to square it , then | |
| 02:55 | I have to do a squaring operation . It's gonna | |
| 02:56 | be a little more involved to solve it . But | |
| 02:58 | you will get exactly the same answer . So if | |
| 03:00 | you saw literally for any variable in any equation and | |
| 03:04 | then stick it into the other one , you will | |
| 03:05 | get the right answer . But some paths are going | |
| 03:08 | to be easier than others . And I know from | |
| 03:10 | doing this and looking at it that solving one of | |
| 03:12 | these things for why . Uh either this one or | |
| 03:15 | this one and putting it in is gonna be a | |
| 03:16 | little easier easier . So let's take this guy and | |
| 03:19 | solve for why ? So why is going to be | |
| 03:22 | if we move the two X . Over it'll be | |
| 03:24 | negative two X . Or you can write it as | |
| 03:26 | three minus two X . So now we know that | |
| 03:28 | why has to equal this quantity . So then what | |
| 03:31 | I wanna do is I want to take and I | |
| 03:34 | want to plug it into this this equation right here | |
| 03:38 | when I write P . I . That means plug | |
| 03:41 | in I'm gonna write that down a lot . Or | |
| 03:43 | you could write sub . If you want to write | |
| 03:44 | substitution , what that means is I'm taking this and | |
| 03:46 | I'm putting it into this location . So what I'm | |
| 03:49 | gonna then have is X squared minus Y . But | |
| 03:52 | why is now this whole big thing ? And you | |
| 03:54 | have to be really careful . You don't want to | |
| 03:56 | write it like this three minus two , X equals | |
| 03:59 | five . This is wrong . Why ? Because in | |
| 04:02 | this equation it was x squared minus the thing that's | |
| 04:04 | called . Why the thing that's called , Why is | |
| 04:06 | this whole thing in here ? So this whole thing | |
| 04:09 | has to be wrapped in parentheses and you see it | |
| 04:11 | will make a difference because when you wrap the princes | |
| 04:13 | now this is going to be distributed in , it's | |
| 04:15 | gonna flip the sign here . If I don't have | |
| 04:18 | any parentheses there , then I have an incorrect sign | |
| 04:21 | right out of the gate and I'm going to get | |
| 04:22 | the wrong answer . So when you substitute in its | |
| 04:25 | probably a good idea to rap whatever you're substituting in | |
| 04:27 | inside of parentheses as a unit because as a unit | |
| 04:31 | , that's what Y . Is equal to . And | |
| 04:33 | then you just let the rest of the math take | |
| 04:34 | care of itself . So here you have X squared | |
| 04:38 | minus three plus two , X equals five . Just | |
| 04:42 | distribute the negative one in . Now we cannot really | |
| 04:45 | add these because they're different powers of X , but | |
| 04:47 | we can arrange them appropriately , X squared plus two | |
| 04:50 | X . And then we have the -3 is equal | |
| 04:54 | to five . How do we solve this ? We've | |
| 04:56 | solved equations like this many times . We have to | |
| 04:58 | move the constant over here . So we have X | |
| 05:00 | squared plus two X . When we subtract five it'll | |
| 05:03 | be minus eight equals zero . Now you have a | |
| 05:07 | bunch of options here , this is just a quadratic | |
| 05:09 | equation . You could use the quadratic formula . There's | |
| 05:11 | no shame in using the quadratic formula . In fact | |
| 05:14 | , sometimes you have to use the quadratic formula . | |
| 05:16 | You could also use completing the square to make it | |
| 05:18 | fact arable . But the first thing you should always | |
| 05:21 | try is just try to factor the thing . So | |
| 05:23 | we're gonna open up our parentheses , set it equal | |
| 05:25 | to zero . We have X times X . Giving | |
| 05:28 | us X squared for this . We have two times | |
| 05:31 | for giving us eight . And then we take a | |
| 05:32 | look at the signs and realize the only way it's | |
| 05:34 | going to work is with a minus and a plus | |
| 05:36 | . Because negative two times four is negative eight . | |
| 05:39 | This will give me negative two X . This will | |
| 05:41 | give me positive four X . We add them together | |
| 05:43 | . We're gonna get this . So that was nicely | |
| 05:46 | fact herbal . So then what this tells me is | |
| 05:49 | that X is equal to two because we set this | |
| 05:51 | equal to zero . X is equal to two and | |
| 05:53 | then X is equal to negative four because we set | |
| 05:55 | this one equal to zero . So you see now | |
| 05:57 | I know something about the solution . The solution I | |
| 06:00 | know what the X . Values are . X . | |
| 06:02 | Has to be two and X has to be negative | |
| 06:04 | for but I want points X . Comma Y . | |
| 06:08 | So how do I find the Y . Values after | |
| 06:10 | I have the X . Values in my possession . | |
| 06:14 | Then I start plugging in to either one of the | |
| 06:19 | original equations . It doesn't matter which one I pick | |
| 06:21 | . I could stick this X value in here and | |
| 06:23 | calculate why or I could stick this X . Value | |
| 06:26 | in here and calculate why . But I'm actually gonna | |
| 06:29 | stick it in this because I got this equation from | |
| 06:31 | solving this one , it's the same equation . So | |
| 06:34 | I'm gonna substitute as why is 3 -2 x . | |
| 06:37 | I'm gonna put the X . Value in 3 -2 | |
| 06:40 | times two . So it's going to be 3 -4 | |
| 06:44 | which is negative one . I'm gonna plug this one | |
| 06:48 | into the same equation . Why is 3 -2 x | |
| 06:51 | . Why is 3 -2 times negative 4 ? Right | |
| 06:56 | ? So then why is three plus eight ? Because | |
| 06:59 | this becomes a positive eight . And so why is | |
| 07:02 | 11 right ? 11 then ? Now you have the | |
| 07:06 | X . Value and a Y . Value index value | |
| 07:08 | . And why they noticed that this X goes with | |
| 07:10 | this , why it doesn't go with this . It | |
| 07:12 | came directly from this and this one came directly from | |
| 07:15 | this . So the solution There's actually two solutions X | |
| 07:21 | comma Y two comma -1 and X comma Y negative | |
| 07:27 | for comma 11 . This is what you circle on | |
| 07:30 | your paper . Let me double check myself to common | |
| 07:31 | negative one negative four comma 11 . This process of | |
| 07:35 | substitution is what you do for every single one of | |
| 07:39 | these problems . You just pick an equation , solve | |
| 07:41 | for a variable . It doesn't even matter which one | |
| 07:43 | you solve for . But you got to pick something | |
| 07:45 | solve for that variable , Take what you have and | |
| 07:48 | plug it into the other equation . You follow the | |
| 07:50 | solution all the way to where you get down to | |
| 07:52 | where you saw for the variables . For some either | |
| 07:55 | extra wide depending on what you picked . Then you | |
| 07:57 | take those answers and you have to substitute them back | |
| 07:59 | and it doesn't matter what equation you pick because it's | |
| 08:02 | a system of equations . So no matter which one | |
| 08:04 | I plug it , I'm looking for the intersection point | |
| 08:06 | . So it doesn't matter which when I plug it | |
| 08:08 | into back up here , I chose this one because | |
| 08:12 | there's less work for me if I put it in | |
| 08:13 | here , like if I put X in here and | |
| 08:15 | have to square it and all this other stuff , | |
| 08:17 | if I put it in here and have to multiply | |
| 08:19 | by two and move it over . It's not so | |
| 08:21 | hard , but this one is even easier because it's | |
| 08:23 | already solved for a while , so I'm going to | |
| 08:24 | use that one . Okay , Everything else from here | |
| 08:28 | on out is just simply making the problem is more | |
| 08:30 | complex . Notice that in the beginning here we figured | |
| 08:33 | out that this was a parabola in a line . | |
| 08:35 | We had to solutions and we talked about this in | |
| 08:37 | the last lesson . It might look something like this | |
| 08:39 | with two intersection points here . Now , I haven't | |
| 08:42 | looked to see this is not the graph of this | |
| 08:44 | . I don't know if it's upside down which way | |
| 08:45 | the line is tilted . I'm just showing you that | |
| 08:47 | the line plus a problem can have two solutions , | |
| 08:49 | which is what we have figured out there . Yeah | |
| 08:52 | . All right . Now , the next problem is | |
| 08:56 | like this , why is equal to x squared ? | |
| 08:59 | And then the other equation is x squared plus y | |
| 09:03 | squared Is 12 . And again , it's useful to | |
| 09:07 | think about what you have . I have a parabola | |
| 09:09 | and I have a circle . So parable in a | |
| 09:12 | circle , can have lots of different intersection points . | |
| 09:14 | If the parabola just touched the top of the circle | |
| 09:17 | , that could only have one intersection point . If | |
| 09:19 | the problem , the circle never crossed at all , | |
| 09:21 | it would be zero intersection points , no solutions . | |
| 09:23 | If the parabola intersect and goes down into the circle | |
| 09:27 | and comes out , I can have two solutions . | |
| 09:28 | Or it could go all the way through the circle | |
| 09:30 | and back up again giving me four solutions . So | |
| 09:32 | uh , you know , and there are probably other | |
| 09:34 | possibilities out there . Maybe there is impossible to come | |
| 09:36 | down and touch the other side in one spot and | |
| 09:38 | giving me three solutions . So the point is is | |
| 09:40 | it's it's useful to look at but you still can't | |
| 09:43 | predict what the answer is going to be just from | |
| 09:44 | knowing what you have . I have to pick something | |
| 09:47 | to solve for . But in this case the first | |
| 09:49 | equation is already solved for why ? So it would | |
| 09:51 | be silly not to take advantage of that . So | |
| 09:53 | what we're gonna do is we're gonna plug this in | |
| 09:55 | . So that's what P . I means . All | |
| 09:57 | right , we're gonna plug it into here . It | |
| 09:59 | says X squared plus Y squared . But we now | |
| 10:03 | know that . Why is this ? So we wrap | |
| 10:05 | it in parentheses . That's what I'm gonna teach you | |
| 10:06 | to do . See , I just took what ? | |
| 10:09 | Why was put it in princes ? And I still | |
| 10:11 | have to square because why is squared ? So X | |
| 10:13 | squared squared like this ? So I'm going to have | |
| 10:15 | X squared plus X to the fourth is 12 . | |
| 10:19 | And you say , oh my gosh , how do | |
| 10:20 | I solve that ? So let's rearrange everything X to | |
| 10:23 | the fourth plus X squared minus 12 is equal to | |
| 10:27 | zero . All I did was move the 12 over | |
| 10:29 | . And you might say , how do I do | |
| 10:30 | that ? There's different ways to solve it . But | |
| 10:33 | actually we have solved problems like this . Even though | |
| 10:35 | there's an X to the fourth , we can still | |
| 10:37 | attempt to factor it right ? We can say I | |
| 10:41 | can have an X squared times and X square those | |
| 10:43 | multiply to give me X to the fourth tower . | |
| 10:46 | Okay . And then for the 12 , I can | |
| 10:49 | go through my list and land on three times four | |
| 10:51 | and I can land on minus and plus . And | |
| 10:53 | we should check ourselves . So the negative three times | |
| 10:56 | positive for is negative 12 . This gives me negative | |
| 10:59 | three X squared . This gives me positive for X | |
| 11:01 | squared which add to give me this . So this | |
| 11:03 | is the factor form of this . But the interesting | |
| 11:05 | thing is that now I know that this has to | |
| 11:09 | be equal to zero , X squared minus three is | |
| 11:11 | equal to zero . And this means that X squared | |
| 11:14 | plus four has to be equal zero . So you | |
| 11:17 | see , I'm actually gonna get to solutions from this | |
| 11:19 | side and also to solutions possibly two solutions from this | |
| 11:22 | side . Let's see what we actually have . If | |
| 11:25 | I saw this guy it's going to be X squared | |
| 11:27 | is equal to three , Move it over and then | |
| 11:30 | X is going to be plus or minus the square | |
| 11:32 | root of three . So I have actually two solutions | |
| 11:34 | . I have X is equal to the square root | |
| 11:36 | of three and I also have X is equal to | |
| 11:39 | negative square root of three . Now don't be frightened | |
| 11:41 | by the radicals . The radical just means it's just | |
| 11:44 | a number , it's not it's not rational but it's | |
| 11:46 | a number . What about this 1 ? X squared | |
| 11:49 | is equal to negative four . X is going to | |
| 11:52 | be equal to plus or minus the square root of | |
| 11:53 | negative four . And then I say oh this is | |
| 11:55 | going to be an imaginary number . Of course I | |
| 11:57 | can tell you it's too I plus or minus two | |
| 11:59 | . I but remember back when we talked about comic | |
| 12:02 | sections , we're only looking for the real solutions so | |
| 12:05 | it tells me that there's an imaginary solution here but | |
| 12:07 | that doesn't correspond to an actual intersection point so we | |
| 12:11 | toss it aside so we say you can just write | |
| 12:13 | on your paper , not real . So we don't | |
| 12:17 | pursue this any further because it already gave us an | |
| 12:19 | X value that was imaginary . We're only looking for | |
| 12:22 | the real numbers which we have here . Now I | |
| 12:26 | have to take the square root of three and I | |
| 12:27 | have to substitute it back into one of my original | |
| 12:29 | equations . I could put it in here if I | |
| 12:32 | want . But this one is even easier . It's | |
| 12:34 | already solved for why ? So I'm going to plug | |
| 12:36 | in . Why is able to X squared ? And | |
| 12:40 | I'm gonna put this value in here . It's going | |
| 12:42 | to be the square root of three . That's the | |
| 12:43 | thing that is being squared . So that why is | |
| 12:45 | gonna be equal to the square and square ? It's | |
| 12:47 | going to cancel give me three . I'll take this | |
| 12:50 | guy and plug in same thing . Why is equal | |
| 12:52 | to X squared and then why is equal to negative | |
| 12:55 | square root of three ? That is what is going | |
| 12:57 | into the X . Location . The whole thing has | |
| 12:59 | to be wrapped in parentheses and you square it . | |
| 13:01 | But when you square the negative it becomes positive . | |
| 13:04 | And then when you square the radical , the radical | |
| 13:06 | disappears . So you also get y . Is equal | |
| 13:08 | to three . You get exactly the same answer . | |
| 13:10 | But it doesn't mean that you have wasted your time | |
| 13:14 | here . This value of X goes with this value | |
| 13:17 | of why ? And this value of X goes with | |
| 13:19 | this value of why ? So there's actually two solutions | |
| 13:23 | Right ? Square it of three comma three . And | |
| 13:28 | also uh negative square root of three comma three . | |
| 13:33 | Right ? So it's square 23 coming three . A | |
| 13:36 | negative square 23 coming three CC . These are different | |
| 13:38 | points . Even though I got a duplicate for why | |
| 13:40 | this value of why was tied to this X . | |
| 13:42 | And this value of what I was tied to this | |
| 13:44 | X . So it actually is too unique points . | |
| 13:46 | So I haven't graft this thing , I don't want | |
| 13:50 | to mislead you or anything , but I know that | |
| 13:51 | this is a circle and I know that this is | |
| 13:53 | a parabola , right ? So it probably looks something | |
| 13:55 | like this . There's your circle , there's your problem | |
| 13:57 | , you have to intersection points and there you go | |
| 14:00 | . Right . Of course . Yeah . For this | |
| 14:03 | particular case , I know what X square looks like | |
| 14:05 | . I know the circle is centered on the origin | |
| 14:07 | . So it actually probably does look something like this | |
| 14:09 | and those are the answers that you would have . | |
| 14:11 | Now one thing I want to point out really , | |
| 14:13 | really important is what I did to start this problem | |
| 14:16 | is I took the Y value and I put it | |
| 14:18 | into here because it was already solved for why , | |
| 14:21 | but I want to show you something that is important | |
| 14:25 | for you to understand . So I'm going to kind | |
| 14:26 | of draw a little divider here and I'm gonna try | |
| 14:29 | to squeeze it in . Let's rewrite the system again | |
| 14:34 | , let's say why is equal to X squared ? | |
| 14:37 | And X squared plus Y squared is equal to 12 | |
| 14:40 | . I just substituted in because it was convenient . | |
| 14:43 | But notice I could go the other way . I | |
| 14:45 | could I have an X squared here and I have | |
| 14:47 | an X squared here . So if I wanted to | |
| 14:49 | instead of substituting for why , I could actually I | |
| 14:53 | know that X squared is equal to this . So | |
| 14:55 | I could actually substitute into the X squared location . | |
| 14:58 | You can make any valid substitution you want . So | |
| 15:01 | in other words , for the next step , I | |
| 15:03 | could change this . Uh I could substitute an end | |
| 15:06 | like this , right ? Um in such a way | |
| 15:10 | that the X squared location is actually going to be | |
| 15:13 | y plus Y squared is 12 . So you see | |
| 15:17 | what I did here is the why was equal to | |
| 15:20 | X squared ? So I put it in this location | |
| 15:21 | here , I'm saying I have X squared this identical | |
| 15:24 | terms , they're identical . That means this is equal | |
| 15:27 | to Y . And I can just put the Y | |
| 15:28 | in this location . So it's a little bit of | |
| 15:30 | weird substitution . But what you're going to get is | |
| 15:33 | why squared plus y minus 12 Is equal to zero | |
| 15:38 | . That's what you're gonna get . And we can | |
| 15:39 | try to notice it's a totally different polynomial than we | |
| 15:41 | got here . It's a polynomial . And why we | |
| 15:44 | can try to factor it and that's gonna be why | |
| 15:47 | ? Why ? Three and four ? Just double check | |
| 15:50 | . Make sure I haven't messed anything up . You're | |
| 15:52 | gonna have negative three and positive four multiply this , | |
| 15:56 | you get negative 12 . This is negative three . | |
| 15:58 | Y . This is positive for why ? So that | |
| 16:00 | gives you this . So this is the factored form | |
| 16:02 | of this . What does this tell you why is | |
| 16:05 | equal to positive three ? Why is equal to -4 | |
| 16:10 | ? Okay . And you're looking at this and you're | |
| 16:11 | trying to make it make sense with all this and | |
| 16:13 | you're like well I don't know if this actually is | |
| 16:15 | gonna work or not , so let's try to continue | |
| 16:18 | on , let's substitute substitute it into now . You | |
| 16:21 | can pick whatever you want . But I'm gonna pick | |
| 16:23 | the first equation , I wrote it as Y is | |
| 16:26 | equal to X square . But you can also write | |
| 16:27 | it as X squared is equal to Y . Because | |
| 16:30 | it's the same exact equation , I'm gonna put it | |
| 16:32 | in here as X squared is equal to three . | |
| 16:35 | I'm going to substitute this in here , right ? | |
| 16:37 | Then X is going to be plus or minus the | |
| 16:39 | square root of three , so X is going to | |
| 16:41 | be equal to square root of three and X is | |
| 16:45 | equal to negative square three . Yeah . All right | |
| 16:49 | , let me see if I can keep things separated | |
| 16:51 | here . What's going to happen on the other side | |
| 16:54 | ? I'm gonna stick this guy back into this equation | |
| 16:58 | , X squared is equal to y . I'm gonna | |
| 17:00 | put the Y in here , X squared is negative | |
| 17:02 | four and then X is plus or minus the square | |
| 17:06 | root again of negative force and this is not real | |
| 17:10 | . So I can discard everything here because it led | |
| 17:12 | to a solution that had an imaginary values . So | |
| 17:14 | then I go back to the side that did not | |
| 17:16 | have any imaginary values . And I figure out what | |
| 17:18 | do I have ? I have an x value of | |
| 17:20 | square root of three corresponding to a Y value of | |
| 17:24 | three . I have an X value of negative square | |
| 17:27 | root of three . Also corresponding to a Y value | |
| 17:30 | of three . This solution set is exactly the same | |
| 17:34 | as this one . It's exactly the same thing . | |
| 17:36 | Square 23 comma three square 233 negative square 23 common | |
| 17:40 | three negative square 23 times three . So I'm just | |
| 17:42 | showing you that . It doesn't matter what you solve | |
| 17:44 | for . If you pick something and stick it in | |
| 17:46 | and work through it , it's going to give you | |
| 17:47 | the same thing as if you pick something else and | |
| 17:49 | work through it . Even if your substitution is weird | |
| 17:52 | , I can I have an external X squared term | |
| 17:54 | and an X squared term . So in other words | |
| 17:56 | you don't have to solve for X or solve for | |
| 17:58 | y . You can solve for anything that is present | |
| 18:01 | in the other equation and substituted in and it still | |
| 18:04 | works . All right . All right . So cranking | |
| 18:10 | right along here we have for our next solution set | |
| 18:15 | of equations , X squared minus Y squared is equal | |
| 18:19 | to 15 and we have X plus Y is equal | |
| 18:23 | to one . Alright . This thing looks like a | |
| 18:26 | line and this thing looks like a hyperbole to me | |
| 18:29 | . So we all know that we could have tons | |
| 18:32 | of different types of solutions . But what are we | |
| 18:33 | gonna actually do we have to pick something to solve | |
| 18:36 | for ? So , I'm going to solve this guy | |
| 18:40 | four X and I'm gonna move the white to the | |
| 18:43 | other side . So it's gonna be one minus Y | |
| 18:45 | . Okay . And then once I take this guy | |
| 18:48 | , I'm gonna plug this in to the original equation | |
| 18:51 | . Once I saw this , I gotta stick it | |
| 18:52 | into the other guy , X squared minus Y squared | |
| 18:55 | is equal to 15 . Now , once I do | |
| 18:57 | the substitution , the X . Is gonna go in | |
| 19:00 | here . But you need to wrap it in Princes | |
| 19:02 | one minus Y quantity squared minus Y squared is equal | |
| 19:06 | to 15 . Take this thing into the X location | |
| 19:09 | so it's squared . Now when students get to this | |
| 19:12 | point frequently they throw their hands up and they think | |
| 19:14 | they've done something crazy that it's just not gonna work | |
| 19:17 | because it looks like , well why do I have | |
| 19:18 | to square that ? That looks really hard . But | |
| 19:20 | when you have quadratic systems , almost every single time | |
| 19:23 | you do a substitution , you're going to have to | |
| 19:25 | square something so you might as well get used to | |
| 19:27 | it and we know how to square by . No | |
| 19:29 | means we've done it many many , many hundreds of | |
| 19:31 | times by now . You could of course write it | |
| 19:34 | out as one minus y times one minus why ? | |
| 19:36 | But we all know the shortcut . We take one | |
| 19:39 | and we square it the minus sign means it's two | |
| 19:41 | times one times y two times a times B right | |
| 19:45 | . And then the plus the last term square . | |
| 19:48 | This thing is this thing square but then I have | |
| 19:50 | a minus Y squared here And then I have 15 | |
| 19:53 | . Now the neat thing about this is why squared | |
| 19:56 | minus y squared actually gives me zero . So those | |
| 19:58 | terms drop away completely Here , I'm gonna have one | |
| 20:01 | square which is 1 -2 , I Is equal to | |
| 20:06 | what's on the right hand side 15 . And this | |
| 20:07 | is easy to solve . I'm just going to subtract | |
| 20:10 | this over so I'm gonna have a negative two . | |
| 20:12 | I left 14 because it's 14 minus the one . | |
| 20:16 | And then why is going to be 14 over negative | |
| 20:19 | two ? So why is negative seven just double check | |
| 20:21 | myself up to that point ? So negative step why | |
| 20:24 | became negative seven . Now I have to take this | |
| 20:26 | and substitute it back into something . I could take | |
| 20:29 | this negative seven . I could put it into here | |
| 20:30 | and solve for X . Or I could take this | |
| 20:32 | negative seven for why and put it in here and | |
| 20:35 | saw for this . But I'm just gonna take it | |
| 20:37 | into this version and put put it into this version | |
| 20:40 | . Um because it's already solved for X and that's | |
| 20:44 | what I want . So X is one minus y | |
| 20:48 | . X is one minus negative seven . Always wrap | |
| 20:52 | it in parentheses . It's gonna help you so many | |
| 20:54 | times because if you don't wrap it in , parentheses | |
| 20:55 | are gonna make a sign error . So X is | |
| 20:57 | equal to one plus seven , which is eight . | |
| 21:00 | All right . So you only have one solution . | |
| 21:03 | Notice the X coordinate is eight and the y coordinate | |
| 21:06 | is negative 71 solution . And if you take a | |
| 21:10 | look and examine this hyperbole in this line and detail | |
| 21:13 | and graphic , you'll find out that you can arrange | |
| 21:15 | that line and exactly the right way to only have | |
| 21:16 | one solution . One crossing point . Alright , so | |
| 21:19 | notice that you don't have to figure out ahead of | |
| 21:22 | time how many solutions it has or do some kind | |
| 21:25 | of black magic . The math always works out . | |
| 21:28 | In this case it worked out where I had to | |
| 21:31 | solutions . Um in the previous case it worked itself | |
| 21:36 | out where it had two solutions . But in this | |
| 21:39 | case the way the math worked out because the square | |
| 21:41 | terms dropped away , I only actually had one solution | |
| 21:44 | and later on we do more complicated problems . You'll | |
| 21:47 | see the three solutions or the four solutions will pop | |
| 21:50 | out automatically as well . You don't have to do | |
| 21:53 | anything Herculean . It just it pops out of it | |
| 21:56 | falls out of the math . All right . Our | |
| 21:57 | last problem is x times y is equal to eight | |
| 22:01 | and X plus Y is equal to six . Now | |
| 22:06 | there's a couple of ways you can solve this . | |
| 22:08 | I could solve this for X or Y and plug | |
| 22:11 | it in here , or I could solve this for | |
| 22:13 | X or Y and plug it in here . Either | |
| 22:15 | way works . But if I solve this one for | |
| 22:17 | excellent , say I'm gonna have to have eight divided | |
| 22:19 | by Y . That's going to introduce a fraction . | |
| 22:22 | And when I stick that fraction into this equation , | |
| 22:24 | I'm gonna have a bunch of fraction arithmetic . I | |
| 22:26 | don't like to deal with fractions . I know you | |
| 22:27 | probably don't either . Right . But if I saw | |
| 22:30 | this one for X or Y and put it in | |
| 22:32 | here , I don't have any fractions . So it's | |
| 22:33 | worth taking a second in the beginning to figure out | |
| 22:36 | what the path of least resistance really is going to | |
| 22:39 | be . But I guarantee you if you solve this | |
| 22:40 | for X and put it in here or why and | |
| 22:42 | put it in here , it's gonna give you the | |
| 22:43 | same answer . So what I'm gonna do then is | |
| 22:47 | I'm gonna take this and I'm gonna solve four X | |
| 22:51 | . And that's going to be equal to take the | |
| 22:52 | Y over here becomes negative Y . So it's six | |
| 22:55 | minus Y . And then I'm gonna take this And | |
| 23:00 | I'm going to plug in to the original equation x | |
| 23:03 | times y is equal to eight . But excess six | |
| 23:07 | minus Y . Always wrap it in Princes six minus | |
| 23:09 | Y . Times Y is equal to eight . If | |
| 23:12 | you don't wrap it in parentheses , you see if | |
| 23:13 | you cancel those , you're not gonna have this thing | |
| 23:16 | distributed improperly . It's gonna give you the wrong answer | |
| 23:18 | . It's probably one of the biggest things you can | |
| 23:20 | do wrong . Now . The Y gets distributed here | |
| 23:23 | giving you six Y . The Y gets distributed in | |
| 23:26 | there giving you negative Y squared is equal to eight | |
| 23:29 | . Now I have to arrange terms , I can | |
| 23:32 | do it lots of ways but I'm gonna move everything | |
| 23:34 | to the right hand side because ultimately I'm gonna want | |
| 23:37 | the Y squared term to be positive . So let's | |
| 23:39 | move it to the right . So let's just make | |
| 23:42 | it 100% clear . Let's first move the six Y | |
| 23:44 | over . So it's going to be negative . Y | |
| 23:46 | squared is equal to eight minus six . Y . | |
| 23:50 | I just subtract this guy over then let's move this | |
| 23:52 | guy over by addition . That's gonna give you 08 | |
| 23:55 | minus six Y plus y squared . All I did | |
| 23:58 | was add that . And then I'm gonna kind of | |
| 24:01 | flip the whole thing around and write the Y squared | |
| 24:04 | term first minus six . Y plus eight is equal | |
| 24:07 | to zero . I have to write it in descending | |
| 24:09 | powers of X . Like this . And then I | |
| 24:13 | can now factor it or try to factor it . | |
| 24:16 | You can also use the quadratic formula Y times Y | |
| 24:19 | . And then you can do two times four . | |
| 24:21 | We'll see what did I do Yeah two times four | |
| 24:24 | and it has to be negative times negative . Let's | |
| 24:25 | check ourselves these multiply together . Give you negative eight | |
| 24:29 | . This is negative two Y . This is negative | |
| 24:31 | for y . You add those together . You get | |
| 24:32 | the negative six Y . So what does this get | |
| 24:35 | you ? It tells you that why is equal to | |
| 24:37 | two and why is equal to four ? You set | |
| 24:40 | these equal to zero and move them over . Now | |
| 24:42 | I have to take this and plug it into one | |
| 24:43 | of these . I can plug it into this one | |
| 24:45 | , I can plug it into this one or I'm | |
| 24:47 | just going to use this version because it's the same | |
| 24:49 | as this and it's already solved for X . So | |
| 24:51 | I'm gonna plug in to the equation X is equal | |
| 24:55 | to six minus Y . So X is going to | |
| 24:57 | be the six minus two , X is going to | |
| 25:00 | be equal to four . I'm gonna plug this into | |
| 25:02 | the exact same equation . X is equal to six | |
| 25:04 | -Y . X is equal to six minus four . | |
| 25:07 | X is equal to two . And you can see | |
| 25:10 | then this value of X goes with the this value | |
| 25:13 | of Y four comma two and this value of X | |
| 25:18 | goes with this value of Y , which is two | |
| 25:19 | comma for the X term comes first . Right ? | |
| 25:23 | So there's two solutions . Two solutions . So uh | |
| 25:30 | for common to and to calm before . All right | |
| 25:32 | . So we're not done with solving these systems by | |
| 25:35 | substitution , but we've got a really great start . | |
| 25:38 | And the remaining problems that we're gonna do , they | |
| 25:40 | match the complexity up a little bit by putting circles | |
| 25:43 | with hyper bolos and so on . And sometimes you | |
| 25:44 | have four solutions , you have to do a little | |
| 25:46 | more work , but they all basically boil down to | |
| 25:48 | the same exact technique . You pick something to solve | |
| 25:52 | for . Hopefully the thing that's easiest . Stick it | |
| 25:55 | into the other thing and you solve eventually half the | |
| 25:57 | problem , you're going to get half of the variables | |
| 25:59 | , but then you have to take the variables that | |
| 26:01 | you get the answer to and stick them back in | |
| 26:03 | and it doesn't matter what you put it in . | |
| 26:05 | Uh at that point you just have to pick whatever | |
| 26:07 | is easiest for you . Usually it's whatever you solve | |
| 26:09 | for . Okay . And then you find the pairs | |
| 26:12 | that match up , the thing you have to be | |
| 26:13 | careful about is when you have multiple pairs of things | |
| 26:16 | running around , you have to match the X with | |
| 26:18 | the why and the X with the why . That's | |
| 26:20 | why I draw these arrows everywhere to try to help | |
| 26:22 | myself remember notice I did that for every problem I | |
| 26:25 | said I got these boom plugging them in . That | |
| 26:27 | helps me remember what's going on . A lot of | |
| 26:29 | students would write stuff everywhere and then at the end | |
| 26:31 | they don't even know what goes with what . So | |
| 26:33 | then you get stuck . So make sure you can | |
| 26:35 | solve every one of these problems . Follow me on | |
| 26:37 | to the next lesson . We're going to continue conquering | |
| 26:40 | solving the quadratic systems of equations by substitution . |
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