14 - Solve Quadratic Systems of Equations by Addition - Part 1 (Simultaneous Equations) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra . Were conquering the | |
| 00:02 | topic of solving systems of quadratic equations . Also called | |
| 00:06 | solving quadratic systems . It's a short way of putting | |
| 00:09 | it . Last few lessons . We talked about substitution | |
| 00:11 | that's kind of your go to technique . You're gonna | |
| 00:13 | use that almost all the time and you can use | |
| 00:15 | it for every quadratic system . We've talked about substitution | |
| 00:18 | techniques in great detail . But there is another technique | |
| 00:20 | called addition that we're gonna use here . Now if | |
| 00:23 | you remember back when we solve regular old linear equations | |
| 00:26 | , equations that evolve lines to find the intersection point | |
| 00:29 | , we can add the equations together . And we | |
| 00:31 | went through great discussions of why you're allowed to do | |
| 00:34 | that and how it works and so on . But | |
| 00:36 | the same technique that we used in the past for | |
| 00:38 | lines can be applied to these quadratic systems . But | |
| 00:40 | if you remember back to that technique , the way | |
| 00:42 | it works is you have to add them together in | |
| 00:44 | the process of adding them . You can also multiply | |
| 00:47 | an equation by something and add it . But at | |
| 00:49 | the end of the day , when you add them | |
| 00:51 | together , one of the variables must be eliminated in | |
| 00:54 | order to let you solve for something . So you | |
| 00:56 | have to eliminate the variable when you add them together | |
| 00:58 | . So you're allowed to multiply equations by numbers , | |
| 01:01 | you're allowed to add them . But some sort of | |
| 01:03 | way at the end of that process , you must | |
| 01:05 | eliminate one of the variables in order to solve for | |
| 01:07 | the other one and make any progress . If you | |
| 01:09 | can't eliminate a variable doing that , then you cannot | |
| 01:11 | use the method of addition . So what we're going | |
| 01:14 | to find out is that some of these equations , | |
| 01:16 | you can't actually use this method because sometimes you cannot | |
| 01:20 | eliminate variables by adding equations together . So let's crawl | |
| 01:23 | before we can walk and let's show the first one | |
| 01:26 | here and show how you can do it . And | |
| 01:28 | then we'll show some examples of when you can . | |
| 01:30 | So what about X squared minus Y is equal to | |
| 01:32 | five ? What about two X plus Y is equal | |
| 01:36 | to three . Now this system of equations might look | |
| 01:39 | similar actually all of these problems and all of the | |
| 01:41 | problems in the next lessons . We've already solved them | |
| 01:44 | . We solve them by substitution . Now we're solving | |
| 01:46 | them again by addition . So what you might say | |
| 01:48 | , why are you doing the same problems ? That | |
| 01:49 | doesn't help me . Well it does because when we | |
| 01:52 | learn a new technique with the same equations , it | |
| 01:54 | gives you some confidence that either technique you use is | |
| 01:58 | going to give you the same answer . So that's | |
| 01:59 | number one and number two . The process is so | |
| 02:02 | different that you're going to get the same answer at | |
| 02:05 | the end . But the math in the middle is | |
| 02:07 | so different . That doesn't even matter that the problems | |
| 02:08 | are the same . So these equations we've already solved | |
| 02:11 | , we've already gotten the answers by substitution . Now | |
| 02:13 | we're going to use edition . So the first thing | |
| 02:15 | you do is you say , can I add these | |
| 02:18 | these equations together and eliminate a variable ? And the | |
| 02:21 | answer is you can . And let's see what happens | |
| 02:25 | if you add the two X squared to the two | |
| 02:27 | X . You cannot directly add those because you have | |
| 02:30 | an X squared term in an X . But you | |
| 02:31 | can just say when you add these guys , you're | |
| 02:35 | going to get X squared plus two X . Right | |
| 02:39 | Now , when you add these guys together , you | |
| 02:41 | have a negative Y plus A . Y . So | |
| 02:43 | these can be added together and it just gives you | |
| 02:45 | zero . So you don't have to write anything down | |
| 02:47 | here on the right hand side . You add these | |
| 02:49 | together and get an eight , you might say . | |
| 02:51 | Well that didn't really help me . I still have | |
| 02:53 | an X . Squared and X . But you know | |
| 02:54 | how to solve these equations to X . I'm sorry | |
| 02:57 | X squared plus two X . Move the eight over | |
| 02:59 | by subtraction . And then you factor and solve , | |
| 03:02 | you're never gonna get away from having a factor . | |
| 03:04 | Things when you have quadratic equations like this . So | |
| 03:07 | you go ahead and open up your parentheses and you | |
| 03:10 | say X times X is X squared . You can | |
| 03:12 | do two times four is 8 and the signs are | |
| 03:15 | gonna work out like this negative two X . Positive | |
| 03:18 | forex give you this and the two times before with | |
| 03:21 | a negative sign gives you negative eight . Okay . | |
| 03:23 | The biggest problem up until now that students make is | |
| 03:26 | they start getting confused when you add this to this | |
| 03:28 | , they try to actually add them together . One | |
| 03:30 | plus two equals three . You can't do that because | |
| 03:32 | you cannot add X square two X . They're different | |
| 03:35 | . So you just have to carry it through . | |
| 03:36 | But notice we did eliminate a variable . We eliminated | |
| 03:39 | why And that's what I told you in the beginning | |
| 03:41 | . You must eliminate a variable . Otherwise you cannot | |
| 03:43 | use this method . So here we now know that | |
| 03:47 | when we set this equal to zero we're gonna get | |
| 03:49 | X . Is equal to positive two . When we | |
| 03:51 | set this equal to zero we're gonna get X . | |
| 03:52 | Is equal to negative four . And then we have | |
| 03:55 | to plug these values in to any one of these | |
| 03:58 | original equations . It doesn't matter which one you pick | |
| 04:00 | . Um You can pick this one , you can | |
| 04:02 | pick this one . I'm just gonna plug it into | |
| 04:04 | this one so I'm gonna rewrite this down to X | |
| 04:06 | . Plus Y is equal to three . So I'm | |
| 04:09 | gonna substitute it in there . The value of X | |
| 04:11 | goes in here , so it's two times the to | |
| 04:14 | the value of X going in there . Plus Y | |
| 04:16 | is equal to three . This is four . We | |
| 04:18 | move the four over . We're gonna get a Y | |
| 04:20 | . Is equal to negative one because three minus four | |
| 04:22 | . You get a negative one there , we'll plug | |
| 04:24 | this into the same equation . Two , X plus | |
| 04:28 | Y is equal to three . I do not have | |
| 04:30 | to use this equation , it's a solution so the | |
| 04:32 | X values work in both . So I could plug | |
| 04:35 | it in there . It's the same thing . But | |
| 04:36 | let's just go and put it into this one two | |
| 04:39 | times X . Being negative four plus Y is equal | |
| 04:42 | to three . So we're gonna get negative eight plus | |
| 04:45 | y . Is three and we can add this over | |
| 04:47 | . So three plus eight is going to be 11 | |
| 04:51 | . Okay then you have to values of X that | |
| 04:55 | fell out . Each one corresponding to a single value | |
| 04:57 | of Y . So you have two solutions . The | |
| 05:01 | first x value was two common negative one because it | |
| 05:05 | goes with this why ? And then the next one | |
| 05:07 | was negative four comma 11 X comma Y two solutions | |
| 05:11 | . So two comma negative one negative four comma 11 | |
| 05:14 | . If you dig back into a few lessons ago | |
| 05:17 | when we solve these by substitution , the math in | |
| 05:19 | the middle looks different but the answers are exactly the | |
| 05:22 | same . So either technique is fine substitution . You | |
| 05:25 | can use all the time . There really is no | |
| 05:27 | case which you can't use substitution . But addition it | |
| 05:31 | requires the system to be able to eliminate a variable | |
| 05:34 | either straight away . We didn't have to actually do | |
| 05:36 | anything to eliminate because they just add to zero . | |
| 05:39 | But you're allowed to multiply these equations by whole numbers | |
| 05:42 | or fractions or whatever you want . As long as | |
| 05:43 | their numbers two put the coefficient such that you can | |
| 05:48 | cancel a variable . Like just like we did when | |
| 05:49 | we solve linear equations by addition . Now let me | |
| 05:53 | give you an example of something that you run into | |
| 05:54 | problems with . What if you have the equation X | |
| 05:58 | squared minus Y squared Is equal to 15 ? Yes | |
| 06:04 | . What if you have X plus why Is equal | |
| 06:07 | to 1 ? This is a system of equations . | |
| 06:10 | Of course I can use substitution . We did that | |
| 06:11 | before . In fact this we solve this problem for | |
| 06:14 | solve for X . Put it in here or you | |
| 06:15 | can solve for why I put it in here . | |
| 06:17 | But let's say you want to add them together . | |
| 06:18 | So you start saying okay well what if I add | |
| 06:21 | them ? Well I'm gonna have an X squared plus | |
| 06:22 | an X . And you say oh this will cancel | |
| 06:24 | . But then it doesn't because you have this is | |
| 06:26 | a Y squared in a Y . So you have | |
| 06:28 | X squared terms in one equation , but only X | |
| 06:31 | terms in the other equation . You have y squared | |
| 06:34 | terms in the first equation , but only why terms | |
| 06:36 | in the second equation . So no matter how you | |
| 06:38 | add them together , you cannot eliminate these variables because | |
| 06:41 | you can't even add them together , they're different exponents | |
| 06:44 | . You can multiply the equations by whole numbers but | |
| 06:46 | that's not gonna help you . You multiply this by | |
| 06:48 | two or one or three . You still cannot add | |
| 06:50 | these variables together because the exponents of the problem . | |
| 06:53 | So you're allowed to multiply by whole numbers , you're | |
| 06:56 | allowed to add them together . And there's no way | |
| 06:58 | you can eliminate a variable uh by doing that . | |
| 07:01 | So you can't use addition . It does not mean | |
| 07:09 | there's no solution to this . It does not mean | |
| 07:11 | you cannot solve it . It does not mean you | |
| 07:13 | can't graph it . It does not mean you can't | |
| 07:14 | use substitution . It just means you can't use this | |
| 07:16 | method of addition . All right , let me give | |
| 07:19 | you another one that runs into problems . I have | |
| 07:21 | them on different pages , but let's just go ahead | |
| 07:23 | and and uh let me actually figure out where it | |
| 07:26 | is . Yeah , let me just show you here | |
| 07:28 | . I'm gonna put both of the problem Children in | |
| 07:30 | the same place . What if I give you the | |
| 07:31 | equations , X plus Y is equal to six and | |
| 07:35 | then X Y is equal to eight . We solve | |
| 07:37 | this system of equations already by substitution . Let's say | |
| 07:40 | you want to add them together . You have the | |
| 07:42 | X plus this term . But it's Xy . So | |
| 07:45 | you can't really add X plus x . Y . | |
| 07:48 | You can't really add Y plus X . Y . | |
| 07:50 | Even if you multiply this by whole numbers . You | |
| 07:53 | know , you're not supposed to multiply by variables . | |
| 07:54 | Okay . You're supposed to multiply by whole numbers . | |
| 07:56 | You cannot really make it where I can eliminate those | |
| 08:00 | guys . So since you're sticking to the rules of | |
| 08:03 | multiplied by whole numbers and adding and and so on | |
| 08:06 | and so forth , you really shouldn't use addition as | |
| 08:09 | a method to solve this guy . Okay . All | |
| 08:12 | right . So those are problems problems there . All | |
| 08:15 | right . Um Let's go and solve one . That | |
| 08:22 | does work two times Y squared plus 33 X is | |
| 08:27 | equal to 33 . I'm sorry . three x . | |
| 08:31 | Is he cool . 2 33 and then X plus | |
| 08:36 | four , Y Plus seven is equal to zero . | |
| 08:39 | Okay . So this one works . Why ? Because | |
| 08:42 | I have X terms and I have X terms . | |
| 08:45 | Now this is a Y termina . Why ? So | |
| 08:46 | I cannot add them together but I can add the | |
| 08:48 | excess together so I can eliminate something . So let's | |
| 08:50 | tidy this up a little bit more . So let's | |
| 08:53 | tidy this up a little more and right to I | |
| 08:55 | squared plus three X . Is equal to 33 . | |
| 08:59 | This one we're going to write it with the Y | |
| 09:02 | term first for why ? The X term second . | |
| 09:06 | So I can put it under this guy here is | |
| 09:08 | 33 . Yeah . All right now I cannot I'm | |
| 09:13 | sorry is equal to uh are totally and a moment | |
| 09:19 | there . That's not correct . So you have a | |
| 09:20 | seven . Let's move this negative seven . On the | |
| 09:22 | right hand side . You have two Y squared plus | |
| 09:25 | three X . 33 flip these terms around move to | |
| 09:28 | seven over to make a negative seven . Okay now | |
| 09:30 | if we add these guys as they sit three X | |
| 09:33 | plus X . Is gonna give me four X . | |
| 09:34 | So that's not gonna cancel . It's not going to | |
| 09:36 | eliminate a variable . So what I need to do | |
| 09:39 | is I need to multiply this equation by what ? | |
| 09:44 | Not by three . By negative three Multiply by -3 | |
| 09:50 | and when I do that I'm gonna rewrite the system | |
| 09:52 | again . Two Y squared plus three X . Is | |
| 09:55 | 33 . Multiply the negative three here becomes negative 12 | |
| 10:00 | . Y . Multiply the negative three here becomes negative | |
| 10:03 | three X . Multiply the negative three becomes positive 21 | |
| 10:09 | . All right now you see you can add them | |
| 10:11 | so I can add . And what happens is when | |
| 10:15 | I add these together I can just write it down | |
| 10:17 | . I can't actually perform the addition to Y squared | |
| 10:20 | minus 12 . Why ? Because when I add it | |
| 10:22 | , I can't actually combine the variables together . But | |
| 10:24 | these at 20 And on the right hand side you're | |
| 10:27 | gonna get 54 54 . So now I can solve | |
| 10:31 | this guy , let's move it over two , Y | |
| 10:33 | squared minus 12 . Y will subtract the 54 equal | |
| 10:37 | to zero . Now this polynomial is the same polynomial | |
| 10:40 | we got when we did it by substitution . So | |
| 10:43 | from here right now the solution process is the same | |
| 10:45 | . But we're gonna go in and do it . | |
| 10:46 | We're gonna factor out of two here . Just to | |
| 10:48 | make the factoring easier , Y squared minus six . | |
| 10:51 | Y 54 . 5 by two . Double check myself | |
| 10:54 | , 27 equals zero . The two is ultimately not | |
| 10:59 | going to matter at all . We can divide the | |
| 11:01 | two away , divide by two and cancel divide by | |
| 11:03 | two and you'll start zero . So let's go ahead | |
| 11:06 | and just factor it as it sits , why times | |
| 11:09 | y is y squared nine ? Times three is 27 | |
| 11:13 | ? And the signs are gonna work as -10 . | |
| 11:16 | Yes . All right now we set this equal to | |
| 11:20 | zero in this equal to zero . So we found | |
| 11:21 | that why is equal to nine and why can be | |
| 11:23 | equal to negative three ? We set this equal to | |
| 11:25 | zero and now we have to plug in . We | |
| 11:28 | can choose what equation to plug into . I can | |
| 11:32 | take these values of why and I can put them | |
| 11:33 | into here or I can take the values why I | |
| 11:35 | put them in here but I don't like the squaring | |
| 11:37 | so I'm just gonna pick the second equation . So | |
| 11:41 | what I'm going to get is I'm going to plug | |
| 11:43 | in to the bottom equation which is X plus four | |
| 11:48 | , Y plus seven is equal to zero . I | |
| 11:51 | gotta stick the value of Y in here . So | |
| 11:53 | X plus four times nine plus seven is equal to | |
| 11:56 | zero . So X plus nine times four is 36 | |
| 12:01 | plus seven is equal to zero . X . Is | |
| 12:03 | equal to . Let me just double check myself here | |
| 12:05 | . 43 . Well it's gonna be 43 and when | |
| 12:08 | we move it to the other side you're gonna get | |
| 12:10 | -43 . As an answer negative 43 and we're gonna | |
| 12:15 | take this guy and we're gonna plug in to the | |
| 12:18 | exact same equation . It's gonna be X Plus four | |
| 12:22 | times y . Which is negative three plus seven is | |
| 12:28 | equal to zero . So we have this is gonna | |
| 12:30 | be minus 12 plus seven is equal to zero and | |
| 12:34 | then exits will be negative five equals zero . So | |
| 12:37 | then X . Is five by moving the five over | |
| 12:39 | to the right hand side . So then for this | |
| 12:42 | value of why I get this value of X . | |
| 12:44 | For this value of why I get this value of | |
| 12:45 | X . And so I have two solutions . X | |
| 12:51 | is -43 . Why is nine excess five ? And | |
| 12:59 | why is where's that -3 ? These are the two | |
| 13:03 | solutions negative 43 coming 95 comma negative three . Same | |
| 13:07 | answers that we got when we solve this problem before | |
| 13:09 | . So when you look and try to decide if | |
| 13:11 | you're gonna use addition or not , just see if | |
| 13:13 | I can multiply these by whole numbers . If I | |
| 13:15 | can uh allow one of the variables to cancel and | |
| 13:21 | in this case by lining it all up . That's | |
| 13:23 | the other thing they put the variables out of order | |
| 13:25 | . So it's hard to see . So you need | |
| 13:26 | to write all the X terms in the white terms | |
| 13:28 | on top of each other so that you can see | |
| 13:29 | what you need to multiply by in order to have | |
| 13:32 | it cancel . Right ? So let me just double | |
| 13:36 | check myself . Let's see here . Yeah . Actually | |
| 13:41 | I have one more problem left . It's not as | |
| 13:45 | long as this one actually . So we're gonna do | |
| 13:48 | that one next . It's gonna be as follows why | |
| 13:52 | is equal to X squared and X squared plus Y | |
| 13:56 | squared is 12 . Now we've solved this one before | |
| 13:59 | . We did it by substitution . You can substitute | |
| 14:01 | in for the Y . Substitute for the X . | |
| 14:03 | Squared if you want to . It's kind of begging | |
| 14:05 | to be solved by substitution but you can use addition | |
| 14:08 | . So we're gonna figure out how to do that | |
| 14:10 | . Okay , the first thing you need to do | |
| 14:11 | is write all the variables on top of each other | |
| 14:13 | . So the way I want to write this is | |
| 14:15 | move the X squared over . And it's gonna be | |
| 14:17 | negative X squared . Then I'm gonna add the Y | |
| 14:20 | equals zero , move this over . It makes negative | |
| 14:22 | X squared plus the y zero . This I'm gonna | |
| 14:24 | right underneath it is X squared plus Y squared is | |
| 14:27 | 12 . You see why I added I do it | |
| 14:30 | like this because then I can immediately see I'm gonna | |
| 14:31 | add these and I'm gonna immediately gonna get a cancellation | |
| 14:34 | . So I'm allowed to use this technique This is | |
| 14:37 | going to add to give me zero . But then | |
| 14:38 | I'm going to have over here . Why squared plus | |
| 14:41 | why is he 12 ? Just add these together , | |
| 14:44 | add these together . Now I have to solve this | |
| 14:46 | by moving the 12 over -12 . And then I | |
| 14:51 | factor why times why is why squared And then three | |
| 14:56 | times four is 12 double check , myself and the | |
| 14:59 | signs are going to work as this negative three Y | |
| 15:02 | . Positive four is going to give you the positive | |
| 15:04 | one there . And of course this multiplies to give | |
| 15:06 | you the negative 12 . Yeah . So now what | |
| 15:09 | we have is why is equal to three . Set | |
| 15:12 | that equal to zero . Set this equal to zero | |
| 15:14 | . We're gonna get y is equal to negative four | |
| 15:17 | . So we have two solutions and now we're gonna | |
| 15:20 | have to plug in . I can plug this into | |
| 15:22 | anything I want , I can put it into here | |
| 15:24 | or I can put it into here . But this | |
| 15:26 | one is just a much simpler equation . So I'm | |
| 15:27 | gonna put it into here . Why is equal to | |
| 15:29 | X squared ? But I'm gonna write it is X | |
| 15:31 | squared is equal to Y . The same exact thing | |
| 15:33 | . I flipped it around because I want to solve | |
| 15:34 | the value of X . So X squared is equal | |
| 15:38 | to three . Put the value of Y , N | |
| 15:40 | X is plus or minus the square root of three | |
| 15:43 | . So X can be three , sorry square root | |
| 15:46 | of three and X can be negative square root three | |
| 15:50 | . Mhm . And then I go and do the | |
| 15:51 | same thing here . I'm gonna plug in . X | |
| 15:55 | squared is equal to Why now why is negative four | |
| 15:57 | ? So X squared is negative four and X is | |
| 16:01 | plus or minus the square root of negative four . | |
| 16:03 | So this will give you an imaginary number . So | |
| 16:05 | this you just write down not real . So for | |
| 16:09 | each value of why that had negative four you would | |
| 16:11 | get plus or minus two I which is an imaginary | |
| 16:14 | number . Right ? So remember when I said the | |
| 16:17 | coordinate turns out to be imaginary , you just throw | |
| 16:19 | that solution away . It's not a real number , | |
| 16:20 | it's not a physical crossing point . So you don't | |
| 16:23 | actually write any of this down in your answer even | |
| 16:25 | though this is real , it corresponds to an imaginary | |
| 16:28 | value of X . So there's only in this case | |
| 16:31 | to solutions right to solutions . And the X value | |
| 16:37 | here is the square root of three and the y | |
| 16:39 | value came out to be positive three . Yeah , | |
| 16:44 | negative square root of three was the other value of | |
| 16:46 | X . And positive three was the value of why | |
| 16:48 | ? So that's the answer here . All right . | |
| 16:51 | So I hope you understand now how to do these | |
| 16:53 | uh solving these systems of quadratic equations with the concept | |
| 16:57 | of addition . You need to find an equation to | |
| 17:00 | eliminate by addition or by multiplying my whole numbers and | |
| 17:04 | adding . And that's an example of what we did | |
| 17:06 | right here . We have to multiply the second equation | |
| 17:08 | By a number in this case -3 so that it | |
| 17:11 | would cancel with what was above . And so what | |
| 17:14 | you have to do is arrange the term so that | |
| 17:15 | you can quite easily see that once you add , | |
| 17:17 | you must eliminate a variable and then you proceed throughout | |
| 17:20 | the solutions . We have done now , I was | |
| 17:22 | cautious to tell you that you can always use um | |
| 17:26 | you know uh the method of substitution um but you | |
| 17:30 | cannot always use the method of addition . I kind | |
| 17:32 | of fibbed a little bit there in the beginning . | |
| 17:34 | What most textbooks are going to tell you is that | |
| 17:37 | you can use the system of addition as long as | |
| 17:39 | you're multiplying the equations by whole numbers and adding them | |
| 17:42 | . So in this equation this equation we didn't have | |
| 17:45 | to multiply by anything . We just added them together | |
| 17:47 | and it all worked out and this one over here | |
| 17:50 | , if we just add them , it doesn't work | |
| 17:51 | . So we have to multiply by a number and | |
| 17:54 | of course this one over here . Once we arranged | |
| 17:55 | everything , then we didn't have to multiply by a | |
| 17:58 | number . But typically your book is going to tell | |
| 18:00 | you multiply by a whole number and then you can | |
| 18:03 | use the method of addition . But actually this equation | |
| 18:07 | for for most equations , you can actually multiply by | |
| 18:09 | variables if you want to . And i it works | |
| 18:12 | for the cases that I can show you here and | |
| 18:14 | test in front of you know , for these problems | |
| 18:16 | here . But I cannot guarantee you that it's gonna | |
| 18:18 | work for all cases because depending on what the equations | |
| 18:21 | are , you can run into extraneous roots and other | |
| 18:24 | things that become problems . But in some cases you | |
| 18:27 | can multiply by something other than a number . So | |
| 18:29 | this was one that we did just a minute ago | |
| 18:31 | and I told you you really can't use the method | |
| 18:33 | of addition . Because it doesn't you can't get these | |
| 18:36 | guys to cancel . You cannot add this to cancel | |
| 18:39 | and you cannot add this and you can't multiply by | |
| 18:41 | a number to make it work either . So typically | |
| 18:43 | you would just use substitution . I still think that's | |
| 18:45 | the best way forward . But in this case notice | |
| 18:48 | if you multiply the first equation by the variable , | |
| 18:50 | why multiply this by Y . It will become X | |
| 18:53 | . Y . I'm sorry multiply by not by Y | |
| 18:55 | . Multiplied by negative Y . It will become negative | |
| 18:57 | xy This will become negative Y squared , this will | |
| 19:00 | become negative six Y . So we multiply the first | |
| 19:02 | equation by negative y . Second equation remains unchanged . | |
| 19:06 | Then when we add it um this drops away to | |
| 19:09 | zero . This it comes along for the ride just | |
| 19:12 | adding zero to it . And then this is a | |
| 19:14 | negative six Y plus eight . Now you have a | |
| 19:17 | quadratic and why ? So then you can move the | |
| 19:20 | Y terms over here and you can factor and you'll | |
| 19:22 | get two and four as solutions for Y . And | |
| 19:25 | then you can take these two values and plug them | |
| 19:26 | into either one of the original equations . Sorry , | |
| 19:29 | down here that you want , we'll choose this one | |
| 19:31 | , X y is equal to eight . When you | |
| 19:32 | put the two value in , you get a Y | |
| 19:34 | value of an X . Value of four . And | |
| 19:37 | for when you put the four value in you'll get | |
| 19:39 | an X value here of two . So you get | |
| 19:41 | four come into into come before when you go back | |
| 19:43 | and look at how we did this particular problem in | |
| 19:45 | substitution . This is the exact answer you get . | |
| 19:48 | So I kind of fibbed a little bit when I | |
| 19:50 | said you cannot use the method of addition unless you're | |
| 19:53 | multiplying only by numbers . For certainly for some equations | |
| 19:56 | it works . I cannot sit here and swear to | |
| 19:58 | you that it's going to work for every set of | |
| 20:00 | equations you have because some equations can have singularities , | |
| 20:04 | meaning they go to infinity . Rational equations may get | |
| 20:07 | weird if they have fractions or radicals , it may | |
| 20:10 | be difficult for me to justify multiplying by variables like | |
| 20:13 | this because when you multiply by a variable , you | |
| 20:16 | kind of are making assumptions about what if the variable | |
| 20:19 | is well behaved or not . And so I don't | |
| 20:21 | want to get into all the theory . I'm just | |
| 20:22 | letting you know that for some simple ones like this | |
| 20:25 | , you can multiply by variables and get the things | |
| 20:28 | to cancel . But what I really like you to | |
| 20:30 | do is if you can't multiply by whole numbers and | |
| 20:32 | get it to cancel . Just use substitution method . | |
| 20:34 | I think that's the most straightforward path forward there . | |
| 20:38 | So make sure you understand all of these , solve | |
| 20:40 | these yourself , feel free to do this on your | |
| 20:43 | own to make sure that it works when you actually | |
| 20:44 | multiply by the variable negative y up here and then | |
| 20:48 | follow me on to the next lesson . We're going | |
| 20:49 | to wrap up the concept of solving quadratic systems by | |
| 20:52 | addition . |
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