12 - Solving 3-Variable Linear Systems of Equations - Substitution Method - By Math and Science
Transcript
00:00 | Hello . Welcome back to algebra . The title of | |
00:02 | this lesson is solving systems of linear equations in three | |
00:06 | variables with three variables with substitution . Part one . | |
00:10 | So it's a very lengthy , very complex sounding title | |
00:14 | . The basic idea is we have learned already how | |
00:17 | to solve linear systems of equations before we did that | |
00:19 | a long time ago when we had two lines . | |
00:22 | Remember linear means line right or something that doesn't curve | |
00:26 | . So two lines of course can intersect and they | |
00:28 | can have a crossing point and we found those solutions | |
00:30 | by substitution . We also did addition and we also | |
00:33 | use graphing back then with two lines is very easy | |
00:36 | to graph things and and know what's going on . | |
00:37 | But whenever you have three variables X , Y and | |
00:41 | Z . Then it gets difficult because the math to | |
00:44 | find the solution is it's more involved and also it's | |
00:48 | a lot more involved to visualize what's happening with a | |
00:51 | line is very simple to lines either across or they | |
00:54 | don't if they cross there's an intersection point and we | |
00:57 | find it , that's the solution . But if they're | |
00:59 | parallel means they don't cross , then there's no solution | |
01:02 | is very simple cases . But when you have three | |
01:04 | variables X , y and Z . Then the graphs | |
01:07 | become hard to graph . You can't really graph them | |
01:09 | so easily . And it's hard to visualize what's going | |
01:12 | on . So in this lesson , what we're gonna | |
01:13 | do is introduce what a system of equations in three | |
01:16 | dimensions looks like . As far as the math goes | |
01:18 | , we're gonna sketch some pictures . So you understand | |
01:21 | kind of what's going on in terms of what their | |
01:22 | graphs would look like kind of . And then we're | |
01:25 | gonna solve the problem at the end to show you | |
01:27 | how to actually figure out what the solution to a | |
01:29 | system like that looks like . All right , So | |
01:31 | let's crawl before we can walk . Let's go back | |
01:33 | down memory lane down to a system of linear equations | |
01:36 | with only two variables X and Y . And we'll | |
01:39 | extend that to talk about what happens now that we | |
01:41 | have three variables called Z . The third variable called | |
01:44 | Z . And also I'd like to say if you | |
01:45 | haven't already watched my last lesson , please do it | |
01:48 | . Now . In the last lesson , I told | |
01:49 | you all about three D . Graphing three D . | |
01:52 | Points and three D functions . So if you haven't | |
01:54 | already looked at that , then then it's gonna seem | |
01:56 | confusing here . But I've already talked about the fundamental | |
01:58 | basics of what you need to understand to be with | |
02:01 | me here . So what we're gonna do is talk | |
02:03 | about uh system of equations in two D . So | |
02:06 | we're going to call this a linear system . That's | |
02:10 | what S . Y . S . Means uh linear | |
02:12 | system in two dimensions , two dimensions means X and | |
02:16 | Y . So they're basically going to be lines . | |
02:18 | Lines means linear . Okay , so what would example | |
02:21 | system like that looks like ? Well it might have | |
02:23 | something like this to exp plus Y is equal to | |
02:27 | three and the other equation might be x minus three | |
02:30 | . Y is equal to negative one . Now this | |
02:33 | is a system of equations because there's two equations , | |
02:36 | there's also two unknowns , X and Y . So | |
02:39 | what you're really trying to do is figure out what | |
02:41 | value of X and Y . Well , both satisfy | |
02:44 | both equations at the same exact time . That's gonna | |
02:46 | be the intersection point . Now , what are the | |
02:48 | graphs of these things look like ? I mean I'm | |
02:50 | not a computer , I don't know exactly what they | |
02:52 | look like but I know that their lines and you | |
02:54 | should know that their lines to . And the reason | |
02:56 | that you know that their lines and not some kind | |
02:58 | of crazy curved kind of graph is because if I | |
03:02 | wanted to I could solve the top equation and put | |
03:06 | it into why is equal to mx plus B . | |
03:08 | For how do I know I could do that ? | |
03:10 | Well because this is why I could subtract the two | |
03:13 | X . It would be negative two X plus three | |
03:14 | . That's mx plus B . And I can look | |
03:16 | at the slope and the Y intercept . This means | |
03:18 | this is a line . Some kind of lines . | |
03:20 | In other words , it's not a parabola , it | |
03:22 | doesn't curve , it's not any lips , it's not | |
03:24 | a hyperbole A it's not a cubic function . It's | |
03:27 | not a cortical function . It's not a square root | |
03:29 | radical function . It's not an exponential function , it's | |
03:31 | none of those . It has to be a line | |
03:33 | because of the simple nature of the way the variables | |
03:36 | just have coefficients in front . Now this equation , | |
03:39 | same sort of thing . If I wanted to I | |
03:41 | could solve it and put it in an mx plus | |
03:43 | B form because I could take the X and move | |
03:45 | it over there . I could divide by three . | |
03:47 | I'd have some fractions but it would still be mx | |
03:49 | plus B , which is a line . So any | |
03:51 | time you have a number of times X plus number | |
03:54 | of times Y equals number , number of times X | |
03:57 | plus number of times Y equals number , then you | |
03:59 | automatically know it's a line . These are two lines | |
04:01 | . That's how you know that . Now , if | |
04:03 | you were to have an X squared running around then | |
04:06 | you would know it's not a line . If a | |
04:07 | y squared is running around , you would know it's | |
04:09 | not a line . If there's a radical anywhere on | |
04:11 | a variable like a square root or cube cube root | |
04:13 | or something , you would know it's not a line | |
04:16 | . Those are what the things to look for . | |
04:17 | Think back to all the ellipses . We graft all | |
04:20 | the circles we graph . They always have squares everywhere | |
04:22 | . So you know those aren't lines but these are | |
04:23 | lines . That's why it's a linear system of equations | |
04:26 | . Also notice if there's two variables in this case | |
04:29 | , X and Y . The only way to solve | |
04:32 | that system for X and y is to have two | |
04:34 | equations , you have to have the same number of | |
04:36 | equations as you do variables . Otherwise you can't solve | |
04:40 | it . So we have two equations , X and | |
04:42 | Y . We have to uh variables . So that | |
04:45 | means this set of equations insolvable . Okay . We're | |
04:49 | not gonna solve it because we've done it many times | |
04:50 | in the past , we've had entire lessons on this | |
04:52 | . But you know that this is a line and | |
04:54 | this is a line . So what's in general going | |
04:55 | to happen is these lines are gonna cross somewhere probably | |
04:58 | uh They may or may not but they probably will | |
05:01 | cross somewhere in this intersection point . The single intersection | |
05:04 | point is called the solution . Specifically , there's only | |
05:07 | one solution because the solution is the crossing point , | |
05:11 | it's the point that's common to both lines that satisfies | |
05:13 | both equations . And so you say uh that it | |
05:16 | has one point in common with each with each of | |
05:23 | the equations . Right , That's what that means . | |
05:24 | Now , of course lines do not have to intersect | |
05:27 | . You can have a system of equations where I | |
05:29 | have a line going up like this and a line | |
05:32 | exactly parallel to it . Now I can't draw exactly | |
05:34 | parallel , but you have to pretend these are exactly | |
05:36 | parallel . And if these actually are the two lines | |
05:40 | that you have , these are parallel lines sure . | |
05:43 | Which means there's no solution the word solution in your | |
05:49 | mind . You need to replace with intersection points points | |
05:52 | in common between the two graphs . That's what you're | |
05:55 | looking for . That's what the solution is . It's | |
05:56 | a common point between two graphs . Right ? But | |
05:59 | if the lines are parallel then I could go 65 | |
06:02 | million light years away and you'll still never cross . | |
06:05 | There's never any crossing points for parallel lines , but | |
06:08 | if there ever so slightly not parallel , eventually they | |
06:11 | will cross maybe it's 10 million light years away , | |
06:13 | but they will cross somewhere and the solution is way | |
06:16 | way , way down there . Okay , so this | |
06:18 | is a system of equations in two dimensions , we | |
06:20 | need two equations . We have two variables . We | |
06:22 | know how to solve these systems . All right . | |
06:24 | And we first we solve them by graphing and then | |
06:26 | we solve them by substitution and we solve them by | |
06:28 | what we called addition . Okay . So what we | |
06:32 | want to do now is talk about , what does | |
06:34 | the system of equations look like in three dimensions ? | |
06:37 | And not only what does it look like as far | |
06:39 | as the math , but what does it look like | |
06:41 | in terms of physically if we try to graphic , | |
06:43 | what does it look like ? And what kind of | |
06:44 | solutions can we have for a system of equations in | |
06:47 | three dimensions ? So let's take a look at that | |
06:49 | . What if we have a linear system of equations | |
06:55 | in three dimensions ? What does that look like ? | |
06:57 | Well , up here we had something times X plus | |
07:00 | something times Y is equal to a number for all | |
07:03 | of those . So here we have to have a | |
07:05 | system that looks like this two X plus Y plus | |
07:09 | Z is equal to a number . Notice the form | |
07:12 | of this equation is exactly the same as a form | |
07:14 | of this one . It's something times X plus something | |
07:16 | times Y plus now we have something new . Something | |
07:18 | times E . Is equal to a number . It's | |
07:20 | exactly the same form . It just has a new | |
07:22 | variable in it . All right . So I can | |
07:25 | have another equation underneath it . three x plus two | |
07:28 | , Y minus Z is equal to negative two . | |
07:32 | That's another equation there in the system . And then | |
07:35 | I have a third equation , let's say negative X | |
07:37 | minus Y Plus six times Z is equal to 10 | |
07:41 | . So , I have three equations . Notice up | |
07:44 | here I had two equations and two unknowns that allows | |
07:47 | me to solve this system . Notice here I have | |
07:49 | three equations and I also have three unknowns , X | |
07:52 | , and Y , and Z . So , because | |
07:54 | I have three variables , I must have three equations | |
07:57 | to solve it . That is a , that is | |
07:58 | something that's true of any kind of system of equations | |
08:01 | in algebra . If you only have two variables , | |
08:03 | you need two equations to solve it . If you | |
08:05 | have three equations , you need three variables to solve | |
08:07 | it . If you have four variables , then you | |
08:11 | need four equations to solve it . If you're working | |
08:13 | in quantum mechanics and 11 dimensions , right then you | |
08:16 | have 11 variables , you need 11 equations to solve | |
08:18 | it . If you have 65 variables , which sometimes | |
08:22 | actually happens , believe it or not , for very | |
08:23 | complex problems with electric fields and All kinds of weird | |
08:27 | configurations in different directions , 65 variables , you need | |
08:29 | 65 equations to solve it . That's why predicting the | |
08:32 | weather is so hard . People say , Why can't | |
08:34 | we predict the weather ? The weather is no big | |
08:36 | deal . Well , the weather is one of the | |
08:38 | most complex systems we have on the planet , because | |
08:40 | every point in space has a pressure , it has | |
08:44 | a temperature , it has a velocity because the air | |
08:47 | is moving . And there's also other things , like | |
08:49 | there's heating coming in from the sun , there's all | |
08:53 | kinds of other effects . I don't want to get | |
08:54 | into . There's tons of effects that come into play | |
08:56 | for every point . And there's almost an infinite number | |
08:59 | of points and all of those points influence all of | |
09:02 | their neighboring points . So there's tons of variables . | |
09:05 | And so because of that , you need tons of | |
09:06 | equations . So in order to it's really solve the | |
09:09 | weather systems , you have to have computers to crunch | |
09:11 | through all of those equations . Now we only have | |
09:13 | three equations and three unknowns . But for something really | |
09:16 | complex , you might need 1000 equations seriously . And | |
09:19 | when you get into gravity and black holes , you | |
09:21 | could easily have thousands of equations to solve what's going | |
09:23 | on near a black hole with thousands of variables , | |
09:25 | right ? It's true . So we need we have | |
09:29 | three variables . We have three equations . Now the | |
09:31 | question is , if these things look like lines , | |
09:34 | what do these things look like ? So these things | |
09:36 | can't look like lines , but when you have three | |
09:39 | variables like that and they're linear meaning , there's no | |
09:41 | squares , no terms have squares anywhere . These do | |
09:44 | not look like lines . These look like planes . | |
09:47 | When you think about it , this pencil is a | |
09:49 | line , it just goes like this . If you | |
09:51 | take and stretch this thing in the other dimension , | |
09:54 | then this line then becomes a plane . It's still | |
09:58 | kind of flat . There's no curve venous to it | |
10:00 | . There's no beautiful like elliptical shaped or anything , | |
10:03 | it's still flat , it's just flat in another dimension | |
10:06 | . Other than this one , it goes this way | |
10:08 | and it goes this way , but it's flat . | |
10:09 | So these things are lines . These things are planes | |
10:13 | . You kind of have to kind of take my | |
10:14 | word for it a little bit because these are lines | |
10:16 | and this is by extension , these are planes . | |
10:18 | But you can you can convince yourself of that . | |
10:20 | If you were to plot them , I'm not going | |
10:23 | to plot them , it would take too much time | |
10:24 | for us to plot them . But for instance , | |
10:26 | one way in which you could do that is you | |
10:28 | could solve this equation for Z and you can solve | |
10:31 | this for Z and solve this for Z . And | |
10:33 | then you could have an equation in terms of Z | |
10:36 | as a function of X and Y . And we | |
10:37 | talked about that in the last lesson . So for | |
10:39 | instance , here is the X direction , Here is | |
10:43 | the Y direction , here is the Z direction , | |
10:46 | let's pretend I'm not going to use these equations right | |
10:48 | here . Let's say that I solved for some other | |
10:51 | equation of Z . And it came out to be | |
10:53 | something like this , let's say it was X . | |
10:56 | Um Plus to I minus four notice that if I | |
11:00 | solve for Z , I'm going to take and move | |
11:02 | these numbers over to the other side . So I | |
11:04 | I could have used this equation , I just didn't | |
11:06 | , you know , negative two Y negative two , | |
11:08 | X minus Y plus one . Whatever you can see | |
11:11 | there's a number in front of X , a number | |
11:13 | in front of why in a constant . If I | |
11:14 | move these over there's gonna be a number in front | |
11:16 | of X , a number in front of Hawaiian , | |
11:17 | a constant and Z is what's equal to over here | |
11:20 | . So Z is a function of X and Y | |
11:24 | . That means I stick X values in and I | |
11:27 | stick Y values in and I calculate the value , | |
11:29 | the height , the value of Z like this . | |
11:32 | And because there's no squares anywhere . If you actually | |
11:35 | made a table of values and put values of X | |
11:37 | and values of Y and calculate the values of Z | |
11:39 | . What you would figure out is that for a | |
11:42 | given value of X right over here and a given | |
11:47 | value of why ? So an X value in a | |
11:49 | Y value , right ? You would get some value | |
11:52 | of Z . And when you uh so let's say | |
11:56 | this is let's say three for X and four for | |
11:58 | why you would calculate this would give you a value | |
12:00 | of Z for like seven or something . I'm just | |
12:01 | making it up , right ? But then if you | |
12:04 | did it for more and more and more points , | |
12:05 | what would end up happening you would find is that | |
12:08 | this is going to form a plane . Yeah , | |
12:12 | it would form a plane . In other words , | |
12:13 | it would form a surface in three dimensional space where | |
12:16 | the X . And Y plane is underneath it . | |
12:18 | I I take a little points . I calculate the | |
12:20 | value of Z . And that's gonna give me the | |
12:22 | height . Now . The plane might be tilted like | |
12:24 | this , or the plane might be tilted like this | |
12:26 | or the plane might be tilted like this or like | |
12:28 | this . You see the plane can go any which | |
12:29 | way , just like when you have lines up here | |
12:32 | , lines can be any which way as also . | |
12:34 | But when you have a third dimension , the plane | |
12:36 | can be pointed any which way you want . And | |
12:38 | that's why it's almost impossible to graph these things on | |
12:40 | paper . But you can put them into computers and | |
12:42 | see beautiful pictures like that . I'm just showing you | |
12:45 | that if you have equations that has something times X | |
12:47 | plus something times Y plus something times Z is equal | |
12:50 | to a number and there's no squares anywhere , no | |
12:53 | radicals . Nothing crazy . Just the linear things . | |
12:56 | It's always going to look like this if you saw | |
12:58 | for Z something like this , and that's always going | |
13:00 | to yield a plane which is just a line which | |
13:03 | has been stretched in another direction . So , you | |
13:06 | have to accept for a minute that these linear systems | |
13:08 | of equations are always going to form planes . They're | |
13:11 | always gonna look like planes . You cannot predict by | |
13:13 | looking at them how they're going to be oriented , | |
13:15 | but they will always form planes . All right . | |
13:19 | So because they can form planes the way that these | |
13:22 | things make solutions are difficult to predict because this is | |
13:25 | a plane . This equation represents some planes somehow in | |
13:28 | space . This is a separate equation that forms some | |
13:32 | different planes somehow in space . This third equation isn't | |
13:35 | yet a third plane oriented somehow in space . So | |
13:38 | how do we find solutions what the solutions look like | |
13:41 | for three planes that intersect like this ? Because up | |
13:44 | here it was easy for lines . They intersect band | |
13:47 | . There's there's a point , it's easy to see | |
13:48 | , oh , they parallel . Okay , there's no | |
13:50 | solution . Okay , So how do they look when | |
13:52 | you have planes ? Three planes ? What did the | |
13:54 | solutions look like ? So it's gonna get cumbersome here | |
13:57 | , but I'm gonna try to use a prop . | |
13:58 | Okay . Because it's the best I can do . | |
14:01 | Let me see if I can put this down here | |
14:04 | . Um There's a couple of different cases we need | |
14:06 | to look at . All right . The first case | |
14:09 | , if you go back to linear systems of lines | |
14:12 | , we had the parallel line case when there was | |
14:14 | no solution . So , it is possible when you | |
14:16 | have three planes like this to have no solution . | |
14:20 | Sometimes you're going to solve the system of equations so | |
14:23 | there's not gonna be any solution at all . What | |
14:25 | do you think that means if there's no solution , | |
14:27 | that means that the plane's never intersect each other , | |
14:30 | but it means they never intersect at a common point | |
14:33 | . Because remember , a solution is a common point | |
14:35 | between all three . So , if this is a | |
14:37 | plane and this is a plane and this is a | |
14:40 | plan , I don't have three hands unfortunately . But | |
14:42 | if you can visualize this is a plane coming to | |
14:44 | you , the yellow plane is coming parallel exactly parallel | |
14:47 | to this one . And the third plane is also | |
14:49 | exactly parallel . Maybe there's like this , Maybe they're | |
14:52 | angled like this , but they're all parallel , then | |
14:54 | they never intersect . Those planes will never have a | |
14:57 | common points , so they'll never have a solution . | |
14:59 | So , no intersection . I can spell intersection , | |
15:08 | which means you have parallel . That's what that means | |
15:11 | planes . So basically , how do you draw that | |
15:14 | ? How do I draw that on ? You know | |
15:17 | , on a board ? I don't know . You | |
15:19 | could say here's a plane , Here's a plane , | |
15:21 | Here's a plane . I'm drawing them as lines . | |
15:23 | This is a top view . In other words , | |
15:28 | you have to use your imagination , but this is | |
15:30 | a plane right here . Then this is a plane | |
15:32 | right here . Then this is a plane right here | |
15:33 | . And you're looking down on them plain plain plain | |
15:35 | . They never intersect . So there's no solution . | |
15:37 | So , that's one way that you can get no | |
15:39 | solution . All right . And then what is the | |
15:43 | other thing that you can have ? You can have | |
15:44 | no solution . Let me see . We'll check my | |
15:46 | notes here . You can have no solution . They | |
15:48 | don't have to be parallel to have no solution . | |
15:51 | They can also have no solution if they just don't | |
15:53 | intersect at a common point . So let's take a | |
15:56 | look at another example of this . What if I | |
15:58 | had one plane that went over like this ? Looking | |
16:02 | again down like one plane that's like this and then | |
16:05 | another plane that intersects it like this . But then | |
16:08 | the third plane goes like this . So you see | |
16:11 | I have three planes . I have one plane like | |
16:14 | this . One plane like this in one plane like | |
16:16 | this . But you see I do have two of | |
16:18 | the planes intersecting and two of the planes intersecting , | |
16:21 | and two of the planes intersecting , but they never | |
16:23 | all three intersected the same spot . Right ? So | |
16:27 | here's a plane , right ? I can't do this | |
16:29 | without being crazy awkward . There's an intersection point between | |
16:33 | planes , right ? But then the third one intersects | |
16:35 | the other two , but never at the same spot | |
16:37 | because a solution has to be common to all three | |
16:40 | of them . So this is there's no point here | |
16:42 | that's common to all three of them . You know | |
16:44 | like this . You could also have a plane like | |
16:46 | this , A plane like this , A plane like | |
16:48 | this . So yes , you have intersection points between | |
16:50 | two of the three , but never among all three | |
16:53 | . So , it's very common to have no solution | |
16:55 | for planes because it's very easy to have them oriented | |
16:59 | where they're not all going in the same spot . | |
17:02 | Okay , now , you can also have an interesting | |
17:06 | case where you have infinite solutions . All right . | |
17:16 | And it's going to be a little bit easier for | |
17:18 | me to gesture this by pretending that one of the | |
17:22 | planes is actually the chalk board , the board here | |
17:24 | . So pretend that one of the three of the | |
17:27 | planes , one of the three planes actually this board | |
17:29 | right here . If I take another plane intersect it | |
17:33 | , then you can see that what's going to happen | |
17:35 | . I mean , just with these two planes , | |
17:36 | I mean you got to pretend that this yellow plane | |
17:38 | is going through an intersecting . You see , there's | |
17:41 | an infinite number of points here that are common to | |
17:44 | both of this plane . This plane is here and | |
17:46 | this plane is right here . There's an infinity number | |
17:48 | of points right here that are common to this . | |
17:51 | So then if I take my third plane , let's | |
17:53 | say here is one of the three planes . Here's | |
17:55 | one of the three planes . Here is one of | |
17:57 | the three planes and it intersects exactly on this line | |
17:59 | . You see , this one also goes through the | |
18:02 | board . This one also goes through the board and | |
18:03 | they go through it exactly the same spot . There's | |
18:06 | an infinity number of points right along this line that's | |
18:09 | common to this plane , to this plane , and | |
18:11 | also to this plane all at the same spot and | |
18:14 | there's an infinite number of them . So if you | |
18:16 | had , if you wanted to draw that , it's | |
18:18 | very difficult . But from a top view , you | |
18:21 | could say you had one plane , here's another plane | |
18:23 | and the third plane goes right through top view , | |
18:28 | looking down on the plains , it looks like a | |
18:30 | single point . But really these are all planes . | |
18:33 | So this thing forms a line that goes down there | |
18:35 | and that's why there's an infinite number of points . | |
18:37 | So sometimes you'll solve your system of equations in three | |
18:40 | dimensions and you won't get a single answer . You'll | |
18:42 | actually get an infinity of answers . And that's because | |
18:45 | the three planes came together in such a way that | |
18:47 | they all just kind of found one infinity of one | |
18:49 | line , which forms an infinite number of points like | |
18:53 | this . Now , finally I saved the best for | |
18:56 | last . There is a way in which you can | |
18:58 | have one single solution . You can also have one | |
19:00 | solution right now . Again , it's gonna be easier | |
19:06 | for you to pretend that the board is one of | |
19:10 | these planes . What if the board was one of | |
19:12 | the three planes and then I have one of this | |
19:17 | guy coming in like this . So you see if | |
19:19 | I have him going through this forms an infinity of | |
19:22 | commonality between them . But then if I take this | |
19:24 | one and it's really hard to do because I can't | |
19:26 | cut through . But what if I have this plane | |
19:28 | going through as like this as well , cutting through | |
19:32 | the orange one and cutting through the board down here | |
19:35 | ? So you see what's going to happen is this | |
19:37 | is an infinity number of points . But once this | |
19:39 | one slices through , there's only gonna be one point | |
19:41 | where all of them kind of touch , right in | |
19:44 | the middle , I can try my best to draw | |
19:46 | it , but it's gonna be different . Let's difficult | |
19:48 | . Let's say I have a plane here in a | |
19:50 | plane here . I'm looking at a top view . | |
19:53 | Mhm . So there's an intersection point plane here and | |
19:57 | a plane here just like this and this forms a | |
20:00 | line of infinite solutions . But then the third plane | |
20:04 | is actually uh is actually kind of like perpendicular to | |
20:10 | those . So , in other words , it's a | |
20:12 | flat board and you have these two planes cutting through | |
20:15 | it . So , even though it would have been | |
20:18 | an infinity a line of infinite points , it cuts | |
20:21 | through a board . And so there's really only one | |
20:24 | point right here in the center where everything crosses criss | |
20:27 | cross and the other one cuts through one point one | |
20:32 | solution , it's a single point , It would be | |
20:35 | like X comma y comma z , it would be | |
20:37 | some number and that would be the single solution . | |
20:40 | So again , it's difficult because everything is in three | |
20:43 | dimensions and it's very hard to gesture , but you | |
20:45 | can very easily have no solutions if you have three | |
20:48 | parallel planes , or if the three planes cut in | |
20:50 | such a way that they don't all go in the | |
20:52 | same , cut in the same exact location , that's | |
20:55 | gonna give you no solution at all . Or if | |
20:57 | they cut through each other where they all go through | |
21:00 | a common line , then you have an infinity of | |
21:01 | solutions common to both . Or if you have to | |
21:04 | that form kind of a line of solutions and have | |
21:06 | the third one slice through that . Then you're only | |
21:08 | gonna have one point right at the intersection of all | |
21:11 | three of them . That's going to form a single | |
21:12 | point X comma , Y commas e that's going to | |
21:15 | be your solution . So what we need to do | |
21:17 | now is keep this in the back of our mind | |
21:21 | while we solve our first set of equations , because | |
21:25 | the type of solution you get is going to be | |
21:27 | one of these essentially . So our first system is | |
21:31 | going to look like this . Let's say we have | |
21:33 | two X plus three , Y plus two . Z | |
21:38 | Is equal to 13 . The next equation is two | |
21:41 | , Y plus Z is equal to one . And | |
21:44 | the third equation is Z is equal to three . | |
21:48 | So you might say this does not look like you | |
21:51 | told me it would look , you said it would | |
21:53 | look like this , but notice this is something times | |
21:55 | X plus something times Y plus something times Z is | |
21:57 | equal to a number , something times X plus something | |
21:59 | times Y plus something times . He is , he | |
22:01 | will know the same thing for here here , it's | |
22:03 | the same thing . It's still something times X . | |
22:05 | It's just zero . And then something times why in | |
22:08 | something times for this one , it's still something times | |
22:10 | X . It happens to be zero X , zero | |
22:12 | Y . And something times the so you see all | |
22:14 | of these still form planes . Um uh it's just | |
22:18 | that some of the variables are set equal to zero | |
22:20 | . So they all still form three planes in a | |
22:23 | space and they're all going to intersect in some kind | |
22:25 | of way . Now I want to point out to | |
22:27 | you that the shape of this thing kind of looks | |
22:31 | like a triangle like this . Notice has got a | |
22:34 | point in here goes down like this . This is | |
22:36 | called triangle for when a system of equations in three | |
22:44 | variables in general a system of equations . Let me | |
22:47 | just back up for a second . If you wanted | |
22:50 | to solve this thing by substitution , you can definitely | |
22:53 | do it . But it's very hard to do . | |
22:54 | It's just it's not crazy heart . It's just a | |
22:57 | lot of work on the paper to do it . | |
22:59 | Why is that the case ? Because if I wanted | |
23:01 | to solve this by substitution , I have to pick | |
23:05 | two equations to solve for different variables and then substitute | |
23:08 | into the third one . So I could solve this | |
23:10 | one for X . X . Is equal to some | |
23:12 | junk . And then I would have to pick this | |
23:14 | one in salt for why ? Why is equal to | |
23:16 | some junk ? I move everything over once . I | |
23:18 | have X . And Y . I take both of | |
23:19 | them and put them into here . And then I'm | |
23:22 | gonna have and be able to solve for something and | |
23:23 | then I have to back substitute several times . And | |
23:26 | so you're taking two equations solving for different things . | |
23:28 | You're plugging in your rearranging your plugging it again and | |
23:31 | it's just it's kind of a spaghetti mess . Okay | |
23:33 | If your system of equations already looks like this , | |
23:37 | it's much , much easier to solve . And let's | |
23:40 | look at why that is the case . We call | |
23:42 | this triangular form . It makes it easy to solve | |
23:44 | . Why ? Because then I can just take the | |
23:46 | Z . Value which is already given to me and | |
23:48 | I can plug it in directly into the equation above | |
23:52 | . I don't want to put it into this one | |
23:54 | . If I do put it into this one that's | |
23:55 | fine . But I don't have X and Y , | |
23:57 | which is still unknown and I cannot solve for them | |
24:00 | . But if I put it into here then I | |
24:02 | can solve for something . So the way I do | |
24:04 | that , as I say two times Y plus Z | |
24:09 | . But nano Z is equal to three is why | |
24:12 | I just take and plug it into that second equation | |
24:14 | To Y is equal to subtract to get -2 . | |
24:17 | Why is equal to negative one ? So now I | |
24:19 | know that Y is equal to a number . It's | |
24:22 | one of the three numbers I need I need X | |
24:23 | . I need Y and N . E . Z | |
24:25 | . And actually I know what Z is equal to | |
24:27 | and now I know why is equal to . So | |
24:29 | actually all I need is to know what X is | |
24:30 | equal to . How do you think I figured that | |
24:32 | out ? Well , now I know what she is | |
24:34 | now I know what why is I take both of | |
24:36 | those and stick them back in here . So what | |
24:38 | I do is I plug in , why is equal | |
24:42 | to negative one and Z is equal to three in | |
24:46 | to the following equation . The big one at the | |
24:48 | top two X plus three . Y plus two . | |
24:51 | Z is equal to 13 . I just stick these | |
24:54 | values in so I get to X plus three times | |
24:58 | wide which is negative one plus two times Z which | |
25:01 | is three , it's 13 . So I have two | |
25:05 | X . This becomes negative three , this becomes six | |
25:10 | like this . I add these together to X Plus | |
25:16 | three , add these together . Get three is equal | |
25:17 | to 13 and now i subtract 13 minus three is | |
25:22 | 10 and then I divide by two and I get | |
25:25 | five I get X . Is equal to five . | |
25:27 | So now I know X is five and why is | |
25:28 | negative one ? N Z is equal to three . | |
25:30 | So now I can write my solution , it's going | |
25:32 | to be only one solution is just one point that | |
25:35 | I found . The X value was five , the | |
25:38 | y value is negative one and the z value was | |
25:42 | three . And this is of course X comma Y | |
25:45 | comma Z double check myself five , comma negative one | |
25:48 | , comma three . All right now this was so | |
25:52 | easy to solve because the system was already in what | |
25:56 | we call Triangle form , triangle forms really easy because | |
25:59 | it basically means that one of the variables has already | |
26:01 | given to you , essentially you take that single variable | |
26:04 | that was given to you and you put it into | |
26:05 | the next bigger equation right above . And because it's | |
26:09 | a triangle , if you stick it in here , | |
26:10 | you're always going to be able to find the next | |
26:12 | variable . Once you have those two , you stick | |
26:14 | them both into the triangle above . So you kind | |
26:16 | of work above the up and up and up like | |
26:18 | this , eventually getting to where you find that third | |
26:21 | variable . Now this system of equations is not in | |
26:24 | triangle form . So what we're gonna do is we | |
26:27 | solve these is we're going to get a little more | |
26:28 | practice with solving these in triangular form to make sure | |
26:31 | you kind of get a little more practical , what | |
26:32 | I just showed you here , and then we're going | |
26:35 | to do these kinds of equations here where the system | |
26:37 | is not in triangular form like this , but guess | |
26:40 | what we're gonna do ? We're going to learn a | |
26:42 | technique to take this system and put it into triangular | |
26:46 | form . We're gonna be able to take any system | |
26:48 | like this and manipulate it so that it always looks | |
26:51 | like a triangle like this . So then we can | |
26:54 | do the easy solution method . Now , if you | |
26:55 | look at other algebra books , even the ones I | |
26:58 | learned on way back in the day , we may | |
27:01 | have learned this , maybe we didn't , maybe we | |
27:02 | did , but we learned other ways to solve these | |
27:04 | systems . But almost all the other ways are actually | |
27:06 | way more cumbersome trying to substitute backwards , substitute all | |
27:10 | these different ways , trying to add them and eliminate | |
27:13 | to everything all at once . It's very , very | |
27:16 | difficult to do the way that we're gonna learn here | |
27:18 | is bulletproof and it works for everything if it's already | |
27:21 | in triangular form , do this method and it works | |
27:23 | every time . If it's not in triangular form , | |
27:25 | we're gonna manipulate it to put it into triangular form | |
27:28 | and then we're gonna do the same technique and we'll | |
27:29 | get the solution correct every single time with a minimum | |
27:32 | of heart of heart Farm . So we did a | |
27:36 | lot in this lesson . We learned about the concept | |
27:39 | of linear systems . In two dimensions , they form | |
27:41 | lines , either they intersect and they have one solution | |
27:43 | or their parallel and they have no solutions . A | |
27:46 | linear system in three dimensions do not form lines . | |
27:48 | They form planes , three planes which can be oriented | |
27:51 | and intersect in very weird and interesting kind of ways | |
27:54 | . You can have them not intersect at all or | |
27:57 | intersect in ways where they don't all come together at | |
27:59 | once and that has actually no commonality among all three | |
28:02 | . So there's no solution or they can intersect in | |
28:05 | such a way that they form an infinity of solutions | |
28:07 | that run along the intersection line . So you have | |
28:09 | an infinite solution . This is also called by the | |
28:11 | way , a dependent set of equations . It's called | |
28:14 | dependent set of equations . When you have infinite solutions | |
28:17 | like that , or you can just simply have one | |
28:19 | point is a solution where you have a plane where | |
28:22 | two more planes crisscrossing go into it . You only | |
28:24 | have one point and that is the exact kind of | |
28:27 | solution that we had or the type of system that | |
28:29 | we had here , we had one point , you | |
28:31 | cannot figure out by looking at your system if it's | |
28:34 | going to have one solutions , no solutions or infinite | |
28:36 | solutions , you have to try to solve it . | |
28:38 | So follow me on to the next lesson after you | |
28:40 | have practiced this , we're gonna get some more practice | |
28:43 | with solving these triangle systems by substitution . And then | |
28:46 | we'll work on the more general , more complex uh | |
28:50 | linear systems and three variables . As I mentioned , | |
28:52 | we're gonna make them into triangular form before we solve | |
28:55 | them . So we'll work on that at the very | |
28:56 | last part . So make sure you can do this | |
28:58 | . Following on to the next lesson will continue building | |
29:01 | your skills right now . |
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