12 - Solving 3-Variable Linear Systems of Equations - Substitution Method - Free Educational videos for Students in K-12

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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