06 - Solve Quadratic Systems of Equations by Substitution - Part 1 (Simultaneous Equations) - Free Educational videos for Students in K-12

06 - Solve Quadratic Systems of Equations by Substitution - Part 1 (Simultaneous Equations) - By Math and Science

Transcript


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

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