05 - Quadratic Systems of Equations (With Lines, Circles, Ellipses, Parabolas & Hyperbolas) - Free Educational videos for Students in K-12

05 - Quadratic Systems of Equations (With Lines, Circles, Ellipses, Parabolas & Hyperbolas) - By Math and Science

Transcript


00:00	Hello . Welcome back to algebra . The title of
00:02	this section is called quadratic Systems of equations . Now
00:05	it sounds very , very complex but actually it's not
00:09	at all hard to understand what's going on here .
00:11	What we're gonna do in this lesson is given overview
00:13	of what a quadratic system really is . We're gonna
00:15	draw lots of pictures in this lesson . We're not
00:17	gonna have any , hardly any equation , certainly no
00:19	difficult problems to solve . This is a concept lesson
00:23	. We're going to understand the concept here and then
00:24	in the next lesson , I'm gonna show you how
00:26	to solve quadratic systems by the techniques that will learn
00:29	in the next lesson . So the next lesson will
00:31	be a lot more math . This lesson will be
00:32	a lot more pictures which is fun sometimes to make
00:35	sure you understand the concepts . Now if you remember
00:37	back , we already talked about what a system of
00:40	equations is but in the past we called it a
00:43	system of linear equations , linear is the word there
00:46	. In the past that we use linear means line
00:48	. So when you have a system of linear equations
00:51	, it means you have more than one . Usually
00:53	we were talking about two lines and the system is
00:56	basically the solution of that system is where the lines
00:59	cross . If the lines have a crossing point ,
01:02	they're only gonna be one crossing point in that intersection
01:04	point common to both lines is what we call the
01:06	solution . So there's one solution if there's one crossing
01:09	point . But you all know that lines can also
01:12	be parallel where they never intersect at all . In
01:14	that case we say that there is no solution of
01:17	that linear system of equations . So we've done all
01:20	that in the past , but now that we have
01:22	under our belt , the comic sections , we have
01:24	circles and parabolas , ellipses and hyperbole . And we
01:28	also have lines that we can graph also . So
01:30	we can have systems of equations that involve these quadratic
01:33	functions . Quadratic just means it has a square term
01:36	and of course , you know all the circles and
01:38	ellipses , they have x squares and y squares everywhere
01:41	. So all of those we call them comic sections
01:42	. There also quadratic in nature because they have squares
01:46	running around the variables . All right , so the
01:49	bottom line is we now have systems of equations called
01:51	quadratic systems where we graph more than one comic section
01:55	, like a circle and the ellipse or a circle
01:57	and a hyperbole or parabola and a circle or something
02:02	like this . And we're looking to solve by finding
02:04	those intersection points from among those uh quadratic equations ,
02:09	among those uh ellipses , hyperbole and so on .
02:11	Also we can have lines thrown in there . So
02:14	we can have you know , no intersection points or
02:17	one or two or three up to four intersection points
02:20	. So now we need to start to draw pictures
02:22	because it's actually very easy to understand . So let's
02:25	recall things that we already know . We already know
02:27	that there's a thing called a line out there .
02:29	Right ? What is the general equation of a line
02:32	? Of course I can give you the the most
02:34	general version , but I'm just talking about give me
02:35	an example of a life . Well , a lion
02:37	might be something like why is equal to three X
02:40	plus two ? How do you know it's a line
02:42	? Well , because the X . Term does not
02:44	have a square or any other higher power . So
02:46	it has to be aligned . Same with the Y
02:48	. If the X . And the Y . Variable
02:49	don't have any squares or higher powers , then it
02:52	has to be pretty much be a line . All
02:54	right , so this is some line , this is
02:56	the Y intercept , this is the slope . Now
02:58	, I'm not gonna graph this this line that's going
03:00	to take too much time . But in general ,
03:02	lines can have uh that can have slants up into
03:05	the right like this . They can have slants down
03:07	uh and down into the right like this . So
03:10	this is a positive slope line . This is a
03:11	negative slope line . Right ? And you can also
03:15	of course have horizontal lines . You can have vertical
03:18	lines , Right ? So that is what we studied
03:21	in the past when we had two of those lines
03:23	graft on the same graph paper , we called it
03:25	a system of linear equations . And we were looking
03:28	for the intersection points , but now we have a
03:31	much richer set of of Connick sections that we know
03:37	how to talk about the 1st 1 . Let's talk
03:39	about a problem . Just reminding you what we've already
03:43	learned , what would be an example equation of a
03:45	parabola . You know , it might be something like
03:49	y is equal to three parentheses , X minus two
03:52	quantity squared . How do you know it's a problem
03:55	? Well , because the X term is the one
03:57	that squared in the UAE term , isn't that pretty
03:59	much always means it's going to be a parabola .
04:01	The shift in here tells you where the center of
04:03	the vertex is going to be . There is no
04:05	shift and why . And this tells you if it's
04:07	opened up or down now , as you know ,
04:09	when you graph these problems because we've done it so
04:11	many times , you might have a problem that goes
04:13	down and up like this , or you might have
04:15	a problem that opens upside down , so a smiley
04:18	face or a frowny face in general . Those are
04:20	the shapes of the problems that we care about .
04:23	Okay . And then of course we all we also
04:26	know that we can have uh parabolas , left and
04:29	right . Also , I'm not having drawn those .
04:31	That's if the Y term is squared , but the
04:33	X term is not squared . We study those in
04:35	the past as well . All right , so for
04:37	the simplest case of a circle , what does that
04:41	look like ? Just give a simple equation of a
04:43	circle , right ? You might have something like X
04:46	squared plus Y plus four , quantity squared is equal
04:51	to four . So it's the X . Turn that
04:53	squared and the Y term that squared . Then it's
04:57	either gonna be a circle or a any lips .
05:00	It's gonna be a circle or any lips . And
05:02	in the form of an ellipse look slightly different .
05:04	So we know that this is a circle . The
05:06	radius is equal to to the square root of the
05:08	right hand side . This is the shift in the
05:10	center and the shift on the X direction has no
05:13	there's no shift at all . But in general ,
05:15	what does the circle look like ? Again ? Not
05:17	drawing a real graph of this , but a circle
05:20	looks like a circle . And of course , we
05:22	can move all of these all over the xy plane
05:24	, depending on with the shifting values that we have
05:28	. All right , So let's crank along here after
05:30	a circle we studied in the lips , which ,
05:34	as you know , is very similar to a circle
05:36	. It's just a stretched version of that . So
05:39	, an example of an equation of lips might be
05:41	X plus three , quantity squared over four plus y
05:47	minus three , quantity squared over two is equal to
05:51	one . How do you know it's an ellipse What
05:53	you have an X squared term plus Y squared term
05:56	. But you have numbers on the bottom that determine
05:58	how it stretched in the X . And the Y
06:00	. Direction . And the right hand side is equal
06:01	to one . And we've studied many , many ellipses
06:03	and in the shift in the X . Direction ,
06:06	in the Y direction is read directly off of the
06:08	graph like this . So what can an ellipse ?
06:09	More or less look like ? Well , we've studied
06:12	the fact that you can have horizontal ellipses right ?
06:15	And you can also have vertically oriented lips . Is
06:18	that all depends on the numbers that are on the
06:20	bottom here . So you see , I'm drawing all
06:22	these because I want to remind you all the different
06:24	shapes we have to play around with because when we
06:26	have our quadratic systems , we're gonna mix them all
06:28	together . Now . The last one we have that
06:30	we've studied is the hyperbole . What does the general
06:35	equation of hyperbole look like ? All right . Well
06:39	, it might look something like X -4 quantity squared
06:43	over two minus y plus three quantity squared over four
06:51	is equal to one . How do we know this
06:53	is a hyperbole and not an ellipse ? Well ,
06:55	it's because there's a minus sign here , it still
06:57	has an X squared term in a y squared term
06:59	but it's linked with a minus sign . Whereas the
07:00	ellipse is linked with a plus sign . The numbers
07:03	on the bottom determine the assam tops , which kind
07:05	of helps the sketch the thing but more or less
07:07	. What does this thing look like ? We said
07:08	? Hyperbole can look kind of like this horizontal versions
07:12	of the hyperbole . Or we could have vertical versions
07:16	of the hyperbole as well . The center of the
07:17	hyperbole is right between the two curves . All right
07:20	. So take a look at what we have on
07:21	the board . We have lines , we have parabolas
07:23	, we have circles , we have ellipses and we
07:25	have hyperbole is right . So those are all the
07:28	context sections . Of course the line is not a
07:30	comic section , but we can still use we still
07:33	have systems of equations that involve lines as well ,
07:35	so we just throw it in there . All right
07:37	. So a linear set of equations is just equations
07:40	of lines . We've learned how to solve them .
07:43	We said we can solve them graphically . That's when
07:45	you graph them and look for the intersection point .
07:47	You can use addition and you can use substitution .
07:50	So in this lesson we're not doing any of that
07:52	stuff , we're just sketching some things to show you
07:54	how you can have a different number of solutions .
07:56	A quadratic system can have uh let me go and
08:01	write that down . Actually quadratic system , which means
08:07	a system that involves one of these two of these
08:09	equations that we've written on the board there , it
08:12	can have for solutions , it can have 0123 or
08:19	up to four solutions . Now , when I talk
08:21	about solutions , I'm talking about real solutions , so
08:24	I'm gonna talk about real solutions right in this class
08:30	, we're not going to be focusing on imaginary solutions
08:33	of systems of equations . We're just not going to
08:35	talk about that if they intersect , we say that
08:38	those are the real solutions , the actual intersection points
08:40	. If there's no intersection point , we're not discussing
08:43	any imaginary solutions or anything else because that's beyond the
08:45	scope of this class . So we're just looking at
08:47	the intersection points . So let's go down a trip
08:50	down memory lane here , let's talk about some really
08:52	, really simple cases . First of all , forget
08:55	about the rest of the comic sections . Let's say
08:56	we have two lines that are just lines and they
08:59	cross like this . How many solutions are there to
09:02	this ? Well , there's one solution why ? Because
09:06	there's only one intersection point . The point here is
09:08	common to both lines . So because it shares commonality
09:11	with both , it satisfies both equations . And so
09:14	because of that , it is a solution . Now
09:16	. What if you have a line like this and
09:19	then a line parallel to it like this ? So
09:22	you see these lines never cross . So we say
09:24	that we have zero solutions . It's very , very
09:28	common when you saw the system of equations to not
09:30	have any solution at all . It doesn't mean it's
09:32	magical or mystical or what does that mean ? It
09:35	just means that the graphs don't cross . So there's
09:37	no commonality between the two . So there's no solutions
09:40	that satisfy both of the equations . Now let's crank
09:42	up the complexity a little bit . Actually , none
09:44	of this is hard , but uh , I want
09:47	to make sure you really understand what if you had
09:49	. And we're just gonna give some examples here .
09:51	What if I had a circle ? All right .
09:54	And then a line that goes through the circle like
09:58	this . How many solutions do I have here ?
10:00	So I can have an equation , a system of
10:02	equations that has a equation of a circle might look
10:05	something like this and then right next to it .
10:07	The other equation might be a line that looks like
10:09	this . So if I were to plot them ,
10:10	I would say I have two intersection points of there's
10:13	two solutions . What solutions . If I were to
10:17	graph them on a sheet and a graph paper ,
10:19	of course I could figure out the intersection points by
10:21	biographical methods and I could , you know , go
10:25	to town . However , I might have um let's
10:29	draw a little dividing line here . I might have
10:33	a circle with a line that never crosses it .
10:37	Like this . This is zero solutions , Right ?
10:43	zero solutions . Because there's no just like there's no
10:45	solutions here . There's no solutions here . Now let
10:48	me ask you this . You might think a line
10:49	in a circle is always going to have to two
10:52	solutions . But what if I have a special case
10:53	where I have a circle like this , right ?
10:56	But then the line , just see if I can
10:59	draw it , just grazes the surface right here .
11:03	Only one location . If I took a microscope and
11:05	zoomed into this thing , I would find that that
11:07	line only touches the edge at one exact location .
11:10	Because I can surely shift this line up and up
11:13	and up and up to . It just touches the
11:14	surface in one place that's called a tangent line ,
11:18	a line tangent to the circle . So if you
11:20	only touch it in one location , there's only one
11:22	point of commonality . There's only one solution . So
11:26	you see even in the case of the circle you
11:28	can have a circle in the line , you can
11:29	have zero solutions , You can have one solution if
11:32	it just touches the edge and you can also have
11:33	two solutions . If it goes through the center or
11:36	not even through the center , just some goes through
11:38	both sides of the circle like this . All right
11:41	. Um And then of course you can play around
11:43	with the other possibility to that's just a circle and
11:45	a line . But let's take a look at the
11:47	possibilities for a problem . Let's say I have a
11:49	problem . I joined it like this but it could
11:51	be flipped upside down and I have a line that
11:54	goes through like this . Okay , so there's an
11:56	intersection point here and here . So there's two solutions
11:59	. If I can draw , I cannot draw solution
12:02	two solutions , right ? But of course I could
12:04	have a different Perabo one , maybe that goes upside
12:07	down like this and I can have a line that
12:10	just touches the very tippy top of that parable like
12:14	this only at one location . That could be one
12:16	solution . This tangent line . I could draw it
12:20	down here just touching the problem in one location .
12:22	I can touch it here , touching them in one
12:24	location . This parable is curved all the time .
12:26	So I can always find a line to just touching
12:28	in one location . By the way , this concept
12:30	of a tangent line , it's never gonna go away
12:33	. In fact , almost the whole subject of calculus
12:36	when you get into calculus is all about tangent lines
12:39	, I know you might think well who cares about
12:40	tangent lines . It seems so completely worthless as a
12:43	concept . But just trust me when we get into
12:45	calculus you'll see the and understand the necessity of studying
12:49	this stuff . Lines that are tangent two curves .
12:52	It really does cover about half of calculus one .
12:54	So I kind of got to get used to the
12:56	idea . So that's a parable on the line .
13:00	Um and then of course we can have a parabola
13:05	like this and then we can have a line up
13:07	here . So there's no solution at all . So
13:10	you can see the idea here , I can keep
13:11	drawing things and I do have a few more I
13:13	do want to draw . But the bottom line is
13:15	I said that quadratic systems can have real solutions that
13:19	can have zero solutions . One solution to Solutions three
13:22	solutions or even four solutions . And we've already seen
13:24	on the board we can have zero , we can
13:26	have one , here's 1102 and no solution . So
13:30	we've already drawn quite a bit of possibilities already but
13:33	we have some more possibilities . I like to draw
13:36	for you um just so that you know , I
13:38	do the thinking kind of for you ahead of time
13:40	. Let's draw something a little more complex . Let's
13:42	say we have a circle that we have plotted and
13:46	any lips that just cuts into that circle something like
13:48	this . You see there's two intersection points , one
13:50	right here and one right here solution to solutions right
13:54	? You can pick up the pace a little bit
13:56	. What if I move that ellipse over to the
13:57	right , just a little bit so that it only
14:00	touches at one location . Now , the way I
14:01	drew , this is not the best , it looks
14:03	like I intersected , but I'm trying to draw it
14:05	touching just in one location . That's a terrible ellipse
14:08	by the way . And this is only one solution
14:11	. So you can see you can have one solution
14:12	even in the case of ellipses and um and circles
14:17	. Okay , what if I have two ellipses ?
14:19	What if I have a horizontal lips that goes around
14:22	like this ? And then I have I can have
14:23	a vertical lips that goes here . Now you can
14:25	see that's how you get to your four solutions ,
14:28	Right ? Because I can have circular objects , meaning
14:31	in the lips is a circular kind of shape to
14:33	it even though it's stretched and they can intersect in
14:35	such a way that you can have four um intersection
14:38	points . Now let me ask you a question here
14:40	is kind of a trick question you have to think
14:41	about . So I can see how I can have
14:44	one solution if if the circle and the ellipse touch
14:47	in one location , I can have four solutions if
14:49	they fully cross each other , how can I have
14:52	a circle and ellipse cross in only three locations .
14:55	It actually takes a minute of you thinking about that
14:57	. How can you have a circle ? And the
14:58	lips intersect ? But it only in three places and
15:02	it's kind of weird at first . But the way
15:04	that can happen is something like this , I can
15:06	have for instance a circle like this and I can
15:10	have any lips that looks something like this . It
15:12	only goes and touches this border of the circle in
15:14	one location , but then it comes up and crosses
15:17	and of course that's a terribly lips but you get
15:19	the idea it crosses 12 and this is only touching
15:22	in one location . This is tangent right here .
15:25	So this is three solutions . Okay ? And then
15:29	of course you can have special cases when you have
15:31	two circles that can have two circles just kissing each
15:34	other and where they just touch in one location .
15:36	So this is one solution . Mhm . And I'm
15:39	gonna pick up the pace a little bit since we're
15:41	getting the point of it here , I can have
15:43	hyperbole as I can have what happens when you have
15:44	hyperbole . Some lines and hyperbole is um circles ,
15:46	you can have all kinds of things so I can
15:48	have a hyperbole to and I can have a line
15:50	going through it cutting into locations , right ? I
15:54	can have a parabola , right ? Plus a hyperbole
15:59	. A parable . Plus a hyperbole . I can
16:00	have the hyperbole . Let me draw the hyperbole in
16:02	another colour . I can have the hyperbole come in
16:04	like this and the hyperbole come in like this ,
16:07	you see it crosses and that's not symmetric , it's
16:09	not drawn properly , but you can see that you
16:11	can have four crossing points . Four solutions ? Yes
16:15	. Right , so I can have two solutions for
16:17	a line in hyperbole . I can have four solutions
16:19	for something like this . And actually I can have
16:22	a parable and hyperbole that actually only has three solutions
16:26	as well . So I can actually draw the hyperbole
16:28	instead of drawing up just horizontally like this . I
16:31	can draw the hyperbole to like this if I want
16:33	to . There's nothing saying I can if you kind
16:36	of tilt your head sideways , you can see the
16:38	hyperbole comes in like this , It's just I've drawn
16:40	it uh in a different kind of direction , right
16:43	? And I can have a um a parabola come
16:49	in and then down like this . So this is
16:51	a parabola . Plus the hyperbole have +123 crossing points
16:55	, three solutions . Okay . And then there's only
16:59	one more I'm going to do for you . And
17:01	that is I can have any lips that could look
17:05	something like this and then I can have a hyperbole
17:09	to that comes in like this . And then the
17:12	hyperbole doesn't even the other side of the hyperbole doesn't
17:15	even touch this guy . So there's two solutions here
17:18	. Yeah . All right . I could have simplified
17:20	this entire lesson if I wanted to . I could
17:22	have just said , hey guys , there's these things
17:24	called comic sections . Plus , you know , we
17:25	have our lines , we can have what we call
17:27	a quadratic system . We have at least two of
17:29	these things plotted on one sheet of graph paper and
17:32	you can have 0123 or four crossing points . And
17:36	I could have just left it at that and I
17:38	would have been fine . But I really like sketching
17:40	a few because what's going to happen in the next
17:42	lesson is we're going to start solving these systems mathematically
17:44	solving them . Sometimes you're gonna get no solution .
17:47	Sometimes you're gonna get one solution . Sometimes you're gonna
17:49	get three , sometimes you're gonna get four . And
17:51	if you don't have this in your mind and you
17:53	don't even know why you're getting different answers sometimes right
17:56	? But now you can see why because it's just
17:58	a physical nous of how the things are outlined .
18:01	If you or how the things are graft . If
18:02	you were to take the problems in the next few
18:04	lessons and graph them all you would immediately see the
18:06	crossing points . But we're not gonna be graphing them
18:08	in the next lesson . You know when we learn
18:11	system of equations the first time the first thing we
18:13	did is graphed them to find the solutions . But
18:15	you see how hard that would be for quadratic systems
18:17	because you know , we've sketched hyperbole is and parabolas
18:21	, you can sketch them of course and you can
18:23	see roughly how many there are . But if you
18:24	wanted to get the exact values here is really hard
18:27	to do on graph paper . I mean you have
18:29	to really plot a lot of points and get exact
18:31	because the curving nature of it with lines , it's
18:33	very simple . Everything is very easy to kind of
18:36	line up so we can do it graphically . But
18:38	for quadratic systems , graphical just doesn't get you anywhere
18:41	. So in the next lesson , what we're gonna
18:42	do is we're going to learn how to solve these
18:46	things by substitution , which we've also done for the
18:48	linear equations and we're also going to learn how to
18:50	solve them by addition . Which we've done in the
18:54	in the linear system as well . So we're gonna
18:56	be using those techniques and we're gonna be applying them
18:59	the quadratic systems . Sometimes you're gonna get zero ,
19:01	sometimes you're gonna get one solution , sometimes you're gonna
19:03	get to solutions sometimes three sometimes four solutions in the
19:07	back of your mind when those solutions pop out of
19:09	your math , I want you to remember why they
19:11	pop out that way because of the physical nature of
19:14	whatever it is you're trying to solve . We're looking
19:16	for the intersection points and those are going to yield
19:19	the real solutions to the system . So follow me
19:21	on to the next lesson and we're gonna start conquering
19:24	how to solve the quadratic systems mathematically .

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