01 - Direct Variation and Proportion in Algebra - Part 1 (Constant of Variation & More) - Free Educational videos for Students in K-12

01 - Direct Variation and Proportion in Algebra - Part 1 (Constant of Variation & More) - By Math and Science

Transcript


00:01	Hello . Welcome back to algebra . The title of
00:02	this lesson is direct variation and proportion . This is
00:06	part one of two lessons . So we're talking about
00:09	in this case the concept of direct variation . Now
00:12	this is going to be in contrast , a couple
00:14	of lessons down the road , we're gonna talk about
00:16	inverse variation . So in the back of your mind
00:18	, when we talk about direct variation , just keep
00:20	in mind that we also have kind of a cousin
00:23	of this which is called inverse variation , which will
00:25	they will kind of go hand in hand together .
00:27	But in this case we're gonna be talking about direct
00:29	variation in a nutshell , when you have two variables
00:33	that are related to one another , right ? Then
00:36	oftentimes we say that they vary directly with respect to
00:40	one another . And what direct variation means in this
00:42	case , when you have two variables that have a
00:44	direct variation relationship between them all , it means is
00:48	very simple to understand is that when you increase one
00:51	of the variables , the other variable also increases ,
00:54	so they increase in direct correspondence to one another .
00:57	Right ? Alternatively , if one variable goes down then
01:00	the kind of like the partner variable goes down with
01:03	it . So when you have two variables that both
01:06	go up together or both go down together , we
01:08	call it direct variation and we're gonna find out in
01:11	several lessons down the road . The inverse variation is
01:14	when one variable might go up while that would cause
01:17	the other variable to go down so they go opposite
01:19	of each other . That's what we call inverse variation
01:21	. So in math terms we want to talk about
01:23	when we talk about direct variation is the following thing
01:27	. And let me tell you right now , it's
01:28	very easy to understand because we've learned so much about
01:31	lines and you're going to see in a second that
01:33	this is all related to the equation of a line
01:35	. So , direct variation . And the other thing
01:38	I'll say before we get started is we talk about
01:41	direct variation so much in algebra , because there are
01:44	many , many equations in physics and chemistry and engineering
01:48	that have a direct variation relationship . So I'm gonna
01:51	actually show you some of them in just a second
01:53	. But just keep in mind that this is actually
01:54	useful because a lot of the physical laws we have
01:57	in the universe actually obey a direct variation kind of
02:00	kind of relationship . So what does direct variation means
02:03	? It means we have two variables . I'm gonna
02:05	call one of them X and I'm going to call
02:07	the other one why ? And their related to one
02:09	another , right ? There's lots of lots of variables
02:12	and I'll give you examples in a second that are
02:14	related to one another . So a direct variation means
02:16	that as the variable X increases , that's what the
02:19	up arrow means . If it goes up up up
02:21	up up up up then a direct variation means that's
02:25	what this this little arrow means . It means as
02:27	X goes up , it means that the variable Y
02:29	also goes up now , it does not mean that
02:33	X and Y have to go up at the exact
02:35	same rate . Okay ? You could have X going
02:37	up really , really fast and you could have why
02:39	going up in a slower fashion or you could have
02:42	X going up really , really slow and why going
02:44	up even faster ? A faster variation . So they
02:48	don't have to go at the same speed up ,
02:50	but they both have to go the same direction .
02:52	So what it means is that as X goes up
02:54	why must also go up ? All right . And
02:57	in the case of inverse variation , we'll talk about
02:59	later as X goes up , the variable Y goes
03:01	down in the opposite way . So , um what
03:05	this means is that when you when you boil it
03:08	down and start graphing pairs of X and Y points
03:10	and you see that as X goes up implies that
03:12	why goes up , then what it basically means is
03:15	the variable Y is equal to some number M .
03:19	Which we're using a letter in because it's going to
03:21	end up being the slope of the line times X
03:24	. So you might look at this and say wait
03:26	a minute , doesn't this look familiar ? Y equals
03:28	mx Because we talked about lines a whole lot mx
03:31	plus B , right , M was the slope of
03:33	the line and B is the Y intercept . So
03:37	when we talk about direct variation , literally , it
03:40	is the exact same stuff that we learned and we
03:42	talk about the equation of a line . But uh
03:45	the Y intercept B is always set equal to zero
03:49	and you'll see why in just a second . But
03:51	what this means is that X go up , Why
03:53	goes up in in terms of this relationship , notice
03:57	how the equation is set up , even if M
03:59	was no matter what M . Is , let's say
04:01	M . Is equal to to let the slope of
04:03	this uh line here as X goes up . If
04:07	you make X one and two and three and four
04:09	, when you multiply by M , then why must
04:11	also go up ? So this is the form of
04:13	a direct variation . And we say in words that
04:17	why varies directly as X . And what this wording
04:26	means is when X goes up , why goes up
04:29	, that's what it means . And this is what
04:30	the equation tells you , even if M . Is
04:31	one half . Uh If X increases from 1 to
04:35	2 to three of the four , then why must
04:38	also be increased by whatever the numbers are . You
04:41	just multiply what M . Is and you're gonna get
04:43	larger and larger values of why ? Now we have
04:45	a special name for this . By the way .
04:48	You might also see this in your textbook . You
04:51	might see this , I'll do this , you might
04:54	see uh why is equal to K . X .
04:57	So some books don't use the letter M . They
05:00	use the letter K . This uh variables , this
05:03	uh entity , him the M . Here . Or
05:06	if you see it in a book as K .
05:07	Whatever you label it there , that thing uh In
05:10	this case I'm using them . So I'm gonna say
05:12	M . Is equal to the constant of variation .
05:21	So M . Can be whatever you want it to
05:22	be , depending on the problem or depending on the
05:24	equation that you're looking at . But this thing M
05:26	. Is equal to a constant is called a constant
05:29	variation . So if M were equal to one for
05:32	instance , then you just have one here . Then
05:35	as X goes up 1234 then you multiply by one
05:38	. Why would be going up 1234 ? But if
05:40	M were to , let's say then every time I
05:43	increase X and I'm multiplying by two . So you
05:46	see why is actually would be increasing faster than X
05:49	. Because if M is one , I multiplied by
05:52	two . I'll give me too . If if X
05:55	is too , I multiplied by two , I would
05:56	get four . If X would be three , I
05:58	multiplied by two , I get six . So you
06:00	see the constant variation determines the exact relationship between X
06:05	and Y . But in all cases when X goes
06:08	up , why must go up ? But the constant
06:10	variation determines how fast why goes up when I increase
06:14	X , how much faster is why increasing ? Remember
06:17	I told you why can be going up faster than
06:19	X ? Or why can be going up slower than
06:22	X . But they must be going in the same
06:23	direction and what determines how fast they move with respect
06:26	to one another is called the constant variation . Now
06:29	you might see this as K . So , in
06:31	your textbook you might see K is the constant variation
06:34	. So you might see or K little bit lower
06:37	case . K would be the constant variation . All
06:40	right . So , we've done a little bit of
06:41	theory here , and it's a little bit abstract .
06:43	So , I want to talk to you about some
06:44	real examples of direct variation . So , examples .
06:50	And these are examples that you all know Right ?
06:52	I like giving examples that you all know . So
06:54	let's take a famous example . What about this equation
06:57	? Do you recognize this ? Whoops . C is
07:00	equal to pi times deep . I think you recognize
07:04	that from fifth grade right ? The circumference of the
07:06	circle is equal to pi times the diameter of the
07:10	circle . So notice the way this thing is set
07:12	up . It isn't exactly the same form as this
07:15	equation . Have a variable equal to a constant notice
07:19	it's a constant , this is not a variable .
07:21	It's a constant variation means it's just a number ,
07:24	Pi is just a numbers 3.14159 it goes on and
07:27	on with non repeating decimals and then time some other
07:31	variable . So notice that as the diameter of the
07:33	circle increases you make the circle larger , you multiply
07:37	by pi . The circumference also gets larger . So
07:39	as the diameter increases , the circumference also increases ,
07:43	which is exactly what we said . The relationship has
07:45	to be here . Pie , Is that special number
07:49	that governs how much the circumference increases when the diameter
07:53	also increases ? It's the constant variation . You might
07:56	also see it as something called the constant of proportionality
07:59	. So this the title of this lesson is direct
08:01	variation in proportion . So this is a direct variation
08:04	relationship . But you can see that the reason we
08:07	use the word proportional is that the diameter of the
08:09	circle and its circumference are proportional to one another .
08:12	And proportion just means that as one goes up then
08:15	the other variable goes up as well . The constant
08:17	of proportionality are also called . The constant variation just
08:20	determines how much that other variable goes up . All
08:25	right , What are some other examples ? This one
08:27	you might or might not know , but it's one
08:29	of the most famous equations in all of physics .
08:31	F is equal to M . A . So this
08:34	is force is equal to mass times acceleration . This
08:38	means if I take a baseball or anything with some
08:42	mass , the mass of the ball is constant ,
08:44	right ? It's just a fixed thing . The mass
08:46	of the ball doesn't change . So even though it
08:48	looks like a variable here , the mass is really
08:50	the constant of variation here , and you can see
08:53	that the acceleration in the force on that ball have
08:55	to be there related directly as the acceleration on the
08:58	ball goes up . In other words , I have
09:00	to throw the ball with some force in order to
09:02	accelerate , I have to push the ball with some
09:05	uh some force in order to get it to accelerate
09:07	to go faster and faster . But you can see
09:10	that as the acceleration is bigger , bigger , bigger
09:12	, bigger , bigger . The force applied to the
09:14	ball must also be bigger , bigger , bigger ,
09:15	bigger , bigger . And the mass is that special
09:18	quantity that governs how how the force and the acceleration
09:23	are linked . Because you all know that if you
09:25	have a bowling ball it's much harder to accelerate a
09:28	bowling ball with the same force , right ? If
09:30	I apply the same force to a ping pong ball
09:34	and a bowling ball , same force , the acceleration
09:37	will be different . We call it inertia , right
09:38	. Things are harder to push harder to get going
09:41	. So the masses , that thing that governs and
09:43	links the force and the acceleration together . It's the
09:46	constant of variation . So you can see as acceleration
09:48	goes up , mass goes up . What's one more
09:52	? We'll talk in physics , we'll talk about something
09:54	called the potential energy of something . The potential energy
09:57	is equal to M . Times G . Times H
10:00	. So this is a fancy sounding thing but it's
10:02	it's not fancy . What it basically means is the
10:05	energy when you the potential energy when you hold something
10:08	above the ground is equal to the mass of the
10:11	thing . Like a bowling ball times G , which
10:13	is the gravity on Earth 9.8 m per second .
10:16	It's a gravitational acceleration 9.8 m per second squared times
10:21	the height above the ground . The height above the
10:22	ground is the most important thing . As I raise
10:25	the thing higher above the ground , we say has
10:26	more potential energy . If you climb to the top
10:29	of a building , you have a lot more potential
10:31	energy than if you're on a step ladder . If
10:34	I jump off the step ladder , it's not gonna
10:35	be a problem . But if I jump off the
10:37	building , that potential energy makes you go split into
10:41	the ground . So you have a lot more potential
10:42	energy , the higher you are . So you see
10:44	, the higher you go above the ground , the
10:46	more energy you have . And these are directly related
10:49	. Now , you might say , well , wait
10:50	a minute . There's two things listed here and here
10:53	. I only had one constant variation . But really
10:55	the mass of the ball is never changes and the
10:59	and the acceleration of earth never changes . So ,
11:02	these things multiplied together , they become like one constant
11:05	. These are like , like one constant when you
11:08	multiply there just one number , right ? The mass
11:10	of the ball times gravity is one number . So
11:12	, this is really the constant variation . These two
11:14	things kind of multiplied together , and you can see
11:16	they govern how much energy you get when you go
11:19	higher and higher off the ground . All right .
11:22	So those are some examples of what a direct variation
11:27	is , and that's why I told you in the
11:28	beginning , it's so very important . These are just
11:30	three equations . Um They look so simple , but
11:34	it turns out you can solve a lot of problems
11:36	with these kinds of equations . So this direct variation
11:38	is extremely important . Now let's dive a little bit
11:41	deeper into what direct variation is . It looks like
11:45	the equation of the line . However , the plus
11:49	B is gone . Now let's go and investigate exactly
11:52	why that's the case . Let's go and look what
11:54	I want to do this . I think I want
11:56	to do it on the next board over here .
11:58	All right . What I want to get the same
12:00	, get across to you is this is the same
12:04	as what you've already learned before . Why is equal
12:07	to M X plus B ? However we're saying that
12:12	B is equal to zero , the y intercept is
12:13	equal to zero . Why is that the case ?
12:15	Let's just take a look at an example . Let
12:17	me draw a little graph here , right ? Um
12:21	This is X . And this is uh why now
12:25	here I've given examples of what I could use here
12:28	. Force I could use circumference and diameter , I
12:30	can use potential energy or whatever . But I'm gonna
12:32	give you an example that I think even easier to
12:34	understand , imagine that you're swimming in a pool ,
12:37	you jump under the water and you start to swim
12:40	down under the surface of the water . You immediately
12:43	start feeling pressure on your ear when you're near the
12:46	surface , like right under the surface . You don't
12:48	feel much , You could say the pressure is really
12:50	zero , right at the surface , but if you
12:52	go one m down you feel quite a bit of
12:55	pressure pushing on you . And if you go 10
12:58	m down you're gonna it's gonna be very very painful
13:02	. And if you go really really really really far
13:03	down you can actually get hurt by swimming so deep
13:06	down because the pressure of the water is pushing on
13:08	you . So you see as you go deeper ,
13:10	deeper , deeper , deeper deeper the pressure goes up
13:13	up up up up so five m below the surface
13:15	means more pressure . 10 m below the surface means
13:18	more pressure . But right at the surface there's really
13:21	no pressure at all . So you can kind of
13:24	imagine this being a line , right ? That goes
13:27	through the origin here . Something like this , A
13:30	straight line . I know it's not perfect , but
13:32	that's my best guess at a straight line and here
13:35	on the X axis , I'm gonna call this the
13:37	dive depth in meters right ? So this is going
13:43	deeper and deeper and deeper under the water . And
13:45	as that happens this is the water pressure . And
13:52	it goes of course up here . So you see
13:54	as the dive depth goes deeper , deeper , deeper
13:57	, more and more meters under the water then the
13:59	pressure gets more and more and more and more .
14:01	And that's why this thing looks like a line .
14:02	Right ? So you can see that this direct variation
14:06	, it looks like a line with B is equal
14:08	to zero . And that's an exact in words this
14:11	in terms of a graph , that's exactly what we're
14:13	seeing here . We have a direct variation . It
14:15	is a line that goes through the origin here .
14:18	Why does it go through the origin ? Well it's
14:20	because when you're at the surface and the depth of
14:23	zero there's no pressure , the pressure zero . So
14:26	these direct variation problems go through the origin because at
14:30	zero there's no there's no there's no change in the
14:34	other variable . It's the same thing here with the
14:36	circle . If I have zero diameter , my circumference
14:38	is zero . So it goes through the origin .
14:40	If I have zero acceleration , my force must have
14:43	been zero . The things not even moving at all
14:45	. If I'm zero m above the ground , I
14:47	have no energy at all . All of these things
14:49	go through zero when the variable is set to zero
14:52	. And so the same thing here with the pressure
14:53	here . Now you can imagine there being two points
14:58	on this line here . In fact there's an infinite
15:01	number of points . I can go one m 2
15:03	m three m and so on . But let me
15:05	just pick a random point right here and I'll pick
15:07	another random point right there , right ? This point
15:10	is that X one comma Y one . It could
15:13	be like five m down and like three uh atmospheres
15:17	of pressure or whatever . I have to give you
15:18	the units of pressure . I don't want to get
15:20	into units of pressure right now . But this could
15:22	be however many meters down and whatever the pressure is
15:25	kind of reading off the graph here and then this
15:27	is X two more meters down . In other words
15:30	, and I have some higher value of the pressure
15:33	. Y two . I would just read it right
15:34	off of the graph all right now , because this
15:38	is a line that goes through the origin , that
15:40	means the Y intercept is zero . So it's Y
15:43	equals mx plus B but B a zero , right
15:45	? Because it goes through the origin like this ,
15:47	then all of the points on this line , you
15:50	know , you can calculate the slope of any line
15:52	by looking at any two points and the slope of
15:55	the line never changes . So no matter what points
15:57	I pick anywhere on this line , I'm gonna get
15:59	the same slope because the slope never ever changes for
16:03	a line . Right ? So what I want to
16:06	do is I want to say , let me calculate
16:08	the slope of kind of this line segment between this
16:11	this guy in zero here . So what would be
16:13	the slope of this line ? The constant variation is
16:16	what I'm trying to calculate . I'm trying to show
16:18	you here , what is the slope of the line
16:19	? Well , it's y tu minus 11 divided by
16:23	X two minus x one . So what I have
16:24	here is this y value here is why one minus
16:28	zero . I'm going to use these two points .
16:30	So here is zero comma zero and here's X .
16:33	One comma Y one . So it's the subtraction of
16:35	the Y values . Uh The subtraction of the X
16:38	values X . Is X . One and then zero
16:40	right here . So what do I get if I
16:42	calculate the slope there is just Y one over X
16:46	. One , that's going to be what it is
16:47	. So the slope of the line , in other
16:49	words is just the ratio of Y two X .
16:52	That's all it is . Now let's calculate the slope
16:56	using this point and let's let's go calculated between this
16:59	point and also zero . Right ? Because we know
17:02	the slope is gonna be the same everywhere . Right
17:04	? So let's calculate the slope there . It's gonna
17:07	be y two minus at this point which is the
17:10	Y . Value is zero and then X two minus
17:13	the X point here is zero . So what do
17:15	I get if I calculate the slope there ? Why
17:17	two divided by X two . So you see what's
17:21	interesting is we know it's a line , but if
17:23	I use this point in the origin , I have
17:26	two points . I get a slope . Here's what
17:27	it is . If I use this point and calculate
17:31	the slope of the origin , I get this .
17:33	So the slope here is equal to this . The
17:36	slope calculated over here is equal to this . But
17:38	we know that these slopes have to be the same
17:40	thing . They are because the slope of the line
17:43	is always the same . It never changes no matter
17:45	what points you use to figure out the slope .
17:48	So why am I doing all this ? Because I
17:50	can set these things equal to one another . Set
17:54	equal and what you get out of that is something
17:58	you will see in your textbook and that's the following
18:00	why one over X one . We know it's equal
18:03	to end , but that's also equal to this .
18:05	So that's going to be equal to Y two over
18:08	X two . So this is useful to solve direct
18:16	variation . That's devi direct variation problems . So what
18:21	I could have done is I could have said ,
18:23	hey guys , this thing is called direct variation .
18:25	This is what the equation is . I could have
18:27	not even given you any examples at all . I
18:29	could have not even drawn this thing on the board
18:31	. And I could have said , this is what
18:32	you do here . It's with any two points ,
18:34	it's Y over X . Y two over X and
18:37	then go for and solve your problems . But you
18:39	would have no idea . How did he know this
18:40	was the case ? Well , how could he do
18:42	that ? All this is telling you , is that
18:45	the points X one and Y . One and X
18:48	. Two and Y to their related to one another
18:51	. Specifically the ratio of the Y values that to
18:54	the X . Value at one point is equal to
18:56	the ratio of the Y . Value to the X
18:58	. Value at the other point . And the reason
19:00	that's the case is because of the slope of the
19:02	line is calculated using those points and because the line
19:05	always goes through the origin for direct variation , problems
19:08	like this . Okay , so we can say that
19:11	why here is directly proportional to X . And you
19:15	can use this in here as the constant of proportionality
19:18	or the constant variation . Mhm . Both terms are
19:22	used now . That's enough I think theory , right
19:26	. I've tried to illustrate what a direct variation is
19:28	and all that but we really need to do a
19:29	problem or two for you to understand how to do
19:32	these things and they're not hard . So , problem
19:34	number one says something like this . If why varies
19:41	directly , you have to have that word directly as
19:45	X . If why varies directly as X . And
19:50	Why is equal to six when X is equal to
19:55	15 ? Find no why when X is equal to
20:03	25 . So this looks really difficult at first because
20:06	there's a lot of weird wording in it . Why
20:08	varies directly as X . And then you're given a
20:10	Y value and then you're given two X values .
20:12	And a lot of times students will look at it
20:14	and they'll all get jumbled and they won't have any
20:16	idea like what do I multiply these do ? I
20:18	divide these ? And then usually you'll start jumping back
20:21	and getting this and start putting things in . But
20:23	you won't really know because you really didn't know where
20:25	the formula came from , you don't really know what
20:27	to do . So what we're gonna do is we
20:30	are going to translate this . First of all you
20:32	look and you say why varies directly as X .
20:35	That means that it has to be of the form
20:38	. Um It has to be of the form .
20:41	Why is equal to some constant variation times X .
20:44	That's what the word directly is later on when we
20:47	learn about inverse variation . If it were to say
20:49	inversely , if it varies inversely , then the equation
20:52	is completely different . So that word tells you it's
20:54	direct variation problem . Right ? And the other parts
20:58	of it is just telling you that why is 61
21:02	X . Is 15 . So the easiest way to
21:04	do this is to draw a little sketch . So
21:06	let's draw a little xy graph . And it's telling
21:09	us that why is 61 X . Is equal to
21:11	15 ? So I don't know , I don't have
21:14	any scale here . I don't need to put tick
21:15	marks , I don't need to be exact . But
21:17	what I want to do is I'm just gonna put
21:18	a point , maybe I'll do it more like this
21:23	like this . So what I know is that let's
21:27	see when X is . Let me do it like
21:29	this over here over here , X is equal to
21:33	15 . Why is equal to six ? So this
21:35	point right here is 15 comma six , right ?
21:39	That's all I know . It's telling me that when
21:41	why is equal to 61 X is equal to 15
21:44	. But it's asking me to find why when X
21:46	is 25 . Now , if this is 15 and
21:48	25 is somewhere over here , I don't know exactly
21:50	where but I know that this has to be a
21:52	line . So I can draw through this point and
21:55	through the origin of this straight line and that tells
21:58	me right there at this point here must be up
22:01	here At X . is equal to 25 . And
22:04	why is equal to something ? I don't know what
22:06	it is . I can put I can put why
22:09	there I guess . But I think it's better to
22:11	put a question mark . Let's put a question mark
22:12	right there . So here's the translation of what the
22:15	problem is telling you . It's basically telling you another
22:17	way I could have done it is I could have
22:18	said you have a line going through the origin .
22:20	One point on the line is 15 comma six .
22:23	The other point on the line is 25 comma something
22:26	. Find the value of why that goes with 25
22:29	. That's all you have to do . Right ?
22:31	So you don't have to make it more complicated than
22:33	it is . So there's there's two ways to do
22:35	it . I'm gonna call , I'm gonna do both
22:37	ways for your method . What you know this thing
22:41	is direct variation . So you know that y .
22:43	Is equal to mx if you know what M .
22:48	Is , then all I would have to do is
22:49	put the 25 for X . In . Multiply by
22:51	the M . And I would get the value of
22:53	Y out but I don't know what M . Is
22:55	. However , I can find what M . Is
22:58	because I have another point on the line . And
23:00	I also know that all of these things go through
23:02	zero comma zero . So I can find the slope
23:06	of this thing by saying that uh it's going to
23:10	basically be y tu minus Y one over X two
23:15	minus x one . And I'm gonna use this point
23:17	in the zero point to figure it out . So
23:19	six is the Y value six minus zero . Because
23:22	I'm going to this point here and then 15 minus
23:26	zero . So it's six divided by 15 . 6/15
23:30	. And when you simplify that I can divide the
23:32	top by three . And that's gonna give me to
23:34	divide the bottom by three and that's gonna give me
23:35	five . So now I know that the slope is
23:37	2/5 . I didn't know that when the problem started
23:40	but now I know that this direct variation form looks
23:44	like this . In other words , this line is
23:48	2/5 X . So think about mx plus B .
23:51	Right ? If I told you graph 2/5 X plus
23:54	B , you would say why intercept is zero and
23:57	the slope is to over five so rise over run
23:59	, that's what's happening . And these points both go
24:02	through this line . But now that I know what
24:05	the line is , I'll just take the x value
24:07	of 25 and stick it in here . So why
24:10	is 2/5 times 25 Right now ? Of course I
24:15	have . Let me mark it up separately to 5th
24:19	25 . I can say I defy divided by five
24:24	is 1 , 25 by five is five . So
24:27	what I really have is two times five . And
24:29	so what I'm going to have is why is equal
24:32	to 10 . So the question says , find the
24:35	value of why when X is 25 . So what
24:38	you have to do is use the other point ,
24:40	they give you to find the slope . You always
24:42	have to know for direct variation . That it always
24:45	goes through zero comma zero through the origin . That's
24:48	what allowed us to find the slope . Once we
24:51	know the slope we know direct variation always looks like
24:54	this . Why is equal to some constant variation times
24:57	X . Once we know that we put in the
24:59	other point and calculate the corresponding value of Why ?
25:03	Now that's the way I like to do it because
25:06	that makes sense to me . But a lot of
25:08	students , uh a lot of teachers try to push
25:11	the formulas that come out of the book . So
25:13	notice what we did is we we found what this
25:16	relationship was by looking at two points on the line
25:19	. We know it also goes through zero finding a
25:21	slope from one point down to zero and the other
25:23	point down to zero , setting people equal . And
25:25	we know this relationship is valid why one over X
25:29	. One , Y two over X . Two .
25:32	So I'm gonna do it that way for you ,
25:34	but I'm gonna show you that you're gonna get the
25:37	same thing . So what we get is why um
25:40	you just double check myself why one over X .
25:42	One Y . Two over X . Two . So
25:47	what you give here is you go back to the
25:50	problem statement , you gotta find the pairs of variables
25:53	that go together . So we know that . Why
25:55	is 61 X . Is 15 those go together .
25:58	So why is 61 X . Is 15 ? That
26:01	corresponds to this one here , The other one here
26:04	we're trying to find the value of Y when X
26:07	. Is 25 . So we'll call it why 20/25
26:10	. So you see what happens , you just fill
26:12	in the blanks here and this is what a lot
26:13	of teachers try to do and it works . It's
26:16	just you don't often know what you're doing , you're
26:17	just plugging stuff and I don't really like that too
26:19	much . But how would you solve this ? Um
26:22	well you multiply both sides by 25 to get why
26:24	by itself . So what you would have is why
26:28	would be 25 Times 6/15 ? Right ? Um You
26:37	have 25 times 6/15 . All I've done is multiply
26:40	25 on both sides . Now , let's simplify those
26:43	. Uh Let's simplify it by I'm trying to think
26:46	of the best way to do it , let's do
26:47	it this way . Let's leave the 25 alone for
26:49	now . And the 6/15 , I'm gonna simplify this
26:52	fraction divided by two is gonna give me I'm sorry
26:55	, divide by three is gonna give me two divided
26:56	by three , is gonna give me five . Right
26:59	? But then I see right here , let me
27:01	go in . Right , one more time . 25
27:03	times 2/5 5555 is 1 , 25 divided by five
27:07	is five . And so I end up with five
27:09	times two , which gives me 10 . That's exactly
27:11	the same thing I got here . But I want
27:13	you to really understand what you've done here by using
27:16	this formula is exactly the same math that we did
27:18	with Method one , which I actually think makes more
27:21	sense finding the slope of the line and then using
27:23	that to solve the thing . Because when you plug
27:26	all the values in here , notice what you're done
27:28	. You're basically saying 6/15 times 25 . But notice
27:32	what we did over here . We did 2/5 times
27:34	25 . But the 6/15 actually is 2/5 times 25
27:39	. So all we did in the first way was
27:41	find the slope , simplify it and then put it
27:43	in , multiplied by 25 got the answer here ,
27:46	we put it into our equation . We end up
27:47	with exactly the same math 25 times here . This
27:51	is 2/5 when you simplify . So you actually get
27:53	the exactly the same answer . I prefer using method
27:56	one because it makes more sense and it leads to
27:59	less errors when you know what you're doing . All
28:01	right , So we have one more here . I
28:03	want to do with you uh with direct variation and
28:06	it's gonna be a similar format . Most of these
28:08	are all the same . And it goes like this
28:10	if P is directly proportional to the variable Q .
28:24	And uh P is equal to nine , that is
28:29	not a Q . That's a nine P is equal
28:31	to nine win Q . It's a bad looking Q
28:36	. Sorry , seven point when Q is equal to
28:37	75 find Q when P is equal to 24 .
28:46	So again you've got lots of numbers mixed together and
28:48	all this other kind of stuff . So we're not
28:49	going to draw a picture this time . But I
28:52	do want you to keep in mind the picture that
28:54	we use in the last lesson . There is a
28:56	line , it's a direct variation line , it goes
28:58	through the origin and we have two points on the
29:00	line . I'm given one of the points on the
29:02	problem completely . The other point I've only given I've
29:04	only been given part of that's what you have to
29:07	keep in the back of your mind , that's what
29:09	you're really doing . So I know that this is
29:11	a direct direct variation between P and Q . It
29:13	says P is directly proportional to queue . So I
29:16	then know right away that P is equal to directly
29:18	proportional two Q . Some constant variation times Q .
29:23	Right ? Um And it's asking me find Q .
29:28	When P is equal to 24 right ? Um And
29:32	it's also telling me that P . Is equal to
29:35	nine when Q . Is equal to 75 . So
29:37	if I knew what M . Was , if I
29:39	knew what M . Was , all I would have
29:41	to do is put the P value in here ,
29:44	I would know the M . Value and I would
29:45	find the Q . Value . But the problem is
29:47	I don't know what he is , but they give
29:49	me another point on the line . I know That
29:52	P is nine when Q . is seven .5 .
29:54	So I can put in nine , that's not a
29:57	Q . That's a nine in times the Q .
30:00	Which is 75 And from this I just calculate the
30:02	slope , I could have calculated the slope , You
30:05	know uh separately differently like I did back before over
30:09	here , of course I can calculate the slope between
30:12	this point and the origin . I'm showing you kind
30:13	of an alternative way of doing it when you know
30:16	one point on this line and you know the equation
30:18	on the line here , I can just put the
30:19	two values and nine and 7.5 . And then when
30:22	I'm gonna get is nine divided by 7.5 . And
30:24	so the value of the slope is going to be
30:28	one too for the slope . And then once I
30:32	know the slope , I know that they are directly
30:34	proportional or they very directly with respect to one another
30:38	. So I know that P is equal to MQ
30:41	. But I now know what the slope is .
30:44	It's 1.2 times Q . So now I have the
30:46	equation . It would be very simple if I gave
30:48	you this line and I said , hey find Q
30:50	. When P is 24 you would know that how
30:53	to do that . But we had to find the
30:54	slope first . So then we put a P .
30:57	Of 24 in 1.2 times cute . And then Q
31:02	. is 24 divided by one point , sorry 1.2
31:08	. And so then Q . When you take 24-5
31:11	x 1.2 you get 20 . Yes . So if
31:14	we were to draw this line through the origin ,
31:17	we would see that the 0.9 comma 7.5 was on
31:22	the line and we would also see that become a
31:24	QP being 24 And Q being 20 would also be
31:29	on the line . This pair points would also be
31:31	on the line because we calculated the value of this
31:33	point using this line . Now just for giggles ,
31:37	we're going to go back and do the other do
31:38	it the other way in case you want to use
31:41	the equation given in the book . And that's that
31:43	if you have two points very directly , why one
31:46	over X one is Y . Two is equal to
31:49	Y . Two over X . Two . Now the
31:52	pair of points I'm given uh in this case actually
31:55	why why an X . Are really not in the
31:57	problem statement . So it's going to be useful for
31:59	you to say instead of that say Q one ,
32:01	P . One . Q two . P . Two
32:06	. Right ? And what are what are Q 1
32:09	? Q 2 ? So we know that uh P
32:13	. is nine when Q is 7.5 . So what
32:15	we can say is we can say 7.5 is Q1
32:21	, P . Is nine because these points go together
32:24	and then the other Q . Is what I'm trying
32:26	to find . So I'm gonna leave it as Q
32:28	. Two . And that p when that happens is
32:30	24 . So all I've done is put all these
32:32	values in . Look what I'm going to get .
32:33	Q . two is equal to this fraction . I'm
32:36	gonna multiply by the 24 . So I'm gonna say
32:38	24 times 7.5 over What's not over Q . over
32:43	nine like this . And when you go in your
32:46	calculator type 24 times 7.5 divided by nine . What
32:50	you're gonna get is Q two is equal to 20
32:55	Which is exactly what you get now . Why do
32:57	you think it's the same thing ? Look at the
32:58	math you actually did when you plugged it all in
33:00	and you got it you took 24 and you multiply
33:02	by a fraction . The fraction was 7.5 over nine
33:06	. Look what we did here and you can see
33:08	that that's basically what you were doing before . We
33:10	calculated Cube by taking 24 and divided by 1.2 .
33:13	But the 1.2 was the slope which was a fraction
33:16	. So you can kind of think of this fraction
33:17	being on the bottom , you flip it over ,
33:19	it's gonna be 24 times 7.5 divided by nine ,
33:22	which is exactly what we got there . I'm not
33:24	trying to throw a bunch of different ways that you
33:26	to do things to confuse you . I'm trying to
33:28	show you that there's always more than one way to
33:30	get the answer to a problem . Personally I like
33:33	the more logical way of doing it . I don't
33:35	like grabbing a formula out of a book and throwing
33:37	it in there . All I need to know is
33:39	that these things very directly Soapy has got to be
33:41	equal to m times Q . I'm given to one
33:44	point on the line in its entirety which lets me
33:46	find the slope . Once I know the slope I
33:49	can then put it in for the value of the
33:50	slope and then I can use this equation of the
33:52	line to find any other point I want . In
33:54	this case I was given the p value here ,
33:56	so then I go backwards and find the Q value
33:59	. Um not rocket science but extremely important because direct
34:03	variation pops up everywhere . As I said , potential
34:08	energy and physics , force and acceleration , even geometry
34:11	and I could go on , I can have a
34:12	whole lesson in direct variation . So for now just
34:15	make sure you solve these problems yourself and you understand
34:17	them . Then follow me on to the next lesson
34:19	. We'll get a little more practice with direct variation
34:21	before moving into kind of the opposite of that ,
34:23	which we call inverse variation in algebra .

Summarizer

DESCRIPTION:

Quality Math And Science Videos that feature step-by-step example problems!

OVERVIEW:

01 - Direct Variation and Proportion in Algebra - Part 1 (Constant of Variation & More) is a free educational video by Math and Science.

This page not only allows students and teachers view 01 - Direct Variation and Proportion in Algebra - Part 1 (Constant of Variation & More) videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.