Recognizing Direct and Inverse Variation Examples of variables varying directly and inversely
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- I've written some example relationships
- between the two variables,
- in this case between 'm' and 'n',
- between 'a' and 'b', between 'x' and 'y',
- and what I want to do in this video is
- [to] see if we can identify whether the relationships
- are a direct relationship,
- whether they vary directlly,
- or maybe they vary inversely,
- or maybe it is neither.
- So let's explore it a little bit.
- So over here we have m/n = 1/7.
- So let's see how we can manipulate this.
- If we multiply both sides by 'n',
- what are we going to get?
- And in general, you want to separate them
- so that the two variables are on different
- sides of the equation,
- so you can see-- is going to be the pattern--
- Let me write it this way:
- is m=kn, this would be direction variation,
- or is it going to be the pattern
- m = k * 1/n -- this is inverse variation.
- And you see in either one of these,
- they are on different sides of the equal sign.
- So let's take this first relationship right now,
- let's multiply both sides by 'n',
- and you get m -- these cancel out--
- is equal to 1/7 * n.
- So this actually meets
- the direct variation pattern.
- It's some constant times 'n'.
- m = some constant * n
- So this right over here is direct variation.
- Let's see ab = -3.
- So if we want to separate them,
- and we could do it with either variable,
- let's divide both sides by 'a'.
- We could've done it by 'b'.
- If we divide both sides by 'a',
- we get b = -3/a,
- or, you could also write this as
- b = -3 * 1/a.
- And once again,
- this is this pattern right here.
- One variable is equal to
- a constant times 1 over the other variable.
- In this case, our constant is -3.
- So over here, they vary inversely.
- They vary inversely.
- Let's try this one over here.
- xy = 1/10
- Once again, let's try to separate the variables,
- isolate them on either sides of the equation,
- and let's divide both sides by 'x'.
- You could divide by 'y',
- because you're really trying to find
- an inverse or direct relationship.
- So divide both sides by 'x',
- you get y = 1/10 over x,
- which is the same thing as 1/10x,
- which is the same thing as 1/10 * 1/x.
- So y = some constant * 1/x.
- Once again, 'y' and 'x' vary inversely.
- Let's do this one over here:
- 9 * (1/m) = n
- So this one's actually already done for us.
- And it might be a little bit clearer
- if we just flip this around.
- If we just flip the left and the right-hand side,
- we get n = 9 * 1/m.
- n = some constant * 1/m.
- So n varies inversely with m.
- And, remember, if I say that n varies that m,
- that also means that m varies inversely with n.
- Those two things imply each other.
- Now let's try it with this expression over here.
- This one's a little bit of a trickier one.
- Because we've already separated
- the variables on both hands
- and we have this kind of--
- if this [were] b = 1/3 * a,
- then we would direct variation,
- and b would vary directly with a.
- But in this case, we have 1/3 - a.
- And you say, "Hey, maybe they're opposites,
- or whatever."
- And it actually turns out
- that this is neither.
- And to make that point 100% clear,
- let's look at 2 of these examples.
- In direct variation,
- if you scale up one variable in one direction,
- you would scale up the other variable
- by the same amount.
- So if x doubles from one to two,
- when x is one, actually I should do this
- with 'm' and 'n',
- so m and n.
- And the way I've written it here,
- although you could algebraically manipulate it
- so that one looks more dependent that the other,
- but in this situation
- where n is 1, m is 1/7.
- And when n is 7, m is going to be 1.
- So you have the situation
- that if n is scaled up by 7,
- then m is also scaled up by 7.
- Or vice versa.
- This is more of a relationship.
- I could've expressed n in terms of m.
- But when you scale one variable up by 7,
- you also have to scale up the other variable by 7.
- Or if you scale it up
- by some amount,
- you have to scale up the other variable
- by the same amount.
- This is direct variation.
- Let's take the inverse,
- or when two variables vary inversely:
- this situation right over here.
- Let's take a and b.
- When a = 1, b = -3.
- Or we could do it explicitly right over here,
- we could even go to the original one.
- When a = 1, we have 1b = -3, b=-3.
- If I were to take 'a', and I were to triple it,
- so if I were to multiply it by 3,
- so now a = 3, we have 1/3 * -3.
- So now b = -1
- Notice! We didn't multiply b times 3 here!
- Now we divided by 3.
- Or another way is
- [that] we multiplied by 1/3.
- So if you scale up 'a' by 3,
- you're scaling down 'b' by 3.
- So they're varying inversely.
- What you're going to see in this 'neither'
- is that neither of these
- are going to be the case.
- So let's try it out.
- I'll do it in that same green color.
- That same green.
- So we have 'a' and 'b'.
- So when 'a' is, I don't know,
- when 'a' is 1, what is b?
- 1/3 - 1, that's 1/3 -3/3,
- that's -2/3.
- And then let's divide,
- just for fun,
- 'a' by 3.
- So 'a' goes to 1/3.
- So we're dividing by 3,
- or you could say that we're multiplying by 1/3.
- So if 'a' is 1/3,
- then 'b' is equal to 0.
- If a = 1/3, b = 0.
- So notice: if this [were] direction variation,
- we would be multiplying this by 1/3 as well,
- which clearly we didn't!
- And if this [were] inverse variation,
- if they varied inversely,
- we would be multiplying by 3,
- which clearly we didn't,
- we just got some other number!
- The scaling actually didn't matter;
- what happened is that things
- really just got shifted by some amount.
- They got shifted by 2/3.
- So these vary neither directly nor inversely--
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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