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I've written some example relationships

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between the two variables,

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in this case between 'm' and 'n',

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between 'a' and 'b', between 'x' and 'y',

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and what I want to do in this video is

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[to] see if we can identify whether the relationships

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are a direct relationship,

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whether they vary directlly,

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or maybe they vary inversely,

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or maybe it is neither.

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So let's explore it a little bit.

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So over here we have m/n = 1/7.

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So let's see how we can manipulate this.

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If we multiply both sides by 'n',

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what are we going to get?

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And in general, you want to separate them

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so that the two variables are on different

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sides of the equation,

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so you can see-- is going to be the pattern--

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Let me write it this way:

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is m=kn, this would be direction variation,

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or is it going to be the pattern

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m = k * 1/n -- this is inverse variation.

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And you see in either one of these,

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they are on different sides of the equal sign.

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So let's take this first relationship right now,

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let's multiply both sides by 'n',

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and you get m -- these cancel out--

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is equal to 1/7 * n.

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So this actually meets

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the direct variation pattern.

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It's some constant times 'n'.

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m = some constant * n

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So this right over here is direct variation.

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Let's see ab = -3.

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So if we want to separate them,

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and we could do it with either variable,

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let's divide both sides by 'a'.

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We could've done it by 'b'.

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If we divide both sides by 'a',

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we get b = -3/a,

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or, you could also write this as

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b = -3 * 1/a.

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And once again,

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this is this pattern right here.

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One variable is equal to

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a constant times 1 over the other variable.

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In this case, our constant is -3.

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So over here, they vary inversely.

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They vary inversely.

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Let's try this one over here.

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xy = 1/10

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Once again, let's try to separate the variables,

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isolate them on either sides of the equation,

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and let's divide both sides by 'x'.

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You could divide by 'y',

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because you're really trying to find

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an inverse or direct relationship.

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So divide both sides by 'x',

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you get y = 1/10 over x,

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which is the same thing as 1/10x,

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which is the same thing as 1/10 * 1/x.

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So y = some constant * 1/x.

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Once again, 'y' and 'x' vary inversely.

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Let's do this one over here:

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9 * (1/m) = n

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So this one's actually already done for us.

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And it might be a little bit clearer

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if we just flip this around.

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If we just flip the left and the right-hand side,

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we get n = 9 * 1/m.

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n = some constant * 1/m.

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So n varies inversely with m.

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And, remember, if I say that n varies that m,

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that also means that m varies inversely with n.

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Those two things imply each other.

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Now let's try it with this expression over here.

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This one's a little bit of a trickier one.

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Because we've already separated

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the variables on both hands

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and we have this kind of--

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if this [were] b = 1/3 * a,

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then we would direct variation,

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and b would vary directly with a.

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But in this case, we have 1/3 - a.

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And you say, "Hey, maybe they're opposites,

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or whatever."

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And it actually turns out

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that this is neither.

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And to make that point 100% clear,

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let's look at 2 of these examples.

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In direct variation,

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if you scale up one variable in one direction,

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you would scale up the other variable

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by the same amount.

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So if x doubles from one to two,

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when x is one, actually I should do this

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with 'm' and 'n',

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so m and n.

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And the way I've written it here,

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although you could algebraically manipulate it

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so that one looks more dependent that the other,

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but in this situation

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where n is 1, m is 1/7.

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And when n is 7, m is going to be 1.

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So you have the situation

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that if n is scaled up by 7,

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then m is also scaled up by 7.

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Or vice versa.

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This is more of a relationship.

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I could've expressed n in terms of m.

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But when you scale one variable up by 7,

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you also have to scale up the other variable by 7.

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Or if you scale it up

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by some amount,

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you have to scale up the other variable

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by the same amount.

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This is direct variation.

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Let's take the inverse,

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or when two variables vary inversely:

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this situation right over here.

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Let's take a and b.

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When a = 1, b = -3.

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Or we could do it explicitly right over here,

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we could even go to the original one.

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When a = 1, we have 1b = -3, b=-3.

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If I were to take 'a', and I were to triple it,

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so if I were to multiply it by 3,

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so now a = 3, we have 1/3 * -3.

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So now b = -1

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Notice! We didn't multiply b times 3 here!

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Now we divided by 3.

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Or another way is

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[that] we multiplied by 1/3.

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So if you scale up 'a' by 3,

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you're scaling down 'b' by 3.

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So they're varying inversely.

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What you're going to see in this 'neither'

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is that neither of these

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are going to be the case.

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So let's try it out.

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I'll do it in that same green color.

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That same green.

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So we have 'a' and 'b'.

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So when 'a' is, I don't know,

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when 'a' is 1, what is b?

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1/3 - 1, that's 1/3 -3/3,

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that's -2/3.

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And then let's divide,

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just for fun,

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'a' by 3.

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So 'a' goes to 1/3.

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So we're dividing by 3,

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or you could say that we're multiplying by 1/3.

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So if 'a' is 1/3,

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then 'b' is equal to 0.

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If a = 1/3, b = 0.

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So notice: if this [were] direction variation,

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we would be multiplying this by 1/3 as well,

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which clearly we didn't!

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And if this [were] inverse variation,

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if they varied inversely,

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we would be multiplying by 3,

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which clearly we didn't,

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we just got some other number!

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The scaling actually didn't matter;

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what happened is that things

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really just got shifted by some amount.

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They got shifted by 2/3.

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So these vary neither directly nor inversely--

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--this last one right over here.