Understanding Direct, Inverse, and Joint Variation

In this video, we're going to go over direct, inverse, and joint variation word problems. So the first thing that you need to be able to do is you need to be able to write the equation. So we're going to focus on that and then we'll apply that in solving word problems. So let's begin. If y varies directly with x, the equation that you need is y is equal to kx. Now let's say if R varies directly with S. R is equal to KS. If Z varies directly with L, the equation would be... z equals kl. So you can see a pattern here. That's direct variation. Inverse, let's say if y varies inversely with x, y is going to be k divided by x. with S. R is equal to K over S. If Z varies inversely with L, Z is equal to K divided by L. Now what if Y varies jointly with X and Z? How would you write that equation? So it's going to be Y is equal to K times X and Z, if it's jointly with X and Z. So if Y varies jointly with X and Z, what would you write? jointly with R and S, the equation is Y is equal to K times R times S. Now what if Y varies directly with X but inversely with Z? So if it's directly with X, X is on top. If it's inverse, then that's going to be on the bottom, so Z is on the bottom. Now what if Y varies directly with R but inversely? with s. So r is going to be on the top side, on the right side of the equation, and s is on the bottom. So it varies directly with r, but inversely with s. Now what if y varies directly with the square of X. How would you write that? So this would be Y is equal to KX squared. It varies directly with the square of X. Now what if Y varies inversely with the root or the square root of R? If it's inverse, it's on the bottom and we have the square root of R. Now, what if y varies directly with the cube of x, but inversely with the square of z? So, directly, x is going to be on top, but it's the cube of x. and inversely with the square of z. So it's z to the second power, but on the bottom. So now you know how to design the equations given a sentence. So let's try a few word problems. So let's start with a very simple problem. Y varies directly with X. So if X is 3 when Y is 12, what is Y when X is 9? So first write the equation. So Y is equal to KX. The next thing you need to do is solve for K. So we know that X is 3 when Y is 12. So let's plug in those values. To solve for K, we need to divide both sides by 3. 3 divided by 3 is 1, so we only have have K on the right side. 12 divided by 3 is 4. So that's K. Now once you have K, you can answer the second part of the problem. So what is Y when X is 9? So starting with the equation Y equals KX, let's plug in 4 and let's replace X with 9. So 4 times 9 is 36. So therefore, Y is equal to 36 when X is 9. Since we have a directly proportional situation, if we multiply X by 3 3 times 3 is 9 therefore we should multiply y by 3 12 times 3 is 36 this works if y varies directly with x In the last problem, since y varied directly with x, what that meant was that as x goes up, y goes up. So in the last problem, when x was 3, y was 9. So if we triple the value of x to 9, y in... increased actually why was 12 not 9 I don't know why I said 9 so we had to be in 12 when X increased to 9 y increase of 36 so when y varies directly with X these two So, if you were to double the value of x, y would double. If you triple the value of x, which is what we did, the y value would triple. Now, in this problem, we have an inverse variation. So, what does that mean? Well, as x goes up, y is going to go down. So, let's see if we can get the answer conceptually, and then we'll use the equation to confirm it. So, x is 4 and y is 48. Thank you. So, if we increase x to 8, what's going to happen to y? Notice that we increase x by a factor of 2. Since it's inverse, as x goes up, y goes down. So we can't multiply 48 by 2 because that's going to go up. We need to divide it by 2. So y should go down to 24. That's an inverse variation. When x goes up, y goes down proportionally. So now let's solve it. Y varies inversely with X, so the equation that we need is Y is equal to K divided by X. After you write the equation, use the first part to solve for K. So Y is 48, and X is 4. So how can we solve for k in this situation? Let's cross multiply. 1 times k is k, and 4 times 48, that's 4 times 40 is 160, because 4 times 4 is 16. And 4 times 8 is 32, so this gives us 192. Now that we have the value of k, we can find the value of y when x is 8, so we can answer the second part of the problem. So starting with the equation, let's replace k with 192 and x is 8. with 8 so what's 192 divided by 8 well we know that 192 is 48 times 4 and 48 let's make some space 48 is 8 times 6, and we still have the 4 on top. So notice that we can cancel an 8. So 6 times 4 is 24. So therefore, 192 divided by 8 is 24. When x is 8, y is 24, which is the answer that we predicted earlier. Because it's inverse, we said that as x goes up, y goes down. So if we double the value of x from 4 to 8, y decreases from 48 to 12. So, X went from 4 to 8, we increased it by a factor of 2, and for Y, it decreased by a factor of 2. It went from 48 to 24. Y varies jointly with X and Z. If Y is 36 when X is 2 and Z is 3, what is the value of Y when X is 4 and Z is 6? So whenever you hear the word jointly, it's direct variation but with two variables. So because it's direct variation, that means that as X goes up, Y goes up. And as Z goes up, Y also goes up. So let's see if we can answer conceptually first. So let's focus on x. x increases from 2 to 4. So it increases by a factor of 2. Now y increases from 3, I mean not y, excuse me, z increases from 3 to 6. So how much should y increase by? If y is proportional to x, that means that as x doubles, y should double. And y is also proportional or directly related to z. So as z doubles, y doubles. So what about the combined effects of doubling x? 2 times 2 is 4, so therefore, y should increase by a factor of 4. So, the original value of y is 36. If we increase it by a factor of 4, it should now be 144, so this should be our answer. So, for direct variation, you need to multiply the combined effects of x and z. So now let's solve it using the equation. So the first thing we need to do is write the equation. So if y varies jointly with x and z, we have the equation y equals kx times z. So we need to solve for k. using the information that we have here so why is 36 access to Z is 3 2 times 3 is 6 and 36 divided by 6 is 6 so K is equal to 6 Now, let's solve for y. So y is kxz. Let's plug in the value of k. So k is 6. And let's plug in the new values for x and z. So x is 4 and z is 6. So, 6 times 4 is 24. And what's 24 times 6? 2 times 6 is 12. So, 20 times 6 is 120. And 4 times 6 is 24. you add 120 and 24 you get 144 which is the answer that we had before so Y is going to be 144 when X is 4 and Z is 6 so we get the same answer as we did last time try this problem conceptually and then see if you can solve it algebraically. So y varies directly with the square of x. If y is 8 when x is 2, what is the value of y when x is 4? So let's analyze x and y. x changes from 2 to 4. So x increases by a factor of 2. y is currently 8 when x is 2. What is the new value of y? So, y varies directly with the square of x. The equation is y is equal to kx squared. So, y is proportional to x squared. So, if x doubles, y is not going to double. Now, we do have the word directly. That means that as x goes up, y is going to go up. But, y varies directly with the square of x. So, if x increases by a factor of 2, y is going to go up. y is going to increase by a factor of 4. The reason for that is because 2 squared is 4. You have to incorporate the square. So 8 times 4 is 32. So the answer should be 32. But now let's calculate it conceptually, I mean algebraically. So let's start with the equation, y equals kx squared, and let's solve for a k. So y is 8, x is 2. 2 squared is 4, and 8 divided by 4 is 2. So k is equal to 2. So now let's find the value of y when x is 4. So at this point... replace K with 2 and plug in 4 into X 4 squared is 16 16 times 2 is 32 so Y is indeed 32 as we got it conceptually before Try this one. Y varies inversely with the cube of X. If Y is 108 when X is 2, what is the value of Y when X is 6? So the equation is Y is equal to K over X to the 3rd. So, x is currently 2, and it's going to increase to 6. That means it's going to increase by a factor of 3, because 6 divided by 2 is 3. y is currently 108. So, what's the new value of y? So, for an inverse variation problem, we know that as x goes up, y is going to go down. So we need to divide 108 by some number. But what number should we divide it by? So what is 3 to the 3rd power? Because we have the cube of x. 3 to the 3rd is 3 times 3 times 3, which is 27. So that means we need to divide 108 by 27. It turns out 108 divided by 27 is 4. So let's see if we can calculate the same answer algebraically. So starting with the equation, Let's solve for k. So y is 108 when x is 2. So 2 to the third power is 8. And let's cross multiply. So we have... 8 times 108 is equal to k. So 8 times 100 is 800, and 8 times 8 is 64. So k is equal to 864. So now, let's plug it back into this equation. And let's find the value of y when x is equal to 6. So k is 8 times 108. We'll leave it like that. And x is 6. So that's 6 to the 3rd power. So we have 800, actually let's break down the larger number into small numbers. So 8, we can write 8 as 2 to the 3rd power. 108 was 27 times 4. And 27 is 2 to the 3rd times 3 to the 3rd. 2 to the 3rd times 3 to the 3rd is the same as 6 to the 3rd. You can multiply the bases if the exponents are the same. For example, 2 to the 1st power times 3 to the 1st power is 6 to the 1st power. Because 2 times 3 is 6. So you can multiply the bases if the exponents remain the same. So 2 to the third times 3 to the third is 6 to the third, which means we can cancel 6 to the third. So the answer is 4, which is what we had in the beginning. Now you can use a calculator, but some of you out there... may have to do these problems without a calculator so I've chosen to do that as well. Here's the last problem of the day so why varies directly with the cube of X and inversely with the square of Z? What's the equation? So directly with X, that means X is on top and it's associated with the cube of X but inversely with the square of Z so we have Z squared on the bottom. Now X X changes from 2 to 4, so X increases by a factor of 2. And Z changes from 3 to 6, so Z also increases by a factor of 2. But notice that Y is proportional to X cubed, since it's directly related to X. As X goes up, Y goes up. So 2 to the third power. Therefore, X will cause Y to increase by a factor of 8. Now what about Z? If we double the value of Z, what effect will that have on Y? So notice that Y is inversely related to Z. That means that as Z goes up, Y goes down. Notice that we have the square of Z. Z increases by a factor of 2. squared is 4. So since z is on the bottom, y is going to decrease by a factor of 4, so we need to divide y by 4. So the y value is currently 8. So we're going to divide it first by 4 and then multiply it by 8. If you multiply it by 8 and then divide it by 4 later, it's going to be the same thing. But it's better to divide first than multiply. 8 divided by 4 is 2. 2 times 8 is 16 so the net effect is we're multiplying y by 2 because if you times 8 and then divide by 4 8 divided by 4 is a net increase of a factor by 2 so the final answer should be 16 now let's solve it algebraically So let's solve for a k using this information. So y is 8, x is 2, and z is 3. 2 to the third power is 8. And 3 squared is 9. So let's cross multiply. So 8 times 9, I'm just going to leave it as 8 times 9. It's 72, but you'll see why. And then on the other side we have 8k. So notice that if we divide by 8, the 8s will cancel. So we can clearly see that k is equal to 9. So now let's take the second part of the problem and let's solve for y. So k is 9, x is now 4, and z is 6. So let's see if we can simplify it. 9 is the same as 3 times 3. And 4 to the 3rd, so we have 4 times 4 times 4. I'm going to change one of the 4's into 2 times 2. The other two 4's I'm going to rewrite it. 6 squared is 6 times 6, and each 6 I can make it 3 times 2. So that's one of the 6's, and here's the other one. So I can cancel a 3, and another one, and I can cancel the 2's. So it's 4 times 4, which is 16, which is the answer we had before. So now you know how to solve these problems conceptually and algebraically.

So let's begin. If y varies directly with x, the equation that you need is y is equal to kx. Now let's say if R varies directly with S.

R is equal to KS. If Z varies directly with L, the equation would be... z equals kl. So you can see a pattern here.

That's direct variation. Inverse, let's say if y varies inversely with x, y is going to be k divided by x. with S. R is equal to K over S. If Z varies inversely with L, Z is equal to K divided by L.

Now what if Y varies jointly with X and Z? How would you write that equation? So it's going to be Y is equal to K times X and Z, if it's jointly with X and Z. So if Y varies jointly with X and Z, what would you write? jointly with R and S, the equation is Y is equal to K times R times S.

Now what if Y varies directly with X but inversely with Z? So if it's directly with X, X is on top. If it's inverse, then that's going to be on the bottom, so Z is on the bottom.

Now what if Y varies directly with R but inversely? with s. So r is going to be on the top side, on the right side of the equation, and s is on the bottom.

So it varies directly with r, but inversely with s. Now what if y varies directly with the square of X. How would you write that? So this would be Y is equal to KX squared. It varies directly with the square of X.

Now what if Y varies inversely with the root or the square root of R? If it's inverse, it's on the bottom and we have the square root of R. Now, what if y varies directly with the cube of x, but inversely with the square of z?

So, directly, x is going to be on top, but it's the cube of x. and inversely with the square of z. So it's z to the second power, but on the bottom.

So now you know how to design the equations given a sentence. So let's try a few word problems. So let's start with a very simple problem.

Y varies directly with X. So if X is 3 when Y is 12, what is Y when X is 9? So first write the equation.

So Y is equal to KX. The next thing you need to do is solve for K. So we know that X is 3 when Y is 12. So let's plug in those values. To solve for K, we need to divide both sides by 3. 3 divided by 3 is 1, so we only have have K on the right side.

12 divided by 3 is 4. So that's K. Now once you have K, you can answer the second part of the problem. So what is Y when X is 9?

So starting with the equation Y equals KX, let's plug in 4 and let's replace X with 9. So 4 times 9 is 36. So therefore, Y is equal to 36 when X is 9. Since we have a directly proportional situation, if we multiply X by 3 3 times 3 is 9 therefore we should multiply y by 3 12 times 3 is 36 this works if y varies directly with x In the last problem, since y varied directly with x, what that meant was that as x goes up, y goes up. So in the last problem, when x was 3, y was 9. So if we triple the value of x to 9, y in... increased actually why was 12 not 9 I don't know why I said 9 so we had to be in 12 when X increased to 9 y increase of 36 so when y varies directly with X these two So, if you were to double the value of x, y would double. If you triple the value of x, which is what we did, the y value would triple. Now, in this problem, we have an inverse variation.

So, what does that mean? Well, as x goes up, y is going to go down. So, let's see if we can get the answer conceptually, and then we'll use the equation to confirm it. So, x is 4 and y is 48. Thank you. So, if we increase x to 8, what's going to happen to y?

Notice that we increase x by a factor of 2. Since it's inverse, as x goes up, y goes down. So we can't multiply 48 by 2 because that's going to go up. We need to divide it by 2. So y should go down to 24. That's an inverse variation.

When x goes up, y goes down proportionally. So now let's solve it. Y varies inversely with X, so the equation that we need is Y is equal to K divided by X. After you write the equation, use the first part to solve for K.

So Y is 48, and X is 4. So how can we solve for k in this situation? Let's cross multiply. 1 times k is k, and 4 times 48, that's 4 times 40 is 160, because 4 times 4 is 16. And 4 times 8 is 32, so this gives us 192. Now that we have the value of k, we can find the value of y when x is 8, so we can answer the second part of the problem.

So starting with the equation, let's replace k with 192 and x is 8. with 8 so what's 192 divided by 8 well we know that 192 is 48 times 4 and 48 let's make some space 48 is 8 times 6, and we still have the 4 on top. So notice that we can cancel an 8. So 6 times 4 is 24. So therefore, 192 divided by 8 is 24. When x is 8, y is 24, which is the answer that we predicted earlier. Because it's inverse, we said that as x goes up, y goes down. So if we double the value of x from 4 to 8, y decreases from 48 to 12. So, X went from 4 to 8, we increased it by a factor of 2, and for Y, it decreased by a factor of 2. It went from 48 to 24. Y varies jointly with X and Z.

If Y is 36 when X is 2 and Z is 3, what is the value of Y when X is 4 and Z is 6? So whenever you hear the word jointly, it's direct variation but with two variables. So because it's direct variation, that means that as X goes up, Y goes up.

And as Z goes up, Y also goes up. So let's see if we can answer conceptually first. So let's focus on x.

x increases from 2 to 4. So it increases by a factor of 2. Now y increases from 3, I mean not y, excuse me, z increases from 3 to 6. So how much should y increase by? If y is proportional to x, that means that as x doubles, y should double. And y is also proportional or directly related to z.

So as z doubles, y doubles. So what about the combined effects of doubling x? 2 times 2 is 4, so therefore, y should increase by a factor of 4. So, the original value of y is 36. If we increase it by a factor of 4, it should now be 144, so this should be our answer.

So, for direct variation, you need to multiply the combined effects of x and z. So now let's solve it using the equation. So the first thing we need to do is write the equation. So if y varies jointly with x and z, we have the equation y equals kx times z.

So we need to solve for k. using the information that we have here so why is 36 access to Z is 3 2 times 3 is 6 and 36 divided by 6 is 6 so K is equal to 6 Now, let's solve for y. So y is kxz.

Let's plug in the value of k. So k is 6. And let's plug in the new values for x and z. So x is 4 and z is 6. So, 6 times 4 is 24. And what's 24 times 6?

2 times 6 is 12. So, 20 times 6 is 120. And 4 times 6 is 24. you add 120 and 24 you get 144 which is the answer that we had before so Y is going to be 144 when X is 4 and Z is 6 so we get the same answer as we did last time try this problem conceptually and then see if you can solve it algebraically. So y varies directly with the square of x. If y is 8 when x is 2, what is the value of y when x is 4?

So let's analyze x and y. x changes from 2 to 4. So x increases by a factor of 2. y is currently 8 when x is 2. What is the new value of y? So, y varies directly with the square of x.

The equation is y is equal to kx squared. So, y is proportional to x squared. So, if x doubles, y is not going to double.

Now, we do have the word directly. That means that as x goes up, y is going to go up. But, y varies directly with the square of x. So, if x increases by a factor of 2, y is going to go up. y is going to increase by a factor of 4. The reason for that is because 2 squared is 4. You have to incorporate the square.

So 8 times 4 is 32. So the answer should be 32. But now let's calculate it conceptually, I mean algebraically. So let's start with the equation, y equals kx squared, and let's solve for a k. So y is 8, x is 2. 2 squared is 4, and 8 divided by 4 is 2. So k is equal to 2. So now let's find the value of y when x is 4. So at this point... replace K with 2 and plug in 4 into X 4 squared is 16 16 times 2 is 32 so Y is indeed 32 as we got it conceptually before Try this one.

Y varies inversely with the cube of X. If Y is 108 when X is 2, what is the value of Y when X is 6? So the equation is Y is equal to K over X to the 3rd. So, x is currently 2, and it's going to increase to 6. That means it's going to increase by a factor of 3, because 6 divided by 2 is 3. y is currently 108. So, what's the new value of y? So, for an inverse variation problem, we know that as x goes up, y is going to go down.

So we need to divide 108 by some number. But what number should we divide it by? So what is 3 to the 3rd power?

Because we have the cube of x. 3 to the 3rd is 3 times 3 times 3, which is 27. So that means we need to divide 108 by 27. It turns out 108 divided by 27 is 4. So let's see if we can calculate the same answer algebraically. So starting with the equation, Let's solve for k.

So y is 108 when x is 2. So 2 to the third power is 8. And let's cross multiply. So we have... 8 times 108 is equal to k. So 8 times 100 is 800, and 8 times 8 is 64. So k is equal to 864. So now, let's plug it back into this equation.

And let's find the value of y when x is equal to 6. So k is 8 times 108. We'll leave it like that. And x is 6. So that's 6 to the 3rd power. So we have 800, actually let's break down the larger number into small numbers.

So 8, we can write 8 as 2 to the 3rd power. 108 was 27 times 4. And 27 is 2 to the 3rd times 3 to the 3rd. 2 to the 3rd times 3 to the 3rd is the same as 6 to the 3rd.

You can multiply the bases if the exponents are the same. For example, 2 to the 1st power times 3 to the 1st power is 6 to the 1st power. Because 2 times 3 is 6. So you can multiply the bases if the exponents remain the same.

So 2 to the third times 3 to the third is 6 to the third, which means we can cancel 6 to the third. So the answer is 4, which is what we had in the beginning. Now you can use a calculator, but some of you out there...

may have to do these problems without a calculator so I've chosen to do that as well. Here's the last problem of the day so why varies directly with the cube of X and inversely with the square of Z? What's the equation? So directly with X, that means X is on top and it's associated with the cube of X but inversely with the square of Z so we have Z squared on the bottom.

Now X X changes from 2 to 4, so X increases by a factor of 2. And Z changes from 3 to 6, so Z also increases by a factor of 2. But notice that Y is proportional to X cubed, since it's directly related to X. As X goes up, Y goes up. So 2 to the third power.

Therefore, X will cause Y to increase by a factor of 8. Now what about Z? If we double the value of Z, what effect will that have on Y? So notice that Y is inversely related to Z.

That means that as Z goes up, Y goes down. Notice that we have the square of Z. Z increases by a factor of 2. squared is 4. So since z is on the bottom, y is going to decrease by a factor of 4, so we need to divide y by 4. So the y value is currently 8. So we're going to divide it first by 4 and then multiply it by 8. If you multiply it by 8 and then divide it by 4 later, it's going to be the same thing.

But it's better to divide first than multiply. 8 divided by 4 is 2. 2 times 8 is 16 so the net effect is we're multiplying y by 2 because if you times 8 and then divide by 4 8 divided by 4 is a net increase of a factor by 2 so the final answer should be 16 now let's solve it algebraically So let's solve for a k using this information. So y is 8, x is 2, and z is 3. 2 to the third power is 8. And 3 squared is 9. So let's cross multiply. So 8 times 9, I'm just going to leave it as 8 times 9. It's 72, but you'll see why.

And then on the other side we have 8k. So notice that if we divide by 8, the 8s will cancel. So we can clearly see that k is equal to 9. So now let's take the second part of the problem and let's solve for y.

So k is 9, x is now 4, and z is 6. So let's see if we can simplify it. 9 is the same as 3 times 3. And 4 to the 3rd, so we have 4 times 4 times 4. I'm going to change one of the 4's into 2 times 2. The other two 4's I'm going to rewrite it. 6 squared is 6 times 6, and each 6 I can make it 3 times 2. So that's one of the 6's, and here's the other one. So I can cancel a 3, and another one, and I can cancel the 2's.

So it's 4 times 4, which is 16, which is the answer we had before. So now you know how to solve these problems conceptually and algebraically.

Transcript for:Understanding Direct, Inverse, and Joint Variation

Transcript for:
Understanding Direct, Inverse, and Joint Variation