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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In the real world, we come across a lot of problems that vary directly with each other. You will learn how to solve these types of direct variation problems in this video lesson.

If you are the owner of an ice cream cart, what do you think will happen when the temperature rises? Why, you would get more business. So, we have our number of customers rising along with the temperature. This is a real life example of direct variation.

We can define **direct variation** as problems where if one thing rises or lowers, another thing also rises and lowers accordingly. In our ice cream example, we see that as our temperature goes up, we get more customers. Also, if the temperature goes down, then we also get fewer customers. Our two things, our temperature and number of customers, vary directly with each other. In this video lesson, we will talk about how to solve these direct variation problems. So, let's keep going.

Of course, this being math and all, we have a formula for direct variation problems. All direct variation problems will follow this formula: *y* = *kx*, where *k* is the constant of variation that tells you how the two things change with each other and *x* and *y* are our two things. We can rearrange this problem for *x* also: *x* = *y* / *k*.

Now, let's see how we can use this formula for our ice cream example. With direct variation problems, the problem will usually give you one complete pair of information that we can use to find our constant of variation, our *k*, and then it will ask you to find the missing information of a second pair of information.

Our problem could say, 'The number of customers that an ice cream cart gets varies directly with the temperature. If there are 50 customers when the temperature is 60 degrees Fahrenheit, how many customers will there be if the temperature is 90 degrees Fahrenheit?'

We know that our ice cream example is a direct variation problem, so we can label our *x* and *y* in the problem. We will label our number of customers as *x* and our temperature with *y*. Our complete pair of information is 50 customers and 60 degrees Fahrenheit.

We will use this to find our constant of variation by plugging in these values in for *x* and *y*, and then solving to find our *k*. Let's see what our *k* equals. We have *y* = *k * x*, so 60 = *k* * 50. Solving for *k*, we get *k* = 60 / 50 = 1.2.

Now, we can use this *k* to find the missing information. The problem wants us to find the number of customers when the temperature is 90 degrees Fahrenheit. Now, we will use the formula again and plug in *k* and *y* to solve for *x*.

We have *y* = *k * x* or *x* = *y* / *k*, so *x* = 90 / 1.2 = 75. Does our answer of 75 sound right? Since this is a direct variation problem, we should expect our number of customers to increase as our temperature increases. Is this what is happening? Yes, it is. So, our answer is correct.

Let's see another problem: Sally makes $200 working for 10 hours at a cupcake shop. If Sally's pay varies directly with the number of hours she works, how many hours will Sally need to work to make $260?

To solve this problem, we first label our *x* and *y*. We will label our *y* with the amount of money Sally makes and our *x* with the number of hours she works. Our first pair of information is $200 and 10 hours.

We use this to find our *k*, our constant of variation. We plug in *x* and *y* into *y* = *k * x* to get 200 = *k* * 10. Solving for *k*, we find *k* = 200 / 10 = 20. Now, we will use this *k* to find our missing information.

We are plugging in the amount of money Sally made to find the number of hours, so we are plugging in *y* and *k*. We have 260 = 20 * *x*. Solving for *x*, we have *x* = 260 / 20 = 13. So, Sally needs to work for 13 hours to make $260.

Look at that! We've solved two problems already. And what have we learned? We've learned that **direct variation** problems are problems where if one thing rises or lowers, another thing also rises or lowers, accordingly.

The formula for all direct variation problems is *y* = *kx* or *x* = *y* / *k*, where *k* is your constant of variation and *x* and *y* are your two things that vary directly with each other. To solve these problems, you usually use the first complete pair of information to find your *k*, and then you will use your *k* to find the missing information in your second pair.

Once you are done with this lesson you should be able to:

- Recall the equation for direct variation
- Solve a direct variation problem

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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