# Solving Equations Involving Variation

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

The concept of direct and indirect variation is quite common in our lives. In this lesson, we will look at the steps involved with solving equations involving direct and inverse variation. We will use an example that could easily be applicable to your life to illustrate the steps of the solving process.

## Variation

You just found a flyer advertising an upcoming concert that your favorite band is going to be playing at in 12 weeks!

You want to buy tickets for you and four of your friends. The only problem is that tickets aren't cheap! For five tickets, you will have to pay \$550! You decide to set aside the same amount of money each week so you can get the tickets. After 4 weeks, you have \$200, and you're hoping that if you continue putting the same amount of money aside each week, you'll have \$550 in 12 weeks.

Here's something that's pretty interesting! The relationship between the amount of money saved and the number of weeks you've been saving is actually an example of something called direct variation. In mathematics, variation is a term used to describe relationships between quantities.

There are two main types of variation - direct and inverse. There's also joint and combination variation, but these are just more involved instances of direct and inverse variation. Therefore, in this lesson, we'll just concentrate on direct and inverse variation. Let's get to it!

## Direct Variation

If a variable y varies directly with a variable x, then y = kx, where k is a constant called the constant of variation. To solve equations of this type, we must first find k, and then we can use the resulting equation to solve problems of variation. In other words, we follow these steps.

1. Set up the variation equation with k in it.
2. Use the information in the problem to find k.
3. Plug k into your variation equation.
4. Use the equation to answer the question posed in the problem.

For example, consider the concert. We said that the amount of money you've saved, call it A, varies directly with the number of weeks you've been saving, call it x. Therefore, we have the variation equation A = kx. That was the first step - not so hard, is it?

The next step is to find k. We said that after 4 weeks, you have \$200. We can find k by plugging these values into our variation equation and solving. We plug in A = 200 and x = 4.

We see that k = 50, which means that you've been setting aside \$50 each week to save up for the tickets. That step wasn't so bad either!

Now, we move on to the third step, which is simply plugging the value we found for k, which is 50, into the variation equation to get A = 50x. Okay, that step was really easy, and you know what? We can use this equation to find out if you'll have \$550 in 12 weeks, which is the fourth and final step! To do this, we simply plug 12 in for x in the equation.

We see that after 12 weeks, you will have saved \$600! You will have enough money for the tickets, plus an extra \$50!

## Inverse Variation

Okay, let's take a look at inverse variation! If a variable y varies inversely with a variable x, then y = k/x, where k is the constant of variation.

What's nice is that the solving process for problems involving inverse variation involves the exact same steps as a problem involving direct variation. The only difference is the equation itself.

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