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Direct Variation: Definition, Formula & Examples

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  • 0:01 Definition
  • 0:57 Example of an Equation
  • 3:08 Example of a Table
  • 4:13 Example of a Graph
  • 4:52 Using All Three Methods
  • 6:41 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Often

Elizabeth has taught high school math for over 10 years, and has a master's in secondary math education.

What is the relationship between the distance a car travels when cruise control is activated and the amount of money you have in the bank when you save regularly? Both are examples of direct variation. Learn more about direct variation in this lesson.

Definition

There are so many quantities that can increase at a consistent rate. If you set your car's cruise control and start driving, you will travel a consistent number of miles per hour, every hour. If you save the same amount of money every week, you will increase your savings by a consistent amount. If we graph these scenarios we get a straight line, and so they are called linear equations. However, there are different types of linear equations. The one we will investigate in this lesson is called direct variation.

In direct variation, two quantities, such as time and distance, or hours and pay, increase or decrease at a consistent rate. The consistent rate of increase or decrease is also called the constant of variation. In this lesson, we will look at three ways of presenting direct variation, an equation, a table, and a graph, through some real life examples.

Example of an Equation

As an example, let's say you want to save some money for an upcoming holiday or event. You have no money saved for this event at the start, but you are going to save $50 per week for 5 weeks.

In this equation, 50 will be your constant of variation, because it will stay the same for the duration of the time you'll be saving your money. You know that at the end of this, you will have $50 x 5, or $250. But what if you want to save for a longer period of time, or you know you can only save $25 per week? Could we come up with a more general way of describing the problem?

We can with an equation:

amount saved = (dollars)(weeks)

In algebra, we are more likely to use single letters as variables, so let's rewrite the equation that way. If a = the amount saved, d = the number of dollars, and w = weeks, your equation would look like this:

a = (d)(w)

Now, once you decide how much money you can save per week, you only need to substitute values into the equation. For example, if you can save $50 per week, how much can you save in 12 weeks? To solve the direct variation equation, we substitute in the values:

a = (50)(12)

Solving this equation, you see that 50 x 12 = 600, so a = 600, meaning you will save $600.

Now, instead of solving for a, we'll solve for a different variable, w. Let's say you can save $40 per week. How many weeks will it take to save $520 dollars?

Again, we can substitute in the values that we have into the direct variation equation. The amount you want to save is $520 and the amount you can save per week is $40, so your equation would look like this:

520 = (40)(w)

Using algebra to solve this, we will divide both sides by 40 to find the value of w:

w = 520 / 40

w = 13

So in 13 weeks, you will save $520

Example of a Table

Another way you can track your savings is by using a table. In this table, we will assume that you save $50 per week. Again, because this is direct variation, we will assume that you start with no money saved.

Week Amount
0 0
1 $50
2 $100
3 $150
4 $200
5 $250

Here are two key points to notice about this table:

  1. When you begin saving at week 0, you have $0. Thus, the first table entry is (0, 0). This will be true in every direct variation table you see.
  2. Each week the amount you have saved increases by $50. Another way to say this would be 'as the week number increases by 1, the amount saved increases by $50.' This is what is meant by 'increase or decrease at a consistent rate.' The $50 is also called the constant of variation, because it tells us how much the savings will go up by each week.

Using the information in the table above, and assuming a consistent rate of increase of $50 per week, how much money will be saved by week 20? Well, if you keep adding $50 to the previous week, by week 20 you'll get to $1,000.

Example of a Graph

Finally, you can represent how your savings grow by using a graph. To make the graph, we can plot some of the points from the table and connect them. Notice how the graph is a line; this is because direct variation is a linear equation.

Graph of Money Saved versus Time

Here are two key things to notice about this graph:

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