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Independent & Dependent Variables in Math

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  • 0:03 Independence & Dependence
  • 0:47 Deciphering Variable Types
  • 1:26 Example
  • 3:27 Equations Without Context
  • 5:55 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Learn what it means for a variable to be independent or dependent. Then explore how we identify which variables are independent and which are dependent in word problems and math equations.

Independence & Dependence

You're probably familiar with what it means to be independent or dependent on something. A person's income is dependent upon their job because without that job they wouldn't get paid. A car's speed is independent of the amount of windshield wiper fluid it has because that has no effect on the speed whatsoever. In general, if you're independent of something, it means it cannot affect you in any way, and if you're dependent on something, changes in whatever it is you're dependent on will affect you.

These concepts of independence and dependence also have meaning in math, specifically with variables. In this lesson, we will learn what independent and dependent variables are and how to identify them.

Deciphering Variable Types

In a math equation, variables are the symbols or letters that represent numbers whose values can change. Variables can either be dependent or independent on other variables. Dependent variables rely on other variables to find their value, and independent variables do not rely on other variables to find their value.

So, to sum it up, an independent variable will change the value of a dependent variable, but a dependent variable cannot change the value of an independent variable, and the value of the dependent variable is determined by the value of the independent variable.

We can best see how the two types of variables differ in word problems, so let's look at one.

Example

Sarah makes $10 an hour and works between 6 to 8 hours a day. How much money does she make in a single day?

Let's start by breaking down the components of our word problem. We have one constant, a value that does not change, and two variables. The constant is her wage, $10 per hour, and the variables are how much money she makes in a day (m), and how many hours she works (h).

So, now which variable is dependent and which is independent? Since we know what the variables physically represent, we can figure this out by determining which variable depends upon the other to find its value. This will be our dependent variable.

  • Does Sarah need to know how many hours she works to find out how much money she makes in a day

- or -

  • Does Sarah need to know how much money she makes in a day to find out how many hours she works?

In this problem, it's the first option. Since Sarah's wage is hourly, how many hours she works determines how much money she makes.

So m, money made in a day, is our dependent variable, and h, hours worked, is our independent variable.

To see how the independent variable changes the value of the dependent variable, we can set up an equation for this word problem and solve it. Here, the amount of money Sarah makes is equal to her wage multiplied by the hours she works.

m = $10 * h

Sarah can work for 6, 7, or 8 hours in a day. We can see how this changing value for the independent variable affects the value of the dependent variable.

m = $10 * 6 = $60

m = $10 * 7 = $70

m = $10 * 8 = $80

When you know what each variable represents in the real world, you can use logic to figure out whether each variable is independent or dependent. There is no one size fits all formula that will automatically tell you what type of variable each is; you must figure it out yourself.

Equations Without Context

So, we've seen that you can figure out which variables are independent and dependent when you know what they represent, but what if you don't know that? Often in a math class you're given nothing more than an equation like the following:

y = x + 1

a = 3b2 + 4c

When you're given an equation with a single stand-alone variable on one side of the equals sign, and a combination of constants and variables on others, the stand-alone variable is often the dependent variable. The variables on the other side of the equals sign are then independent.

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