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 = $80When 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* = 3*b*2 + 4*c*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.

Just like before, a change in the independent variable will affect the value of the dependent variable. You can see this by plugging in a couple different values for the independent variable *x* in the first equation example from this section:*y* = 2 + 1 = 3*y* = 200 + 1 = 201Unfortunately, we cannot absolutely guarantee that the stand-alone variable will always be the dependent variable. Not only can this be untrue, but sometimes you aren’t even given an equation with a stand-alone variable on one side of the equals sign.*y* + 2*x* = 1*a* + 4*b* = 3*a* + *b*In order to be absolutely sure which variable is the dependent variable in an equation where you don’t know what the variables physically represent, you need to know which variable is a function of the others.

When you say one variable is a function of another, i.e. ‘ *x* is a function of *y*,’ what you’re really saying is that *x* is dependent on *y*. Knowing this would let you rearrange the equation so that the dependent variable is by itself on one side of the equals sign.*y* + 2*x* = 12*x* = 1 – *y**x* = (1 – *y*) / 2We can represent which variable is a function of the other visually with **function notation**. In function notation, you replace the dependent variable with a single letter or symbol representing the name of the function with the independent variables in a parenthesis next to it.

*f(y)* = (1 – *y*) / 2*g(a,b)* = 5*a*+ *b*When the equation you’re given uses function notation, it is immediately clear which variable is the dependent variable, and which ones are independent variables.

## Lesson Summary

When doing math equations, you often find yourself working with variables. **Variables** are the symbols or letters in math equations whose values can change. There are two types of variables; **independent variables**, which don’t rely on other variables to find their values, and **dependent variables**, which must rely on other independent variables to find their values.

In order to figure out which variables are independent or dependent in a math equation, we either need to know information about what the variables physically represent, or we need to be told which variable is a function of the others. If we’re given information as to what they represent, we can use logic to determine which variables are dependent on the values of others, and which are not.If we are told which variable is a function of the others, we know that variable is the dependent variable and the others are the independent variables. We can represent this with **function notation**. This notation lets us immediately know visually which variables in an equation are dependent and independent.