# Variables in Math

## What is a Variable in Math?

In math, a **variable** is a letter used in place of an unknown number in equations, expressions, and formulas. The variable is used as a placeholder for the unknown number.

In an equation like 11*x* + 25 = 135, the variable will have a definite, but unknown, value.

In an expression like *x* - 9, the variable can be replaced with any number to produce a different result.

An example of how variables are used in math is the Pythagorean Theorem:

{eq}a^2 + b^2 = c^2 {/eq}

This formula contains three variables, *a*, *b*, and *c*.

Another example of how variables are used in math is the quadratic formula, which is shown here.

There are different categories of variables.

- A
**dependent variable**gets its name from the fact that its value*depends*on something else. - An
**independent variable**, on the other hand, is not dependent on anything else. - A
**free variable**is a variable with no limits on its value. A free variable can represent any number. For example, in the function f(*x*) = 8*x*, the variable*x*is a free variable. It can represent any number, positive or negative, whole number, fraction, or decimal. - A
**bound variable**has parameters that it can not exceed. It is bound within those parameters and can only represent a number within the limitations given. - A
**random variable**, often used in statistics, is a variable that can take on different values.

In a mathematical function, the independent variable is the input, and the dependent variable is the output. In the equation *y* = 6*x* + 9, for example, the variable *x* is the independent variable. The variable *y*, on the other hand, is the dependent variable because its value *depends* on the value of *x*.

### Key Properties of Variables

What is a variable in math? These are important things to keep in mind when using variables in math:

- A variable in math is a letter. Lowercase letters and uppercase letters are both used.
- Any letter can be used, although
*x*and*y*are letters used frequently. - Variables in math are placeholders for unknown numbers.
- In math, variables can be used in expressions, equations, and formulas.
- Variables are often used when the best answer needs to be found.

## Defining Variables

**Variables** are nothing more than a placeholder. They stand for things that you want to find out but don't have the answer to yet. Think back to grade school when you were learning addition and multiplication. Did your teacher ask you, 'What plus two equals five?' Well, when the teacher uses the word 'what,' she actually used it as a variable. The 'what' is something you wanted to know but didn't have the answer to right away.

Now, of course, you wouldn't even blink at that question. You quickly would answer 'three' and not even think twice about the 'what.'

In math lingo, there is a formality when it comes to variables. When you write them, instead of using the word 'what,' you would use letters. You can use any letter you choose. Typically, we stick to the English alphabet but technically, you can use any easy-to-identify symbol as your choice.

The two most common variables you will encounter in your schooling career are the variables *x* and *y*. Don't ask the reason why these two are the most popular. They just are and have been in use for so many years past. They are easy to spot and to identify, which makes them ideal candidates for a wide variety of mathematical applications.

Just because *x* and *y* are the two most popular variables doesn't mean you have to use just those two. You can use anything you wish. You have numerous choices to pick from.

Any letter you can think of can be used as a variable. They can be uppercase or lowercase. When you get into more complex math applications, you might have several variables in one equation, so your ability to use more than one variable at a time will become important.

## How to Find the Variable

Sometimes, in a math equation, finding the value of the variable is the goal. In this case, the variable is being used when the best answer needs to be found. To find the value of the variable, the equation can be manipulated.

When an equation is manipulated, it is very important to perform the same operation on both sides of the equation.

In order to keep it a true equation, the two sides must be equal. This means whatever is done to one side of the equation, the same thing must be done to the other side of the equation.

Here is an example of how to find the value of the variable in a math equation:

- Equation: 5
*x*+ 8 = 23 - Subtract 8 from both sides of the equation: 5
*x*+ 8 - 8 = 23 - 8 - Simplify: 5
*x*= 15 - Divide by 5 on both sides of the equation: 5
*x*/ 5 = 15 / 5 - Simplify:
*x*= 3

The equation was manipulated to isolate the variable, *x*, and find its value.

In this example, the value of the variable is 3.

Some examples require the equation to be manipulated more than others, but the basic steps to follow are always the same.

### Step 1: Isolate All Terms With the Variable

Remember what a variable in math is: A variable in math is a letter that is acting as a placeholder for an unknown value. The goal is to find the value of the variable.

In the first step, all terms *with* the variable are manipulated so they are one side of the equation, and all terms *without* the variable are on the other side of the equation.

Here is an example:

- Equation: 4
*x*+ 3 = 6*x*- 17 - Subtract 4
*x*from both sides: 4*x*+ 3 - 4*x*= 6*x*- 17 - 4*x* - Simplify: 3 = 2
*x*- 17*(Now, all terms with the variable are on one side of the equation.)* - Add 17 to both sides: 3 + 17 = 2
*x*- 17 + 17 - Simplify: 20 = 2
*x**(Now, all terms without the variable are on the other side of the equation.)*

### Step 2: Isolate a Single Value of the Variable

Remember what a variable in math is: A variable in math is a letter that is acting as a placeholder for an unknown value. The goal is to find the value of the variable.

In the second step, here is how to find the value of the variable. Use the same equation started in step one.

- Equation: 20 = 2
*x* - Divide both sides by 2: 20 / 2 = 2
*x*/ 2 - Simplify: 10 =
*x*

In this example, the value of the variable is 10.

A single value of the variable was isolated to find its value.

The example shows how to find the value of the variable.

## Lesson Summary

These are important things to keep in mind when using variables in math:

- A
**variable**in math is a letter. Lowercase letters and uppercase letters are both used. - Any letter can be used, although
*x*and*y*are letters used frequently. - Variables in math are placeholders for unknown numbers.
- In math, variables can be used in expressions, equations, and formulas.
- An example of a variable in use is the variable
*x*in 7x - 9 = 47. - Variables are often used when the best answer needs to be found.

Sometimes, in a math equation, finding the value of the variable is the goal.

- To find the value of the variable, the equation can be manipulated.
- When an equation is manipulated, it is very important to perform the same operation on both sides of the equation.
- Step one is to isolate all terms with the variable.
- Step two is to isolate a single value of the variable.

## Using Variables

Variables have always stood in the place of an unknown answer. Their use remains the same. They always stand for something you want to know but don't have the answer to. With the aid of variables and algebra, you can find out the answer.

Going back to grade school again, let's look more carefully at the question 'What plus two equals five?' We now know that the 'what' stands for a variable. We also know now that we can choose any letter for our variable. So, let us choose *x*. Why not? It's an easy letter to choose and there is a phrase that goes, 'X marks the spot.' That's easy to remember.

We've changed our question into a written mathematical form using variables and math symbols. Now, to finish using our variable to find our answer, we use some algebra and reverse our operations to find our answer. We have a '+ 2,' so we need to subtract 2 to find our answer.

Look at what we found! We found our answer to be 3, what we've known all along. But now we have mathematical proof that 2 + 3 = 5. How cool is that?

## When to Use Variables

Variables can be used anywhere an answer is required but not known. It can be used to find unknown numbers. It can be used to graph equations where you have two variables. To graph, you would choose various numbers to put into one of the variables and calculate the other. This would give you points you can plot on a graph. It is similar with graphs involving three variables. In this case, you would input numbers into two of the variables to calculate the third and then plot the points.

In higher math such as signal analysis, cell phone companies use variables to help them calculate the best signal system to use so that you get the crispest and clearest voice service. Really, variables are used wherever a best answer needs to be found.

## Lesson Summary

**Variables** stand for things that you want to find but don't have the answer to yet. They are useful for finding out answers to something you don't currently know. Writing them mathematically is as easy as picking a letter from the alphabet. Variables have many useful applications, but all of them are when a best answer needs to be found.

## Variables: Key Points

**Variable**- a placeholder for an unknown number in an equation- Represented with lowercase or uppercase letters
- Have many uses and applications, including graphs of equations and signal analysis

## Learning Outcomes

After watching this video, check to see if you can:

- Define variable
- Identify examples of variables in equations
- Solve for variables in simple equations
- Recall some uses of variables in mathematics

To unlock this lesson you must be a Study.com Member.

Create your account

## Defining Variables

**Variables** are nothing more than a placeholder. They stand for things that you want to find out but don't have the answer to yet. Think back to grade school when you were learning addition and multiplication. Did your teacher ask you, 'What plus two equals five?' Well, when the teacher uses the word 'what,' she actually used it as a variable. The 'what' is something you wanted to know but didn't have the answer to right away.

Now, of course, you wouldn't even blink at that question. You quickly would answer 'three' and not even think twice about the 'what.'

In math lingo, there is a formality when it comes to variables. When you write them, instead of using the word 'what,' you would use letters. You can use any letter you choose. Typically, we stick to the English alphabet but technically, you can use any easy-to-identify symbol as your choice.

The two most common variables you will encounter in your schooling career are the variables *x* and *y*. Don't ask the reason why these two are the most popular. They just are and have been in use for so many years past. They are easy to spot and to identify, which makes them ideal candidates for a wide variety of mathematical applications.

Just because *x* and *y* are the two most popular variables doesn't mean you have to use just those two. You can use anything you wish. You have numerous choices to pick from.

Any letter you can think of can be used as a variable. They can be uppercase or lowercase. When you get into more complex math applications, you might have several variables in one equation, so your ability to use more than one variable at a time will become important.

## Using Variables

Variables have always stood in the place of an unknown answer. Their use remains the same. They always stand for something you want to know but don't have the answer to. With the aid of variables and algebra, you can find out the answer.

Going back to grade school again, let's look more carefully at the question 'What plus two equals five?' We now know that the 'what' stands for a variable. We also know now that we can choose any letter for our variable. So, let us choose *x*. Why not? It's an easy letter to choose and there is a phrase that goes, 'X marks the spot.' That's easy to remember.

We've changed our question into a written mathematical form using variables and math symbols. Now, to finish using our variable to find our answer, we use some algebra and reverse our operations to find our answer. We have a '+ 2,' so we need to subtract 2 to find our answer.

Look at what we found! We found our answer to be 3, what we've known all along. But now we have mathematical proof that 2 + 3 = 5. How cool is that?

## When to Use Variables

Variables can be used anywhere an answer is required but not known. It can be used to find unknown numbers. It can be used to graph equations where you have two variables. To graph, you would choose various numbers to put into one of the variables and calculate the other. This would give you points you can plot on a graph. It is similar with graphs involving three variables. In this case, you would input numbers into two of the variables to calculate the third and then plot the points.

In higher math such as signal analysis, cell phone companies use variables to help them calculate the best signal system to use so that you get the crispest and clearest voice service. Really, variables are used wherever a best answer needs to be found.

## Lesson Summary

**Variables** stand for things that you want to find but don't have the answer to yet. They are useful for finding out answers to something you don't currently know. Writing them mathematically is as easy as picking a letter from the alphabet. Variables have many useful applications, but all of them are when a best answer needs to be found.

## Variables: Key Points

**Variable**- a placeholder for an unknown number in an equation- Represented with lowercase or uppercase letters
- Have many uses and applications, including graphs of equations and signal analysis

## Learning Outcomes

After watching this video, check to see if you can:

- Define variable
- Identify examples of variables in equations
- Solve for variables in simple equations
- Recall some uses of variables in mathematics

To unlock this lesson you must be a Study.com Member.

Create your account

#### Which is an example of variable terms?

A variable is a letter, either lowercase or uppercase. For example, in the equation ** x + 2 = 5**, the

**is being used as a variable.**

*x*#### How do you define a variable?

In math, a variable is a letter used in place of an unknown number in equations, expressions, and formulas. The variable is used as a placeholder for the unknown number. Any letter can be used as a variable.

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