# Equivalent Expressions Calculator + Online Solver With Free Steps

The **Equivalent Expression Calculator** is used to find out the equivalent expressions to your algebraic expressions. An **Algebraic Expression** can be expressed in many forms as it represents a relationship between quantities and variables. So there is this thing called **Equivalent Expressions** which could be present for any number of algebraic expressions.

Solving these **Expressions** can be very challenging and that is where this **Calculator** comes in, it is very capable as it can solve such intuitive and not very straightforward problems.

You can simply enter your **Algebraic Expression** into the input box, and at the press of a button, you can have your solution in front of you.

## What Is an Equivalent Expressions Calculator?

**The Equivalent Expression Calculator is an online calculator which can solve your algebraic expression to extract equivalent expressions for the given problem.**

This **Calculator **is special because it goes through all possible combinations to extract the **Equivalent Expression**, as there is no straightforward **method** for solving such a problem.

It is very easy to use, and it can be used an **indefinite** number of times and for free. This works in your **browser** and does not require anything to be downloaded or installed on your device.

## How to Use the Equivalent Expressions Calculator?

To use the **Equivalent Expression Calculator**, you must simply enter your **Algebraic Expression **into the input box, press a button, and you will be provided with the solution to your problem.

Now, the step-by-step guide for getting the best result from your calculator is given below:

### Step 1

First, you must set up your problem, and check whether it is in the right format to be read by the calculator. Once, through that, you can enter your algebraic equation in the input box labeled **Simplify**.

### Step 2

Now, that you have entered your problem inside the box, you can press the button labeled **Submit**. This will open up a new interactable window, where you can access your solution to the problem.

### Step 3

Finally, if you wish to solve more questions of similar nature, then you can simply enter their algebraic expressions into the box present in the interactable new window. And get results for as many problems as you like.

## How Does the Equivalent Expressions Calculator Work?

The **Equivalent Expression Calculator** works by solving the possible equivalent expressions for a given **Algebraic Equation**. We know that **Algebraic equations** represent an expression where variables can have certain values and thus provide certain results.

And this calculator uses the nature of an algebraic equation to calculate the required **Equivalent Expression **for it. Now let’s dig deeper into the Algebra of things and get to know more about **Algebraic Equations** first.

### Algebraic Equations

In crude Mathematical terms, an **Algebraic Equation** is defined as a mathematical expression, where two values are set to be equal. This is more easily understood as an expression setting up a **relationship** between the two different **Representations **of the same thing.

So, let’s assume that there is a number $a$, then we can associate this number with a **Mathematical Operation** between any two numbers:

**c x d = a, e $\div$ f = a, g + h = a, i – j = a **

Thus, all these shown above are an example of Algebraic expressions in a crude definition.

### Equivalent Expressions

Now, this is our main topic, **Equivalent Algebraic Expressions**, and the ways to find them. But first, let’s understand what **Equivalent Expressions** are.

**Equivalent Expressions** can be defined as mirror images of a particular Algebraic Expression but not in terms of **Similarities**, rather in terms of getting the same results. They are also referred to as **Duplicates** of an expression.

They work in such a fashion that the** Results** of both equivalent expressions would be the same, but they wouldn’t be in the most ideal cases. So, one could think of a **Relationship** as follows:

**b = f1 ( x ), b = f2 ( x )**

Here, b would have the same value for both cases, and unless there is a **Limit** applied, it would get the same result for every value of x placed in both functions. Therefore, this is how **Equivalent Expressions** operate and give the same results for the same inputs whilst being different from one another.

### Calculate for Equivalent Expressions

Now, we look into the method for calculating **Equivalent Expressions**, as it still seems like a mysterious process.

We begin by analyzing the **Nature** of the Algebraic Expression, if the expression’s variable is too tied up with **Mathematical Operations,** then, we don’t have a lot of equivalent options. This is shown here:

**b = ax + c, b = a ( x + $\frac { c } { a }$ ) **

So, we saw that there are not many options to deal with in such an expression and we can only get an **Equivalent Expression **by taking one value common.

But we can similarly see that this could be expressed as:

**b = a x + c, b = x ( a + $\frac { c } { x }$ ) **

Or even as:

**b = a x + c, b = c ( $\frac { a x } { c }$ + 1 ) **

Therefore, this is the way we can get equivalent expressions for any given **Algebraic Expression**.

## Solved Examples

Now that we have gone through the theory on the topic, we shall look at some examples to get a better understanding of the subject.

### Example 1

Consider the given Algebraic Equation:

**12 x y + 4 x **

Find all possible Equivalent Expressions for this Algebraic Expression.

### Solution

So we begin by first looking at the** Variables **which can be present in both additive values, and that is x. We can see that x is present in both quantities being added together so, we get one **Equivalent Expression** as:

**12 x y + 4 x = x ( 12 y + 4 ) **

Now, moving forward we see that 4 is a factor of 12, so we can common it too, and then we get another equivalent expression:

** 12 x y + 4 x = 4 x ( 3y + 1 ) **

And finally, we have one more expression we can get where we use y in the equivalent expression too, and this would look like:

**12 x y + 4 x = 4 x y ( 3 + $\frac { 1 } { y }$ ) **

Hence, we have three different equivalent expressions we were able to extract from this one **Algebraic Expression**.

### Example 2

Consider an Algebraic Expression described below:

\[ 3 x y + 9 x ^2 \]

Calculate the Equivalent Expressions for the given expression.

### Solution

We start by first looking at the variable which is **Common** amongst the additional terms. This is important as this will provide us with the term which can be taken as common amongst them. As we can see, this** Variable** is true $x$, present in both values so, we can write one equivalent expression as:

** 3 x y + 9 $x^2 $ = x ( 3 y + 9 x ) **

Now, if we look closer, we can also see that 3 is a factor of 9, so we can common 3 from both values too. Therefore, we get the following result:

**3 x y + 9 $x^2$ = 3 x ( y + 3 x ) **

Here, we could take the y common and create a fraction out of one value, this is another equivalent expression for the same **Algebraic Expression**. This is done as follows:

\[ 3 x y + 9 x^2 = 3 x y ( 1 + 3 \frac {x} {y} ) \]

Now, we present the last but not the least equivalent expression. This one can be calculated with a bit more **Sophisticated** algebra. We can see that the given expression could be of the form:

\[ ( a + b ) ^2 = a^2 + b^2 + 2 ab, \phantom {()} (a + b) ^2 – b ^2 = a^2 + 2 ab \]

So, if we take the a and b values for our original expression we get:

\[ b = \frac {y} {2}, \phantom {()} a = 3 x \]

Hence:

\[ a^2 + 2 ab = ( 3 x )^2 + 2 ( 3 x ) ( \frac {y} {2} ) = ( 3 x + \frac {y} {2} )^2 – \frac {y^2} {4} \]

Therefore, we have our equivalent expressions.