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# Combining Like Terms in an Algebraic Expression

by Ron Kurtus (revised 11 February 2013)

A * term* in Algebra is either a number, a variable, or a constant. Examples of terms are the number 23 and the variable

**y**.

An * expression* is a combination of terms, along with operators and separators of those terms. Simple examples of expressions include

**6y**and

**a**+ 2

**b**

A * like term* is a term in an expression that is similar to—or "like"—another term. For example 2 and 3 are like terms in the expression (3 − 5

**x**+ 2).

Some expressions can be complex and have a mixture of terms. You would like to simplify them as much as possible, in order to perform other operations, solve an equation or such. One method to simplify an expression is to look for terms that can be added together. This includes numbers, variables and sub-expressions.

Questions you may have include:

- How do you combine numeric like terms?
- How do you combine variable like terms?
- How do you combine like sub-expressions?

This lesson will answer those questions.

## Combining numeric like terms

In order to simplify an expression, the first thing to look for are numbers or numeric terms that stand alone and can be combined.

For example, consider the expression** 2 + 3x − 4xy + 7 − y − 5**.

You can see the numbers **2**, **7** and **−5** are like terms that can be combined, resulting in a simpler expression **3x − 4xy − y + 4**.

Notethat typically the numeric terms are listedlastin an expression.Also note that

y − 5can be thought of asy + (− 5).

Another example of combining numbers is **(x ^{2} + 2x + 3) − 6 + (y + 8)**.

Although the parentheses can group sub-expressions, removing them does no harm in this case. The numbers **3**, **− 6** and **8** can be combined, resulting in the expression **x ^{2} + 2x + y + 5**.

## Combining variable like terms

When simplifying an expression, the next thing to look for is like-terms among variables. That means any multiplies of a variable or products of variables.

For example, **x**, **5x** and **−2x** are like terms of **x**.

Also, **2x ^{2}y**,

**−x**and

^{2}y**5x**are like terms of

^{2}y**x**.

^{2}yTo get a little tricky, **3yx ^{2}** is also a like term of

**x**, since you can rearrange terms in a multiplication, such that

^{2}y**3yx**=

^{2}**3x**

^{2}**y**.

**5x** and **5y** are not like terms and do not combine. Likewise, **2x ^{2}y** and

**2x**are not like terms.

^{3}y### Example

Consider the expression **3x − 2y − 4xy + x ^{2} + 4y − x + 2xy − y^{2}**.

The best way to combine like terms is to first group them together.

(3x − x) +(−2y + 4y) + (−4xy + 2xy) + x^{2}− y^{2}

Notethat you can consider an expression like4y − xto be4y + (−x). That makes things easier.

Now combine the like terms.

2x + 2y −2xy + x^{2}− y^{2}

Rearrange the terms in a better format.

x^{2}+ 2x −2xy + 2y − y^{2}

## Combining like sub-expressions

Sometimes an expression contains multiples of like sub-expressions that can be added together.

Consider **3(x + 7) + 7xy + 4(x ^{2} + 7) + 2xy − 2(x^{2} + 7) + (x + 7)**.

Like terms are multiples of **(x + 7)**, **(x ^{2} + 7)** and

**xy**.

Group the multiples of like terms or sub-expressions:

(3 + 1)(x + 7) + (7 + 2)xy + (4 − 2)(x^{2}+ 7)

4(x + 7) + 9xy + 2(x^{2}+ 7)

We can simplify this even more by completing the multiplication and combine like terms again.

4x + 28 + 9xy + 2x^{2}+ 14

4x + 42 + 9xy + 2x^{2}

Rearrange to put in a better form.

2x^{2}+ 4x + 9xy+ 42

## Summary

Complex expressions have a mixture of terms that you would like to simplify. One method to simplify an expression is to look for terms that can be added together. This includes numbers, variables and sub-expressions.

Simplify your life

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## Combining Like Terms in an Expression