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Combining Like Terms in an Expression

by Ron Kurtus (19 January 2008)

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 are:

• 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. There is a mini-quiz near the end of the lesson.

Combining numeric like terms

In an expression, the first thing to look for are numbers or numeric terms that stand alone and then combine them in order to simplify the expression.

Consider the expression 2 + 3x − 4xy + 7 − y − 5.

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

Note that typically the numeric terms are listed last in an expression. Also not that y − 5 can be thought of as y + (− 5).

Another example of combining numbers is (x² + 2x + 3) − 6 + (y + 8).

Although the parentheses group sub-expressions for some reason, removing them does no harm. The numbers can be combined, resulting in the expression x² + 2x + y + 5.

Combining variable like terms

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

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

Also, 2x²y, −x²y and 5x²y are like terms of x²y.

To get a little tricky, 3yx² is also a like term of x²y, since you can rearrange terms in a multiplication, such that 3yx² = 3x²y.

5x and 5y are not like terms and do not combine. Likewise, 2x²y and 2x³y are not like terms.

Example

Consider the expression 3x − 2y − 4xy + x² + 4y − x + 2xy − y².

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

(3x − x) +(−2y + 4y) + (−4xy + 2xy) + x²− y²

Note that you can consider an expression like 4y − x to be 4y + (−x). That makes things easier.

Now combine the like terms.

2x + 2y −2xy + x²− y²

Rearrange the terms in a better format.

x² + 2x −2xy + 2y − y²

Combining like sub-expressions

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

Consider 3(x + 7) + 7xy + 4(x² + 7) + 2xy − 2(x² + 7) + (x + 7).

Like terms are multiples of (x + 7), (x² + 7) and xy.

Group the multiples of like terms or sub-expressions:

(3 + 1)(x + 7) + (7 + 2)xy + (4 − 2)(x² + 7)

4(x + 7) + 9xy + 2(x² + 7)

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

4x + 28 + 9xy + 2x² + 14

4x + 42 + 9xy + 2x²

Rearrange to put in a better form.

2x² + 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.

Answers to Readers' Questions

Simplify your life

Resources

The following resources provide information on this subject:

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Books

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Mini-quiz to check your understanding

1. What is the result of simplifying the expression (2x + 2) + (3y − 6) + 4?

2. What is the result of combining like terms in the expression ax² + bx + c + 2ax² + 3bx + c?

3. What is the result of simplifying the expression 3x(x + 2) + x² − (x + 2)(4 + 1)?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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