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 + 2b
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 − 5x + 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.
Note that typically the numeric terms are listed last in an expression.
Also note 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 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 x2 + 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, 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.
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
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