Combining like terms

Combining like terms refers to the process of simplifying expressions by adding or subtracting variables and their coefficients. Terms are said to be "like" if they have the same variable and exponent. The expression below shows two different terms that are unlike:

Because the terms are unlike, they cannot be added together. If the expression were 4x2 + 7x2, the two terms would be considered like terms. To simplify the expression, the coefficients would then be added together, and the variable and exponent would stay the same:

4x2 + 7x2 = 11x2

One way to think about combining like terms is to think of the variable in terms of a specific object, and the coefficient as the number of objects. For example, if Steve has 1 potato and 1 orange, he cannot combine them because they aren't the same thing.

1 potato + 1 orange = 1 potato + 1 orange

However, if both objects he had were potatoes instead, they could be combined since they are like terms.

1 potato + 1 potato = 2 potatoes

Different variables and exponents are like different types of food - we need to ensure that they are the same object before we can add them together.

The above examples have been relatively simple, but equations and expressions can include many more terms and variables. Generally, the process for combining like terms is as follows:

  1. Look at the expression and determine whether any of the terms are like terms: in order for terms to be considered like terms both the variable and the exponent on the variable must be exactly the same. If there are no such terms, the expression is already simplified, and none of the terms can be combined.
  2. Group like terms together. By convention, terms are often written in descending order based on power.
  3. Combine the like terms, be it by adding, subtracting, etc.

Being able to combine like terms is a fundamental aspect of algebra that allows us to solve algebraic equations. Below are a few other examples of combining like terms in expressions as well as equations.


1. 2x2 + 4x - 3y3 + 6x - 6y3:

There are more than one instance of the variables x and y3, so the expression above does include like terms that can be grouped:

2x2 + (4 + 6)x + (-3 - 6)y3 = 2x2 + 10x - 9y3

Remember that for terms to be considered like, both variable and exponent must be the same. In the above example, 2x2 and 10x cannot be combined even though they share the same variable because their exponents are not the same.

2. 3x3 + 2x2 + 4 - 5x2 + 7x + 8x3:

(3 + 8)x3 + (2 - 5)x2 + 7x + 4 = 11x3 - 3x2 + 7x + 4

3. 7x2 + 12y2 + 4 - 8x:

None of the terms have matching variables and exponents, so none of the terms in the above expression can be combined.