Like Terms

Every algebraic expression is made up of one or more terms. Terms in these expressions are separated by the operators $+$ or $-$. For instance, in the expression $x + 5$, there are two terms; in the expression $2x^2$, there is only one term.

Terms are called like terms if they involve the same variables and exponents. All constants are also like terms.

For example, $5x^2$ and $x^2$ are like terms because they involve the same variable, $x$, raised to the same exponent, 2. Likewise, the following are examples of like terms:

• $3x$ and $25x$
• $y^4$ and $12y^4$
• 13 and 42

Note that terms that share a variable but not an exponent are not like terms. Therefore, $2x^3$and $2x^2$are not like terms because they have different exponents (3 and 2). Likewise, terms that share an exponent but have different variables are not like terms. Therefore, $2x^2$ and $2y^2$ are not like terms, because they have different variables ($x$ and $y$).

Combining Like Terms

Expressions with Two Terms

We can simplify an algebraic expression by combining like terms. For example, let's try simplifying $3x + 6x$.

First, let's write both terms as addition problems:

• $3x = x + x + x$
• $6x = x + x + x + x + x + x$. 

Adding these terms together, we have:

$3x + 6x = x + x + x + x + x + x + x + x + x$

If you count, you'll find that there are 9 $x$s in this expanded expression. Therefore:

$3x+6x=9x$

Note that the expression we started with, $3x + 6x$, had only two terms. When an expression contains more than two terms, it may be helpful to rearrange the terms so that like terms are together. 

Expressions with More than Two Terms

The commutative property of addition says that we can change the order of terms without changing the meaning of the expression (the sum). So, we can rearrange the order of the following expression before attempting to combine like terms: 

$4a + 6b + 2a +b$

We can identify that $4a$ and $2a$ are like terms, as are $6b$ and $b$. We want to rearrange the expression to group like terms together:

$4a + 2a + 6b +b$.

Now we can more easily add the like terms together to simplify the expression:

$(4a + 2a) + (6b +b) = 6a + 7b$

The same rules apply when an expression involves subtraction. However, be careful that when you changing the order of terms you ensure that the minus sign follows the term that it applies to. For example, consider $2x - 3 + 5x$. This expression is properly rearranged and simplified as follows:

$2x-3 + 5x = 2x + 5x - 3 = 7x - 3$.

Summary

In summary, there are three steps to combining like terms:

1. Identify all like terms.

2. Rearrange the expression so the like terms are grouped together.

3. Add or subtract the coefficients of the like terms until there are as few of each kind of term as possible.

Example 1

Simplify the following expression: 

$4x^2 + 3xy - y - 2x^2 + 5xy$ 

First, identify the like terms: $4x^2$ and $-2x^2$,  $3xy$ and $5xy$. Now group these like terms together: 

$4x^2 - 2x^2 + 3xy + 5xy - y $

Add and subtract the coefficients of the like terms:

$2x^2 + 8xy - y$

Example 2

Simplify the following expression: 

$-5x^2 + 3x + 3y - 3x + 21x^2 $

First, identify the like terms: $-5x^2$ and $21x^2$, $3x$ and $-3x$. Now group these like terms together: 

$-5x^2 + 21x^2 + 3x - 3x + 3y $ 

Add and subtract the coefficients of the like terms. Notice that the terms $3x$ and $-3x$ cancel, because $3x - 3x = 0$. The expression therefore simplifies to:

$16x^2 + 3y $