1.12 Like-Term Definitions

Like-Term Definitions in algebra explain terms with identical variable parts, forming the basis for simplifying expressions and solving equations.

Like-Term Definitions establishes the vocabulary needed to determine which terms within an algebraic expression may be combined directly through addition or subtraction, based on the structural matching of their variable parts, and describes the process by which such matching terms are grouped and merged into a single simplified term.

Like Terms

Like terms are two or more terms that share exactly the same variable part—identical variables raised to identical exponents—differing only, if at all, in their numerical coefficients. The terms 4x and 9x are like terms, since both consist of the same variable x raised to the same implied exponent of 1; the terms 3x² and 7x² are likewise like terms, sharing the variable part x².

4 x and 9 x are like terms

Unlike Terms

Unlike terms are two or more terms whose variable parts differ in at least one respect—either in which variables are present or in the exponents attached to them—and which therefore cannot be combined into a single term through addition or subtraction. The terms 4x and 4x² are unlike terms despite sharing the same variable and the same coefficient, because their exponents differ; the terms 5x and 5y are likewise unlike terms, since they involve entirely different variables.

Variable-Part Match

A variable-part match is the specific criterion used to determine whether two terms qualify as like terms: their variable parts must be identical, meaning every variable present, together with its exact exponent, must correspond exactly between the two terms, with no missing or extra variables and no differing exponents. Only when a variable-part match holds exactly can the numerical coefficients of the two terms be combined.

3 x^{2} y and 8 x^{2} y are like terms (matching variable part x^{2} y)

Like-Term Group

A like-term group is the complete collection of all terms within a single expression that share one particular variable part, gathered together regardless of their position in the original expression, in preparation for combining them. Identifying like-term groups is typically the first step of simplifying a multi-term expression, since it separates the expression into independent clusters that can each be reduced on their own.

Combined Coefficient

The combined coefficient is the single numerical coefficient obtained by adding or subtracting the coefficients of every term within a like-term group, according to each term's sign, producing one simplified term that replaces the entire group. Combining 4x and 9x, both members of the same like-term group, produces the combined coefficient 13, yielding the single simplified term 13x.

4 x + 9 x = (4 + 9) x = 13 x

Together, these definitions establish the precise basis for combining terms in an algebraic expression: like terms must satisfy an exact variable-part match, unlike terms cannot be merged regardless of superficial similarity, and once a like-term group is identified, its members are reduced to a single term through addition or subtraction of their coefficients, producing the combined coefficient that represents the simplified result.