12 Combining Like Terms

Combining Like Terms is the process of simplifying an algebraic expression by merging terms that share the same variable factors into a single term, using the distributive property to add or subtract their coefficients. It is one of the most frequently applied simplification techniques in elementary algebra, converting expressions with many terms into shorter, equivalent expressions that are easier to evaluate, compare, and use in equation solving.

Like-Term Structure

Two terms are like terms if and only if they have identical variable factors, including identical exponents on each variable, regardless of what numerical coefficients they carry. The terms 3x² and -8x² are like terms because both consist of a coefficient multiplying x². The terms 3x² and 3x are not like terms, despite sharing the same coefficient, because their variable factors differ (x² versus x). Constant terms, which have no variable factor at all, are like terms with one another regardless of their individual values, since every constant shares the same "empty" variable factor.

Single-Variable Like Terms

When an expression involves only one variable, identifying like terms reduces to matching terms with the same exponent on that variable. In 5x³ - 2x + 7x³ + 4, the terms 5x³ and 7x³ are like terms because both carry x³, while -2x stands alone because no other x¹ term appears, and 4 stands alone as the only constant term. Combining the like terms merges 5x³ and 7x³ into 12x³, leaving the simplified expression 12x³ - 2x + 4.

Multivariable Like Terms

When an expression involves more than one variable, like terms must match on every variable factor and every exponent simultaneously. The terms 4x²y and -9x²y are like terms, since both carry x² and y¹ in identical combination, but 4x²y and 4xy² are not like terms, since the exponents on x and y are swapped between them, making the two terms represent structurally different quantities even though their coefficients and letters match.

Identifying Like Terms Within an Expression

Identifying every group of like terms in a longer expression requires scanning each term's full variable structure — every letter present and its exact exponent — and sorting terms into groups where that structure matches exactly. This identification step should be completed in full before any coefficients are combined, since attempting to combine terms while still scanning for matches is a common source of overlooked or mismatched terms in longer expressions.

Combining Like Terms

Once a group of like terms has been identified, they are combined into a single term by adding or subtracting their coefficients while keeping the shared variable factor unchanged, a direct application of the distributive property in reverse: ax + bx = (a + b)x.

The variable factor itself is never altered by this process — combining like terms changes only the coefficient, never the exponent or the letters present, since 6x + 4x = 10x, not 10x².

Handling Multiple Like-Term Groups

A single expression often contains several distinct groups of like terms that must each be combined separately. In 5x² - 3x + 8x² + 6x - 2, there are two groups (the x² terms and the x terms) plus one standalone constant. Combining each group independently gives 13x² for the x² terms and 3x for the x terms, producing the fully simplified expression 13x² + 3x - 2. A useful strategy for longer expressions is to visually group or underline each set of like terms with a consistent marking before combining, reducing the chance that a term from one group is mistakenly merged into another.

Signed and Fractional Coefficient Cases

Combining like terms with negative coefficients follows the same signed-number addition rules used throughout algebra: the sign of each term travels with its coefficient, so -5x + 2x combines to -3x, not 3x or 7x. Combining like terms with fractional coefficients requires finding a common denominator before adding or subtracting, exactly as with ordinary fraction addition: (1/2)x + (1/3)x becomes (3/6)x + (2/6)x, which combines to (5/6)x. These cases are treated explicitly because sign and fraction handling errors are the most common source of mistakes once the correct like-term groups have already been identified.

Verifying Combined Terms and Diagnosing Errors

A combined expression can be verified by substituting a specific numerical value for each variable into both the original, uncombined expression and the simplified result, and confirming the two evaluate to the same number; if they differ, an error occurred during identification or combination. Common errors in this area include merging terms that share a coefficient but not a variable factor, changing the exponent on the variable factor while combining coefficients, dropping or misapplying the sign of a term when it is moved during grouping, and forgetting to find a common denominator before combining terms with fractional coefficients.