Math101 · Independent Ontario learning library
Math101Combining Like Terms
Combining like terms rewrites repeated copies of the same variable part as one term while preserving the expression’s value.
Like terms have identical variable parts, including every exponent. Their coefficients can be added or subtracted.
Why combining works
Three apples plus five apples make eight apples. Algebra uses the same structure:
The distributive property justifies the step. The variable part $x$ is the shared unit; only the number of copies changes.
Identifying like terms
$4x^2$ and $-7x^2$ are like terms. $4x^2$ and $4x$ are not, because their exponents differ. $3ab$ and $-5ba$ are like because multiplication is commutative, so $ab=ba$.
Constants are like terms with one another. In $2x+7-5x+4$, the variable terms combine, and the constants combine separately.
A reliable method
- Mark each term with the sign immediately before it.
- Group terms with identical variable parts.
- Add or subtract their coefficients.
- Keep the common variable part unchanged.
- Write unlike groups as separate terms.
Colour coding or underlining can help at first, but the algebraic criterion is always the variable part.
Worked example
Check with a test value such as $x=2$. The original expression gives $10-3+4+8-2=17$, and $6(2)+5=17$.
Common mistakes
Adding exponents. $3x^2+4x^2=7x^2$, not $7x^4$. Exponents add when multiplying powers, not adding terms.
Combining unlike terms. $2x+3y$ cannot become $5xy$.
Dropping a coefficient of $-1$. The term $-x$ contributes coefficient $-1$.
Ignoring signs. Rewrite subtraction as addition of a negative if that makes term signs clearer.
Quick self-check
- Do the variable letters and their exponents match exactly?
- Did I include the sign with each coefficient?
- Did I change only coefficients, not variable parts?
- Can a substitution check confirm equivalence?
