Like and Unlike Algebraic Terms — Definition, Examples

TL;DR

Like terms in algebra share the same variables raised to the same powers — only their coefficients can differ. Unlike terms differ in their variable parts or exponents. This article covers the definitions, the addition/subtraction rule that applies only to like terms, three worked examples, and the most common slip students make in simplification.

The Distinction That Decides Whether You Can Add

Most algebra mistakes at the introductory level come from one missed step: trying to add or subtract terms that look similar but are not actually like terms. The distinction between like and unlike terms is the first checkpoint of every simplification.

Two terms are like when their non-numeric parts match exactly. Two terms are unlike when anything in the non-numeric part differs — variable letter, exponent, or both.

What Like and Unlike Terms Are

Like terms. Two or more terms are like terms if they have the same variables raised to the same exponents. The coefficients (numerical factors) can differ.

Examples — 3x and 5x (both have x^1). 4y^2 and −2y^2 (both have y^2). 7ab and −ab (both have a^1 b^1).

Unlike terms. Two or more terms are unlike terms if they differ in the variable part — either the variables themselves differ, or the exponents on the same variable differ.

Examples — 3x and 5y (different variables). 4x^2 and 4x (same variable, different exponents). 7ab and 7ac (same coefficient and variables in common, but the second variable differs).

Quick facts.

• Like terms test: ignore coefficients; the remaining variable-and-exponent part must match exactly.
• Combine like terms: add or subtract their coefficients; keep the variable part unchanged.
• Unlike terms: cannot be combined into a single term. They stay separate.
• Constants: all numbers without variables are like each other (5 and −3 are like terms).
• Order doesn't matter: 3xy = 3yx — variable order in a product is irrelevant.

How to Combine Like Terms

To add or subtract like terms:

1. Identify all terms with the same variable-exponent structure.
2. Add or subtract their coefficients (treat the variable part as a label).
3. Keep the variable part unchanged.

Examples:

• 5x + 3x = (5 + 3)x = 8x.
• 7y^2 - 2y^2 = (7 - 2)y^2 = 5y^2.
• 4ab + 9ab - 2ab = (4 + 9 - 2)ab = 11ab.

If two terms are not like, no combination is possible. The expression stays as-is. (e.g., 3x + 5y cannot be combined.)

Examples of Like and Unlike Algebraic Terms — Quick, Standard, Stretch

Quick. Combine like terms in 4x + 7 + 3x - 2.

Like terms: 4x and 3x (combine to 7x); 7 and -2 (combine to 5).

Final answer: 7x + 5.

Standard. Simplify 3x^2 + 5x + 2x^2 - 4x + 7.

Group by exponent:

• x^2 terms: 3x^2 + 2x^2 = 5x^2.
• x terms: 5x - 4x = x.
• Constants: 7 + 0 = 7.

Final answer: 5x^2 + x + 7.

Stretch. Simplify 7xy + 3yz - 2xy + 5yz - 4xy.

xy terms: 7xy - 2xy - 4xy = (7 - 2 - 4)xy = xy. yz terms: 3yz + 5yz = 8yz.

Final answer: xy + 8yz.

Why Like and Unlike Terms Matter — Beyond Cleanup

The like-vs-unlike distinction is not just for simplification. It is the foundation of every later polynomial operation:

• Adding polynomials. Always works by combining like terms across the two polynomials.
• Subtracting polynomials. Same — combine like terms after distributing the subtraction.
• Solving equations. Move all variable-containing terms to one side, all constants to the other; the moves work because like terms combine and unlike terms separate.
• Function arithmetic. f(x) + g(x) is the polynomial built by combining like terms.

Like and Unlike Algebraic: The Easy-to-Make Errors

1. Treating x^2 and x as like terms

Different exponents → unlike terms.

2. Treating 3x and 3y as like terms

Different variables → unlike terms.

3. Forgetting that constants are like terms with each other

5 and -2 are like terms.

4. Variable order treated as distinct

xy and yx are the same term — like terms.

Conclusion

• Like terms share the same variables raised to the same exponents. Their coefficients can differ.
• Unlike terms differ in variable or exponent and cannot be combined.

Quick Self-Check — Try These

1. Simplify 7a + 4b - 2a + 3b - a.
2. Simplify 3x^2 + 5x + 2x^2 - x + 4.
3. Are 4xy and 4xy^2 like terms? Explain.

Frequently Asked Questions

• What are like terms? Terms with the same variables raised to the same exponents.
• Can a constant be a like term? Yes — to other constants.
• Are xy and yxy like terms? Yes. Variable order in a product does not matter.
• Can unlike terms be combined? No, they cannot be combined into a single term.
• Why do like terms combine but unlike terms don’t? Because addition only collapses terms when they describe the same kind of quantity.