Like Terms- Combining Algebraic Expressions
What Are Like Terms?
Like terms are algebraic expressions that share the exact same variable part. That's it. Nothing fancy. If two terms have the same variables raised to the same powers, they're like terms. If they don't, they're not.
You can combine like terms by adding or subtracting their coefficients. This is the foundation of simplifying algebraic expressions, and if you can't do this, you're going to struggle with everything that comes next.
The Only Rule That Matters
The variable part must be identical. Not similar. Not close. Identical.
For example:
- 3x and 5x are like terms. Same variable (x), same power.
- 3x and 5y are not like terms. Different variables.
- 3x and 5x² are not like terms. Different powers.
- 3xy and 5xy are like terms. Same variables, same powers.
- 3xy and 5yx are like terms. Order doesn't matter—xy equals yx.
Like Terms vs. Unlike Terms
Here's a quick breakdown so you stop confusing the two:
| Like Terms | Unlike Terms |
|---|---|
| 7x and 3x | 7x and 3y |
| 4x² and -2x² | 4x² and -2x |
| 5abc and 12abc | 5abc and 12ab |
| -8 and 3 | -8x and 3 |
| 2x³y and 7x³y | 2x³y and 7x²y |
Notice that constants (plain numbers with no variable) are all like terms with each other. That's a fact most textbooks bury in fine print.
How to Combine Like Terms
Step 1: Identify which terms are alike.
Step 2: Add or subtract the coefficients. Leave the variable part untouched.
Step 3: Write the result.
That's the entire process. No tricks.
Example 1: Simple Case
3x + 5x
Same variable. Add coefficients: 3 + 5 = 8. Result: 8x
Example 2: Watch the Signs
7x - 3x + 2x
All like terms. Work left to right: 7 - 3 = 4, then 4 + 2 = 6. Result: 6x
Example 3: Multiple Variables
4xy + 3xy - 7xy
All share xy. Coefficients: 4 + 3 - 7 = 0. Result: 0
Example 4: Constants Too
2x + 5 + 3x + 2
Group like terms: 2x + 3x = 5x. Constants: 5 + 2 = 7. Result: 5x + 7
Common Mistakes That Will Cost You Points
- Combining unlike terms: You cannot add x + y. You cannot add x + x². Stop trying.
- Forgetting to distribute: In expressions like 2(x + 3) + 4x, you must distribute first before combining anything.
- Dropping negative signs: -5x + 2x is not 7x. It's -3x. Pay attention to your signs.
- Misreading exponents: x and x² are completely different. Treat them that way.
Getting Started: A Step-by-Step Process
When you face a messy expression, follow this order:
- Clear parentheses using the distributive property if needed.
- Rewrite the expression so all terms are visible.
- Circle or highlight groups of like terms.
- Combine each group by adding or subtracting coefficients.
- Write the simplified expression in standard form (usually highest power first).
Practice Problem
Simplify: 3x² + 2x - 5 + 4x² - 3x + 7
Group like terms:
x² terms: 3x² + 4x² = 7x²
x terms: 2x - 3x = -x
Constants: -5 + 7 = 2
Final answer: 7x² - x + 2
Why This Matters Beyond the Worksheet
Combining like terms isn't some isolated skill you'll never use again. It's in every equation you solve, every function you graph, every polynomial you work with. If you can't simplify expressions correctly, you'll fail at everything that follows in algebra—factoring, solving equations, calculus. This is the foundation. Get it solid now.
The Bottom Line
Like terms share the same variable part. Combine them by adding or subtracting their coefficients. That's the whole concept. Practice identifying them quickly, and the rest of algebra gets significantly easier.