Combining Like Terms
Simplify algebraic expressions by identifying and combining like terms — the essential skill for solving any equation.
Before you can solve any equation, you have to simplify it. Combining like terms is the cleanup step that makes every equation solvable. Whether you're a carpenter adding lumber costs, a developer reducing a formula in code, or a student simplifying before solving, this skill is used in every single algebraic problem that follows. Skip it and you'll carry unnecessary complexity through every step.
- Identify like terms by matching variables and exponents
- Combine like terms by adding or subtracting their coefficients
- Simplify multi-variable and multi-exponent expressions correctly
Like terms have the same variable part — the same letter(s) AND the same exponent. Only the coefficients differ. To combine: add or subtract the coefficients, keep the variable part unchanged.
\[ 3x + 7x = (3+7)x = 10x \qquad\qquad 5x^2 - 2x^2 = (5-2)x^2 = 3x^2 \]You can think of it like fruit: 3 apples + 7 apples = 10 apples. But 3 apples + 7 oranges stay separate — you can't combine different kinds. The same rule applies: \(3x + 7y\) cannot be simplified further.
Grouping Like Terms Visually
Simplify: 4x² + 3x − 2x² + 5 − x + 9
Simplify: \(8n - 3n + n\)
All terms have the variable \(n\) (exponent 1). Add/subtract the coefficients:
\[ 8n - 3n + n = (8-3+1)n = 6n \]Note: \(n\) has an implicit coefficient of 1, so \(n = 1n\).
Simplify: \(5x^2 + 3x - 2x^2 - 7x + 4\)
- Group like terms: \((5x^2 - 2x^2) + (3x - 7x) + 4\)
- Combine each group: \(3x^2 + (-4x) + 4\)
- Write the simplified form: \(3x^2 - 4x + 4\)
A contractor charges: \(50h + 30m + 200 + 20h - 10m\) dollars, where \(h\) = hours of labor, \(m\) = materials units, and 200 is a flat fee. Simplify the pricing formula.
- Group: \((50h + 20h) + (30m - 10m) + 200\)
- Combine: \(70h + 20m + 200\)
The simplified pricing formula is \(70h + 20m + 200\).
Simplify each expression:
- \(9y - 4y + 2y\)
- \(6a^2 + 3a - 2a^2 + a - 8\)
- \(4x + 7y - x + 3y\)
▶ Show Answers
- \(7y\)
- \(4a^2 + 4a - 8\)
- \(3x + 10y\) — x and y are different variables; they can't combine with each other.
- Combining unlike terms: \(3x + 5y \neq 8xy\). Different variables mean different terms — leave them separate.
- Combining different powers: \(x^2 + x \neq x^3\) and \(x^2 + x \neq 2x^2\). The exponents must match.
- Forgetting the sign: In \(5x - 3x^2\), the subtraction sign belongs to the \(3x^2\) term. When rearranging, keep signs with their terms.
- Like terms must have the same variable AND the same exponent to be combined.
- Combine by operating on coefficients only — the variable part stays the same.
- Constants combine with constants — they're like terms with no variable.
- Simplifying first makes every equation and word problem dramatically easier to solve.
When developers optimize code, they simplify algebraic complexity expressions — combining redundant operations into fewer steps. A function that calls \(n + 2n + n\) operations can be simplified to \(4n\) (linear), dramatically affecting performance at scale. Combining like terms is the algebraic equivalent of code refactoring.
Calculator Connection
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Combining Like Terms - Quiz
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