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\(x\)
\(x\)
 =
\(2x\)

Combining
Like Terms

Learn to simplify algebraic expressions by grouping terms that share the same variable and exponent: the foundation of every equation you will ever solve.

Pre-Algebra

What Is a Term?

An algebraic expression is built from terms: pieces separated by \( + \) or \( - \) signs. Each term has up to three parts:

Coefficient
The number in front: in \( 5x^2 \), the coefficient is 5.
Variable
The letter: in \( 5x^2 \), the variable is \( x \).
Exponent
The power: in \( 5x^2 \), the exponent is 2.

What Are Like Terms?

Like terms share the exact same variable(s) raised to the exact same exponent(s). Only the coefficients may differ.

LIKE TERMS
\(3x^2\)
\(-7x^2\)
Same variable: \(x\)
Same exponent: \(2\)
 NOT LIKE TERMS
\(3x^2\)
\(3x^3\)
Same variable: \(x\)
Different exponent!

Why Simplify?

Every equation you solve starts with simplifying each side. Combining like terms is the first step: it shrinks complex expressions into something you can actually work with.

\(3x + 2 + 5x - 1\)
\(8x + 1\)
 combine like terms

The Core Rule

$$ax^n + bx^n = (a + b)\,x^n$$

\(4x\)
 +
\(3x\)
\(7x\)
coeff = \(4\)
coeff = \(3\)
\(4 + 3 = 7\)
Variable part stays the same: \(x\)

Think in Groups

Imagine each term as a labeled tile. You can only stack tiles with the same label.

\(x\)
\(x\)
\(x\)
\(y\)
\(y\)
\(1\)
\(1\)
 group by label
\(3x\)
 +
\(2y\)
 +
\(2\)
\(3x + 2y + 2\)

Like vs. Unlike Terms

Pair Like? Why?
\( 5x \) and \( -2x \) ✓ Yes Same variable \( x \), same exponent 1
\( 3x^2 \) and \( 7x^2 \) ✓ Yes Same variable \( x \), same exponent 2
\( 4x \) and \( 4x^2 \) ✕ No Different exponents: 1 vs. 2
\( 6x \) and \( 6y \) ✕ No Different variables: \( x \) vs. \( y \)
\( 9 \) and \( -3 \) ✓ Yes Both are constants (no variable)

Step-by-Step Method

1
Identify each term in the expression (split at \( + \) and \( - \) signs).
2
Group like terms: match terms that share the same variable and exponent.
3
Add or subtract the coefficients within each group.
4
Write the simplified expression: one term per group, in standard form (highest exponent first).

Example 1: Two Variable Types

Simplify: \( 4x + 3y + 2x - y \)

\(4x\)
 +
\(3y\)
 +
\(2x\)
 −
\(y\)
\(x\)-terms: \(4 + 2 = 6\)
\(y\)-terms: \(3 - 1 = 2\)
\(6x + 2y\)

Example 2: Mixed Exponents

Simplify: \( 2x^2 + 5x - 3x^2 + x + 7 \)

1
Identify terms: \( 2x^2 \), \( 5x \), \( -3x^2 \), \( x \), \( 7 \)
2
Group: \( x^2 \)-terms: \( 2x^2 - 3x^2 \) | \( x \)-terms: \( 5x + x \) | constants: \( 7 \)
3
Combine: \( 2 - 3 = -1 \) | \( 5 + 1 = 6 \) | \( 7 \)

Example 3: Two Variables + Constants

Simplify: \( 3a - 4b + 2a + 7b - 5 + 1 \)

\(a\)-terms: \(3a + 2a\)
\(= 5a\)
\(b\)-terms: \(-4b + 7b\)
\(= 3b\)
constants: \(-5 + 1\)
\(= -4\)
\(5a + 3b - 4\)

Watch the Signs

The sign in front of a term belongs to that term. Think of \( -3x \) as having coefficient \( -3 \), not as a separate minus sign floating around.

\( 7x - 9x \)
\( 7 + (-9) = -2 \), so the answer is \( -2x \).
\( -4y + 4y \)
\( -4 + 4 = 0 \), so the terms cancel out.

Why Does It Work?

Combining like terms is really the Distributive Property in reverse:

$$4x + 3x = (4 + 3)\,x = 7x$$

Common Mistakes

✕ Combining unlike terms
Writing \( 3x + 2y = 5xy \). Variables must match: \( x \) and \( y \) are different.
✕ Mixing up exponents
Writing \( x + x^2 = 2x^2 \). The terms \( x \) and \( x^2 \) have different exponents, so they cannot combine.
✕ Dropping the sign
Writing \( 5x - 8x = 3x \) instead of \( -3x \). Always include the negative sign when the result is negative.

Quick Reference

Expression Like Terms Simplified
\( 6x + 2x \) \( 6x, 2x \) \( 8x \)
\( 3x^2 - 5x^2 + x \) \( 3x^2, -5x^2 \) \( -2x^2 + x \)
\( 4a + 7 - 2a + 3 \) \( 4a, -2a \) and \( 7, 3 \) \( 2a + 10 \)
\( y - y + 5 \) \( y, -y \) \( 5 \)
\( 2x + 3y \) None (unlike) Already simplified

Question 1 of 5

Simplify: \( 3x + 7x \)

Question 2 of 5

Which pair are like terms?

Question 3 of 5

Simplify: \( 5x^2 - 3x + 2x^2 + x - 4 \)

Question 4 of 5

Simplify: \( 8y - 8y + 3 \)

Question 5 of 5

A student simplifies \( 2x + 3y \) as \( 5xy \). What mistake did they make?