Combining Like Terms

When an algebraic expression has several terms, some of those terms are "alike" and can be squeezed together into one term. Combining like terms is the process of simplifying an expression by adding or subtracting the coefficients of terms that share the exact same variable part. The result is a shorter, equivalent expression that is easier to work with in later steps such as solving equations.

A term is made of a coefficient (the number in front) and a variable part (the letter, or letters, raised to some power). Two terms are like terms only when their variable parts match exactly, meaning the same letters raised to the same exponents. The coefficients can be different; only the variable part has to line up.

• \(3x\) and \(7x\) are like terms because both have the variable part \(x\).
• \(4y^2\) and \(-9y^2\) are like terms because both have the variable part \(y^2\).
• \(5x\) and \(5x^2\) are unlike terms. Even though the letter is the same, the exponents do not match.
• \(2xy\) and \(6x\) are unlike terms because one term has both \(x\) and \(y\) while the other only has \(x\).

A constant term, like \(8\) or \(-3\), is also its own kind of "variable part": constants can only combine with other constants.

To combine like terms, add (or subtract) their coefficients and keep the variable part exactly the same. Think of the variable part as a "unit," similar to how \(3\) apples plus \(2\) apples make \(5\) apples, not \(5\) apples-squared. The same idea works for algebra:

\(3x + 2x = 5x\)

\(7y^2 - 4y^2 = 3y^2\)

\(2ab + 5ab - ab = 6ab\)

Only the coefficients change. The variable part, including any exponent, stays exactly as it was.

Many expressions include subtraction, which can be rewritten as adding a negative term. This makes it easier to track the sign of each coefficient when regrouping terms.

Simplify: \(4x - 9x + 6\)

Rewrite the subtraction as adding a negative: \(4x + (-9x) + 6\)

Combine the like terms \(4x\) and \(-9x\): \(4x + (-9x) = -5x\)

Result: \(-5x + 6\)

Keeping a term's sign attached to its coefficient prevents a common mistake of dropping or flipping a negative sign while rearranging terms.

Sometimes an expression has parentheses that must be expanded before any terms can be combined. Apply the distributive property first, then look for like terms in the expanded expression. If you have not yet studied how to expand and evaluate expressions in the correct sequence, it helps to review the order of operations (PEMDAS) before tackling multi-step problems like this one.

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

Distribute the \(3\): \(3x + 12 + 2x - 5\)

Group the like terms: \((3x + 2x) + (12 - 5)\)

Combine each group: \(5x + 7\)

1. Expand any parentheses using the distributive property, if needed.
2. Underline or color-code terms that share the same variable part.
3. Rewrite subtraction as adding a negative coefficient so signs are not lost.
4. Add the coefficients within each group, keeping the variable part unchanged.
5. Write the simplified expression using all the combined groups.

Simplify: \(-2x^2 + 5x - 3x^2 + x + 8 - 3\)

Sort into groups: \(x^2\) terms, \(x\) terms, and constants.

\(x^2\) terms: \(-2x^2 - 3x^2 = -5x^2\)

\(x\) terms: \(5x + x = 6x\)

Constants: \(8 - 3 = 5\)

Simplified expression: \(-5x^2 + 6x + 5\)

Combining like terms turns a long, cluttered expression into a shorter equivalent one without changing its value. This skill is a building block for solving equations, factoring, and working with formulas throughout algebra, so being able to reliably identify like terms and combine them accurately is an essential foundation.

Related lessons

• Order of operations (PEMDAS)