Combining Like Terms

Q: Why isn't $3x + 7x$ equal to $10x^{2}$?

Because you are counting, not multiplying. Three $x$'s plus seven $x$'s is ten $x$'s — $10x$. The exponent only changes when you multiply variables together, like $x \cdot x = x^{2}$.

Q: What happens when the terms cancel completely?

You get zero of that variable. $5n - 5n = 0n = 0$, so the term disappears from the expression entirely. If other terms remain, they are the whole answer.

Like terms have the exact same variable raised to the exact same power. $3x$ and $7x$ are like terms. $5x^{2}$ and $2x$ are not, because the exponents differ. Only like terms can be combined — you can add apples to apples, but not apples to oranges.

The move itself is one line: keep the variable, and add or subtract the coefficients. $3x + 7x = 10x$ . That single move is how every long, messy expression in algebra gets shortened, so it is worth doing until it is automatic.

What makes terms alike

Check two things: the variable and its exponent. Both must match exactly. $4y$ and $-9y$ are like terms. $4y$ and $4x$ are not — different variables. $4y$ and $4y^{2}$ are not — different powers.

Plain numbers with no variable, like $6$ and $-7$ , are constants, and all constants are like terms with each other. They combine into a single number.

Coefficients never affect likeness. $100x$ and $\dfrac{1}{2}x$ are like terms, because likeness is about the variable part only.

How to combine them

Keep the variable part exactly as it is, and add or subtract the coefficients. $9p - 2p$ : the variable part is $p$ , the coefficients are $9$ and $-2$ , so the result is $7p$ . The variable never changes — $3x + 7x$ is $10x$ , not $10x^{2}$ .

If a variable has no number written in front, its coefficient is $1$ . So $7c - c$ means $7c - 1c = 6c$ . Forgetting that invisible $1$ is the single most common error on this skill.

Negative results are fine

Sometimes the subtraction wins. $3y - 8y$ gives $3 - 8 = -5$ , so the answer is $-5y$ . A negative coefficient is a perfectly normal answer — resist the urge to flip it positive.

Worked examples

Example 1: add the coefficients

Simplify $9p - 2p$ .

Check the terms are alike — both are

p

to the first power

Subtract the coefficients

9 - 2 = 7

Keep the variable

7p

Answer: $7p$

Example 2: a negative result

Simplify $3y - 8y$ .

Both terms have

y

to the first power — like terms

Subtract the coefficients

3 - 8 = -5

Keep the variable

-5y

Answer: $-5y$

Example 3: the invisible coefficient

Simplify $7c - c$ .

Rewrite the lone

c

with its coefficient

7c - 1c

Subtract the coefficients

7 - 1 = 6

Keep the variable

6c

Answer: $6c$

Try one yourself

Common questions

Why isn't

3x + 7x

equal to

10x^{2}

Because you are counting, not multiplying. Three $x$ 's plus seven $x$ 's is ten $x$ 's — $10x$ . The exponent only changes when you multiply variables together, like $x \cdot x = x^{2}$ .

Are

5x^{2}

and

2x

like terms?

No. The variable matches but the powers do not — one is squared, one is not. $5x^{2} + 2x$ is already as simple as it gets; the two terms just sit next to each other.

Can I combine

4x

and

6y

No. Different variables are different objects — like adding $4$ apples and $6$ oranges. The expression $4x + 6y$ cannot be shortened.

What happens when the terms cancel completely?

You get zero of that variable. $5n - 5n = 0n = 0$ , so the term disappears from the expression entirely. If other terms remain, they are the whole answer.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Combining Like Terms

What makes terms alike

How to combine them

Negative results are fine

Worked examples

Example 1: add the coefficients

Example 2: a negative result

Example 3: the invisible coefficient

Try one yourself

Common questions

Keep learning

Introduction to Expressions

Combining Like Terms (cont)

Distributive Property

Distributive Property (cont)