Combining Like Terms

Quick Answer - How do you combine like terms?

Identify terms that have the same variable raised to the same power, then add or subtract their coefficients while keeping the variable part unchanged. For example: $3x + 5x = 8x$ (add the 3 and 5, keep the $x$).

Picture your desk covered with math homework, pencils, erasers, and maybe some snacks. If you had to organize it, you'd group similar items together — pencils with pencils, erasers with erasers. Combining like terms works the same way: you're organizing the parts of an algebraic expression by grouping similar components.

When you first see an expression like $5x + 3 - 2x + 7$, it might look cluttered. Once you can spot the like terms and combine them, the same expression collapses to $3x + 10$ — cleaner, shorter, and easier to work with.

The skill isn't cosmetic. It's the foundation for solving equations, working with polynomials, and almost every other algebraic move that comes after this. Here's how it works.

Understanding Terms

Before combining anything, get clear on what a "term" is. A term is a single mathematical piece — a number, a variable, or numbers and variables multiplied together. In the expression $4x + 7 - 3y$, there are three terms: $4x$, $7$, and $-3y$.

Terms are separated by addition or subtraction signs, and the sign in front of a term belongs to that term. So $-3y$ is a single term with a coefficient of $-3$, not two pieces.

Each term has a coefficient (the number part) and a variable part (the letter part). In $4x$, the coefficient is $4$ and the variable is $x$. Sometimes the coefficient is hidden: an $x$ by itself has a coefficient of $1$ (because $1 \times x = x$), and $-x$ has a coefficient of $-1$.

Terms without any variables are called constant terms or just constants. In $2x + 9$, the $9$ is a constant term.

What Makes Terms "Like" Terms?

Like terms are terms that have exactly the same variable part. The coefficients can be different, but everything else must match perfectly.

Here's the rule: for terms to be "like" terms, they must have the same variable(s) raised to the same power(s).

Some examples:

Like terms:

$3x$ and $7x$ → both have $x$ to the first power
$5y^2$ and $-2y^2$ → both have $y^2$
$8ab$ and $4ab$ → both have $ab$
$12$ and $-5$ → both are constants (no variables)

NOT like terms:

$3x$ and $3y$ → different variables
$5x$ and $5x^2$ → same variable, different exponents
$2xy$ and $2x$ → different variable combinations
$7a^2b$ and $7ab^2$ → exponents are on different variables

Diagram showing like terms vs unlike terms

The order of variables doesn't matter for multiplication. The terms $3xy$ and $5yx$ are like terms because $xy$ and $yx$ are the same thing. But $3xy$ and $3x + y$ are completely different: one is multiplication, the other is addition.

How to Combine Like Terms

Here's the process for simplifying any algebraic expression by combining like terms:

Step 1: Identify the like terms. Look for terms that have exactly the same variable(s) raised to the same power(s). Remember that constant terms (numbers without variables) are like terms with each other.

Step 2: Group the like terms together. You can rearrange terms using the commutative property of addition. Terms with $x$ go together, terms with $y$ go together, constants go together.

Step 3: Add or subtract the coefficients. For each group of like terms, add or subtract the numbers in front (the coefficients). Keep the variable part exactly as it was.

Step 4: Write your simplified expression. Combine all your results into one clean expression.

Here's how it works on actual problems.

Example 1: Combine $4x + 9x$.

Both terms have $x$, so they're like terms. Add the coefficients: $4 + 9 = 13$.

The variable part stays $x$, so the answer is $13x$.

Think of it this way: if you have 4 apples and someone gives you 9 more apples, you have 13 apples total. Same logic.

Example 2: Combine $8y - 3y$.

Both terms have $y$. Subtract the coefficients: $8 - 3 = 5$.

Answer: $5y$

Example 3: Simplify $5x + 2 + 3x + 7$.

First, identify the like terms. There are two $x$ terms ($5x$ and $3x$) and two constants ($2$ and $7$).

Combine the $x$ terms: $5x + 3x = 8x$

Combine the constants: $2 + 7 = 9$

Answer: $8x + 9$

Step-by-step process showing how to combine like terms: identify like terms, then add or subtract their coefficients

Working With Subtraction

Subtraction trips people up when they don't track the negative signs. The sign in front of a term is part of that term's coefficient.

Example 4: Simplify $7a - 4a$.

The second term is really $-4a$. So you're adding $7a$ and $-4a$: $7 + (-4) = 3$.

Answer: $3a$

Example 5: Simplify $6m - 9m + 2m$.

Combine all three terms: $6 + (-9) + 2 = -1$.

Answer: $-m$ (which means $-1m$)

Example 6: Simplify $10 - 3x + 5 - 7x$.

Combine the $x$ terms: $-3x + (-7x) = -10x$

Combine the constants: $10 + 5 = 15$

Answer: $15 - 10x$ (or you could write it as $-10x + 15$)

Multiple Variables

When expressions have different variables, you handle each variable separately.

Example 7: Simplify $4x + 5y + 2x - 3y$.

Identify like terms:

$x$ terms: $4x$ and $2x$
$y$ terms: $5y$ and $-3y$

Combine $x$ terms: $4x + 2x = 6x$

Combine $y$ terms: $5y - 3y = 2y$

Answer: $6x + 2y$

You can't combine $6x$ and $2y$ any further — they have different variables, so they're not like terms.

Example 8: Simplify $7a + 3b - 4a + b - 2$.

Group by like terms:

$a$ terms: $7a - 4a = 3a$
$b$ terms: $3b + b = 4b$ (remember, $b$ means $1b$)
Constants: $-2$ stays as is

Answer: $3a + 4b - 2$

Expressions with Exponents

When variables have exponents, the exponent is part of what makes terms "like" or "unlike."

Example 9: Simplify $5x^2 + 3x + 2x^2 - x$.

Identify like terms:

$x^2$ terms: $5x^2$ and $2x^2$
$x$ terms: $3x$ and $-x$

Combine $x^2$ terms: $5x^2 + 2x^2 = 7x^2$

Combine $x$ terms: $3x - x = 2x$

Answer: $7x^2 + 2x$

The terms $7x^2$ and $2x$ cannot be combined because $x^2$ and $x$ are different. This is a critical point: $x^2$ means $x \times x$, while $x$ is just $x$. They're as different as squares and line segments.

Example 10: Simplify $8y^3 - 2y^2 + 5y^3 + y^2 - y$.

Group by like terms:

$y^3$ terms: $8y^3 + 5y^3 = 13y^3$
$y^2$ terms: $-2y^2 + y^2 = -y^2$
$y$ terms: $-y$ stays as is

Answer: $13y^3 - y^2 - y$

A More Complex Example

Here's a more involved expression that brings everything together.

Example 11: Simplify $6x + 4y - 2x + 7 - y + 3x - 5$.

Step 1: Identify all like terms.

$x$ terms: $6x, -2x, 3x$
$y$ terms: $4y, -y$
Constants: $7, -5$

Step 2: Combine each group.

$x$ terms: $6x - 2x + 3x = 7x$
$y$ terms: $4y - y = 3y$
Constants: $7 - 5 = 2$

Answer: $7x + 3y + 2$

Using Parentheses and the Distributive Property

Sometimes you'll need to distribute before you can combine like terms. The distributive property says that $a(b + c) = ab + ac$.

Example 12: Simplify $3(x + 2) + 5x$.

First, distribute the $3$: $$3(x + 2) = 3x + 6$$

That leaves $3x + 6 + 5x$.

Combine like terms:

$x$ terms: $3x + 5x = 8x$
Constants: $6$ stays as is

Answer: $8x + 6$

Example 13: Simplify $4(2a - 3) - 2(a + 1)$.

Distribute both: $$4(2a - 3) = 8a - 12$$ $$-2(a + 1) = -2a - 2$$

That leaves $8a - 12 - 2a - 2$.

Combine like terms:

$a$ terms: $8a - 2a = 6a$
Constants: $-12 - 2 = -14$

Answer: $6a - 14$

Watch that negative sign in front of the second set of parentheses. It distributes to both terms inside, which is why $-2(a + 1) = -2a - 2$, not $-2a + 2$.

Combining Terms in Equations

When you solve equations, you'll often need to combine like terms on one or both sides before you can isolate the variable.

Example 14: Solve for $x$: $3x + 5 + 2x = 20$.

First, combine like terms on the left side: $$3x + 2x + 5 = 20$$ $$5x + 5 = 20$$

Now you can solve: $$5x = 15$$ $$x = 3$$

Without combining those $x$ terms first, you couldn't solve the equation efficiently.

Example 15: Solve for $y$: $7y - 2 - 4y = 10$.

Combine like terms on the left: $$3y - 2 = 10$$

Solve: $$3y = 12$$ $$y = 4$$

Practice Problems

Work through these on your own, then check your answers below.

Simplify $8x + 12x$
Simplify $15y - 6y$
Simplify $5a + 3 + 2a + 7$
Simplify $9m - 4m + m$
Simplify $6p + 4q - 2p + 5q$
Simplify $10x^2 + 3x + 4x^2 - 5x$
Simplify $7 - 3t + 2 - 5t + 4$
Simplify $2(x + 4) + 3x$
Simplify $5(2y - 1) - 3(y + 2)$
Solve for $x$: $4x + 6 + 2x = 30$

Check Your Work

$20x$
Both terms have $x$, so add coefficients: $8 + 12 = 20$
$9y$
Subtract coefficients: $15 - 6 = 9$
$7a + 10$
Combine $a$ terms: $5a + 2a = 7a$. Combine constants: $3 + 7 = 10$
$6m$
All three terms have $m$: $9 - 4 + 1 = 6$
$4p + 9q$
$p$ terms: $6p - 2p = 4p$. $q$ terms: $4q + 5q = 9q$
$14x^2 - 2x$
$x^2$ terms: $10x^2 + 4x^2 = 14x^2$. $x$ terms: $3x - 5x = -2x$
$13 - 8t$
Constants: $7 + 2 + 4 = 13$. $t$ terms: $-3t - 5t = -8t$
$5x + 8$
Distribute first: $2x + 8 + 3x$. Combine: $2x + 3x = 5x$, so $5x + 8$
$7y - 11$
Distribute: $10y - 5 - 3y - 6$. Combine $y$ terms: $10y - 3y = 7y$. Constants: $-5 - 6 = -11$
$x = 4$
Combine left side: $6x + 6 = 30$. Subtract 6: $6x = 24$. Divide by 6: $x = 4$

Why This Matters

Common mistakes when combining like terms: don't combine unlike terms, remember the invisible coefficient of 1, and don't add exponents when combining

Combining like terms looks like busy work, but it underlies almost every algebraic move you'll make. Every time you solve an equation, work with polynomials, or simplify a complex expression, this is one of the steps. The simpler version of an expression is also faster to write, easier to graph, and less prone to careless errors later in the problem.

A few specific pitfalls show up over and over. Unlike terms can't be merged: $3x + 4y$ stays as $3x + 4y$, never $7xy$. The invisible coefficient on a bare variable is 1, so $x + 3x = 4x$, not $3x$. Signs matter: $5x - 8x = -3x$, not $3x$, and $-2(x - 3) = -2x + 6$, not $-2x - 6$. And exponents stay put when you combine: $3x^2 + 5x^2 = 8x^2$, not $8x^4$ — you're adding coefficients, not exponents.

As you move forward in algebra, expressions get more complicated: multiple variables, higher exponents, fractions, radicals. The core move of combining like terms stays the same throughout.