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 someone asked you to organize it, you'd probably group similar items together—pencils with pencils, erasers with erasers. Combining like terms works the same way. You're just organizing parts of an algebraic expression by grouping similar components.

When you first see an expression like $5x + 3 - 2x + 7$, it might look cluttered. But once you learn to spot like terms and combine them, you can simplify it down to $3x + 10$. Much cleaner, much easier to work with.

This skill isn't just about making expressions look nicer. It's foundational for solving equations, working with polynomials, and pretty much everything else you'll do in algebra. So let's break it down.

Understanding Terms

Before we can combine anything, we need to know what we're working with. A term is a single mathematical piece that can be a number, a variable, or numbers and variables multiplied together. In the expression $4x + 7 - 3y$, there are three terms: $4x$, $7$, and $-3y$.

Notice that terms are separated by addition or subtraction signs. The sign in front of a term belongs to that term. So $-3y$ is a single term with a coefficient of $-3$, not two separate 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—if you just see $x$ by itself, the coefficient is actually $1$ (because $1 \times x = x$). Similarly, $-x$ 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).

Let's look at 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 $yx$ represent the same thing. But $3xy$ and $3x + y$ are completely different—one is multiplication, one 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.

Let's see how this works with some examples.

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. We have $5x$ and $3x$ (both have $x$), and we have $2$ and $7$ (both are constants).

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 can trip people up because you have to pay attention to the negative signs. Remember, 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 $2y$ together because they're not like terms—they have different variables.

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 lines.

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

Let's put it all together with a more involved expression.

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 $3$: $$3(x + 2) = 3x + 6$$

Now we have: $$3x + 6 + 5x$$

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

Now we have: $$8a - 12 - 2a - 2$$

Combine like terms:

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

Answer: $6a - 14$

Watch that negative sign in front of the $2$ in the second set of parentheses. It affects both terms inside—that's why we get $-2a - 2$, 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: $$3y = 12$$ $$y = 4$$

Let's Try Some 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: $x = 4$

Things to Watch Out For

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

Don't combine unlike terms. The expression $3x + 4y$ cannot be simplified further. You can't turn it into $7xy$ or anything else. If the variable parts don't match exactly, leave the terms separate.

Remember the invisible coefficient. When you see $x$ by itself, it's really $1x$. So $x + 3x = 4x$, not $3x$. Similarly, $-x$ means $-1x$.

Watch those negative signs. When you have $5x - 8x$, the answer is $-3x$, not $3x$. The subtraction matters. And when distributing a negative, it affects all terms: $-2(x - 3) = -2x + 6$, not $-2x - 6$.

Exponents aren't the same as coefficients. The term $x^2$ does not equal $2x$. They're completely different. $x^2$ means $x \times x$, while $2x$ means $2 \times x$. You can't combine them.

Don't add exponents when combining like terms. If you have $3x^2 + 5x^2$, the answer is $8x^2$, not $8x^4$. You're adding the coefficients, not the exponents. The exponent stays the same because you're adding quantities with the same variable part.

Why This Matters

Combining like terms might seem like busy work at first, but it's one of those skills that shows up constantly in algebra. Every time you solve an equation, work with polynomials, or simplify complex expressions, you'll use this technique.

Plus, simplifying expressions makes everything else easier. Would you rather work with $3x + 5 + 2x + 7$ or $5x + 12$? The simplified version is cleaner, takes less time to write, and is less likely to cause errors in later calculations.

As you move forward in algebra, the expressions will get more complicated—multiple variables, higher exponents, fractions, radicals. But the core skill of combining like terms stays the same. Master it now, and you'll thank yourself later.