Combining Like Terms — I can combine like terms to simplify...

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Where Did Algebra Come From?

Have you ever tried to add up a messy list of items? Maybe you had 3 apples, 2 oranges, 5 apples, and 1 orange. You'd probably group the apples together and the oranges together to make counting easier. That's exactly the idea behind combining like terms. People have been simplifying math expressions this way for thousands of years!

~1800 BCE

Babylonian Problem-Solving

Ancient Babylonians carved math problems on clay tablets. They grouped similar quantities together to solve problems about land and trade.

~300 BCE

Greek Geometry Meets Algebra

Greek mathematicians like Euclid used shapes to represent unknowns. Combining areas of the same type was an early form of combining like terms.

820 CE

Al-Khwarizmi Writes the First Algebra Book

The Persian mathematician al-Khwarizmi wrote a book describing how to simplify and solve equations. The word "algebra" comes from the Arabic title of his book!

1600s

Modern Notation Is Born

Mathematicians in Europe began using letters like x and y. This made combining like terms much easier to see and do on paper.

So here's the big question: when you see an expression like 3x + 5 + 2x + 7, how do you know which pieces go together? And why can you add some of them but not others? That's what this lesson is all about.

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Core Definitions & Principles

Before we start combining anything, let's learn the vocabulary. An expression is a math phrase made up of numbers, variables, and operations — like 4x + 3y − 2. Each piece separated by a plus or minus sign is called a term. In the expression 4x + 3y − 2, the three terms are 4x, 3y, and −2.

Term

A single piece of an expression — a number, a variable, or a number times a variable. Examples: 5, x, 3y².

Coefficient

The number in front of a variable. In 7x, the coefficient is 7. If you just see x, the coefficient is 1.

Like Terms

Terms that have the exact same variable(s) raised to the exact same power. 3x and 5x are like terms. 3x and 3x² are NOT like terms.

Constant

A term with no variable — just a plain number, like 4 or −9. All constants are like terms with each other.

Simplify

To rewrite an expression in its shortest, cleanest form by combining all like terms.

✦ KEY TAKEAWAY

Think of like terms as matching socks in a laundry pile. You can only pair socks that are the same color and size. In the same way, you can only combine terms that have the exact same variable part. The number in front (the coefficient) can be different — it's the variable part that must match!

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Seeing Like Terms in Action

Let's look at the expression 3x + 2y + 5x + 4 + y + 1 and see how we sort and combine the like terms. The diagram below color-codes each group of like terms so you can see which ones belong together.

The diagram shows the expression 3x + 2y + 5x + 4 + y + 1 being sorted into three groups: x-terms (cyan), y-terms (violet), and constants (amber). After adding each group's coefficients, the simplified expression is 8x + 3y + 5.

Notice how 3x and 5x are both x-terms. They have the same variable, so we add the coefficients: 3 + 5 = 8. The variable part stays the same. We do the same with 2y and y (which is really 1y): 2 + 1 = 3. And the constants 4 + 1 = 5. We could NOT add 3x and 2y together because they have different variables.

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The Math Behind Combining Like Terms

Combining like terms actually uses a property you already know: the Distributive Property. When you see 3x + 5x, you can factor out the x to get (3 + 5)x = 8x. That's why we add the coefficients — we're really factoring out the common variable!

COMBINING LIKE TERMS RULE

ax + bx = (a + b)x

Here, a and b are the coefficients (numbers in front), and x is the variable part that must match. You add the coefficients and keep the variable the same.

EXAMPLE WITH SUBTRACTION

7x − 3x = (7 − 3)x = 4x

Subtraction works the same way. Think of −3x as adding a negative: 7x + (−3x) = (7 + (−3))x = 4x.

EXAMPLE WITH EXPONENTS

2x² + 6x² = (2 + 6)x² = 8x²

The variable part is x². Both terms share this same variable and exponent, so they are like terms. Note: x² and x are NOT like terms because the exponents are different.

⚠️ Watch Out!

A common mistake is combining terms that look similar but aren't. For example, 3x and 3x² are not like terms. The exponents must match too! Also, 2xy and 2x are not like terms because one has y and the other doesn't.

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Which Terms Are "Like"? A Closer Look

The key rule is simple: two terms are like terms if and only if they have exactly the same variable(s) raised to exactly the same power(s). The coefficient does NOT matter — it can be any number. Let's look at a table of examples to make this crystal clear.

Examples of like and unlike terms

Term 1	Term 2	Like Terms?	Why?
4x	9x	Yes ✓	Both have the variable x to the 1st power.
3y	−7y	Yes ✓	Both have y. The negative sign doesn't change the variable part.
5x²	2x²	Yes ✓	Both have x². Same variable, same exponent.
5x	5x²	No ✗	Different exponents! x¹ ≠ x².
6x	6y	No ✗	Different variables. Same coefficient doesn't matter.
8	−3	Yes ✓	Both are constants (no variable). All constants are like terms.
2xy	5xy	Yes ✓	Both have the same two variables: x and y.

Every term has up to three parts: the coefficient (number), the variable (letter), and the exponent (power). For terms to be "like," the variable and exponent must match. Only the coefficient can be different.

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Worked Example: Step by Step

Let's simplify the expression 4x² + 3x − 7 + 2x² − x + 10 by combining like terms. We'll go through each step carefully.

Simplify: 4x² + 3x − 7 + 2x² − x + 10

Step 1 — Identify all the termsSeparate the expression at each + or − sign. Be careful to keep the sign with the term that follows it. The terms are: 4x², 3x, −7, 2x², −x, and 10.

Six terms total

Step 2 — Sort terms into groups of like termsLook at the variable and exponent of each term. Group the ones that match. The x² terms are 4x² and 2x². The x terms are 3x and −x. The constants are −7 and 10.

Three groups: x², x, and constants

Step 3 — Combine the x² termsAdd the coefficients of the x² terms: 4 + 2 = 6. Keep the variable part x². So 4x² + 2x² = 6x².

6x²

Step 4 — Combine the x termsRemember that −x is the same as −1x. Add the coefficients: 3 + (−1) = 2. So 3x − x = 2x.

Step 5 — Combine the constantsAdd the constant terms: −7 + 10 = 3.

Step 6 — Write the simplified expressionPut all the simplified groups together. Write the highest power first, then the next, then the constant.

6x² + 2x + 3

💡 Pro Tip

Some students like to underline, circle, or highlight like terms with different colors on their paper — just like our diagram! This helps you avoid mistakes, especially with longer expressions.

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Common Mistakes & How to Avoid Them

Even though the idea of combining like terms is straightforward, there are some traps that students fall into. Let's look at the most common mistakes so you can spot them before they happen.

Common mistakes when combining like terms

❌ Common Mistake	Why It's Wrong	✓ Correct Approach
3x + 4y = 7xy	x and y are different variables — these are NOT like terms. You can't add them.	3x + 4y (already simplified!)
2x + 3x = 5x²	Adding coefficients does NOT change the exponent. You add the numbers, not multiply the variables.	2x + 3x = 5x
x² + x² = x⁴	You're adding, not multiplying. The exponent stays the same when you combine like terms.	x² + x² = 2x²
5x − 3 = 2x	5x is a variable term and 3 is a constant — they're not like terms.	5x − 3 (already simplified!)
Forgetting −x = −1x	When a variable has no number in front, its coefficient is 1 (or −1 if there's a minus sign).	Always write the hidden 1. −x means −1x.

🧠 REMEMBER THIS

Think of combining like terms like stacking blocks. You can only stack blocks that are the same shape. Square blocks go on square blocks. Round blocks go on round blocks. You count how many you have of each shape, but stacking doesn't change the shape — it just changes the count. That count is the coefficient!

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From Simplifying to Solving Equations

Combining like terms isn't just a stand-alone skill. It's a building block for almost everything you'll do in algebra. When you solve equations, you'll often need to simplify each side first by combining like terms. Let's see how this skill connects to bigger ideas.

How combining like terms connects to future math

Skill You're Learning Now	Where It Leads Next
Identifying like terms	Simplifying both sides of an equation before solving
Adding and subtracting coefficients	Solving multi-step equations (Algebra 1)
Working with exponents in terms	Adding and subtracting polynomials (Algebra 1 & 2)
Using the Distributive Property	Factoring expressions and solving quadratics

For example, to solve the equation 3x + 5 + 2x = 20, you'd first combine the like terms on the left side to get 5x + 5 = 20. That's a much simpler equation to solve! Every time you see an equation or expression in algebra, your first thought should be: "Can I combine like terms to make this simpler?"

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Practice Problems

Now it's your turn! Try these five problems. They start easy and get more challenging. For each one, try to solve it on your own before checking the answer.

PROBLEM 1 — CONCEPTUAL

Look at these four terms: 5x, 3y, −2x, 7. Which terms are like terms? Group them and explain why.

PROBLEM 2 — BASIC CALCULATION

Simplify: 6a + 4a − 2a

PROBLEM 3 — INTERMEDIATE

Simplify: 3x² + 5x − 2 + x² − 3x + 8

PROBLEM 4 — APPLIED

Maria is saving money. She earns 5d dollars each day from her allowance and 3d dollars each day from walking the neighbor's dog. She spends 2d dollars each day on lunch. She also receives a one-time gift of $15. Write and simplify an expression for her total earnings and gift minus her spending for one day.

PROBLEM 5 — CRITICAL THINKING

A student says: "I simplified 4x + 3x² to get 7x³." Find and explain all the mistakes the student made.

SUMMARY

Putting It All Together

Combining like terms means adding or subtracting the coefficients of terms that share the exact same variable and exponent. A term is a single piece of an expression, and like terms are terms whose variable parts match perfectly. Constants (plain numbers with no variable) are always like terms with each other.

To simplify, follow these steps: identify each term, group the like terms together, add or subtract the coefficients within each group, and write the result. Remember that you use the Distributive Property under the hood — ax + bx = (a + b)x. Never change the exponent when adding, and never combine terms with different variables. This skill is the foundation for solving equations and working with polynomials in algebra!