🧩 Algebra Solver

Combine Like Terms Calculator

Combine like terms in algebraic expressions by grouping matching variables and simplifying their coefficients step by step.

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How to Use & Educational Guide

In-depth guide for Combine Like Terms Calculator

Educational Overview

How to Combine Like Terms in Algebra?

Combining like terms means adding or subtracting the coefficients of terms that have the same variables raised to the same powers. It is a key step when simplifying algebraic expressions and creating an equivalent expression. Constants, which are numbers without variables, can also be combined with other constants.

Combine Like Terms Definition and Rule

Like terms must have the same variable parts and the same exponents. Only their numerical coefficients are added or subtracted:

$$ax^n + bx^n = (a + b)x^n \quad \text{and} \quad ax^n - bx^n = (a - b)x^n$$
✅ Like Terms (Can Combine): $4x$ and $7x$, $3x^2$ and $-5x^2$, $6xy$ and $2xy$, $8$ and $-3$.
❌ Unlike Terms (Cannot Combine): $4x$ and $4x^2$, $3x$ and $3y$, $5x$ and $7$.

Steps to Combine Like Terms

  1. Step 1 (Distribute if Needed): If the algebraic expression contains parentheses, use the distributive property to remove them first: $a(b + c) = ab + ac$.
  2. Step 2 (Identify Like Terms): Find terms that contain the same variables with the same exponents. Keep the sign attached to each term.
  3. Step 3 (Combine Coefficients): Add or subtract the numerical coefficients while keeping the common variable part unchanged.
  4. Step 4 (Create an Equivalent Expression): Write the combined terms together to produce an equivalent but simpler algebraic expression.

How to Distribute and Combine Like Terms

Some algebra problems require you to distribute and combine like terms. First multiply the factor outside the parentheses by every term inside. Then identify and combine any like terms that appear.

For example, $2(x + 3) + 4x$ becomes $2x + 6 + 4x$. The like terms $2x$ and $4x$ can then be combined to give $6x + 6$.

Combine Like Terms Examples

Example 1: Simplify $4x + 7 - 2x + 5$

• Step 1 (Group x-terms): $4x - 2x = (4 - 2)x = 2x$.

• Step 2 (Group constants): $7 + 5 = 12$.

👉 Equivalent Expression: $2x + 12$

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

• Step 1 (Combine x-terms): $5x - 2x = (5 - 2)x = 3x$.

• Step 2 (Combine y-terms): $3y + 7y = (3 + 7)y = 10y$.

• Step 3 (Constants): $-4$ remains unchanged.

👉 Final Answer: $3x + 10y - 4$

Example 3: Simplify $3x^2 + 5x - x^2 + 2x + 4$

• Step 1 (Combine x² terms): $3x^2 - x^2 = (3 - 1)x^2 = 2x^2$.

• Step 2 (Combine x terms): $5x + 2x = (5 + 2)x = 7x$.

• Step 3 (Constants): $+4$ remains unchanged.

👉 Final Answer: $2x^2 + 7x + 4$

Example 4: Distribute and Combine Like Terms in $2(x + 4) - 3x + 5$

• Step 1 (Distribute): $2(x) + 2(4) - 3x + 5 \implies 2x + 8 - 3x + 5$.

• Step 2 (Combine x-terms): $2x - 3x = -x$.

• Step 3 (Combine constants): $8 + 5 = 13$.

👉 Final Answer: $-x + 13$

Example 5: Combine Like Terms in the Polynomial $6x^3 + 2x^2 - 3x^3 + 5x^2$

• Step 1 (Group x³ terms): $6x^3 - 3x^3 = 3x^3$.

• Step 2 (Group x² terms): $2x^2 + 5x^2 = 7x^2$.

• Step 3 (Write the result): Keep the different powers as separate terms.

👉 Final Answer: $3x^3 + 7x^2$

Why Is Combining Like Terms Important?

📐 Simplifying Algebraic Expressions

Combining like terms makes algebraic expressions shorter and easier to work with. For example, $4x + 3 + 2x$ becomes the equivalent expression $6x + 3$.

📊 Algebra & Geometry

Like terms are useful when simplifying perimeter formulas, polynomial expressions, equations, and other algebraic calculations.

Common Mistakes to Avoid

  • Adding Exponents Instead of Coefficients: In $3x + 2x$, add the coefficients to get $5x$. Do not change it to $5x^2$.
  • Losing the Minus Sign: The sign belongs to the term. In $7 - 2x$, the term is $-2x$, not $+2x$.
  • Combining Different Variables: $5x + 3y$ cannot become $8xy$. Different variables make the terms unlike terms.
  • Combining Different Powers: $4x$ and $4x^2$ cannot be combined because their exponents are different.
  • Skipping Distribution: If parentheses are present, distribute first before looking for like terms.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What does it mean to combine like terms?
Combining like terms means adding or subtracting the coefficients of terms that have the same variables raised to the same powers.
What is the definition of combine like terms?
To combine like terms is to group terms with identical variable parts and combine their numerical coefficients to create an equivalent expression.
How do you combine like terms?
Identify terms with the same variables and exponents, add or subtract their coefficients, and leave unlike terms unchanged.
Can you combine x and x²?
No. x has an exponent of 1 while x² has an exponent of 2, so they are unlike terms and cannot be combined.
How do you distribute and combine like terms?
First use the distributive property to remove parentheses. Then identify terms with the same variables and exponents and combine their coefficients.

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