Combining Like Terms - The Math Index (2024)

Combining like terms is a fundamental skill in algebra that simplifies expressions and makes solving equations more manageable. In this article, we will delve into the intricacies of identifying and combining like terms to create more concise algebraic expressions.

We’ll start by discussing how to identify like terms, which are essentially variables with the same exponent or constant factors. Next, we’ll explore methods for simplifying expressions by grouping these similar components together.

As we progress through the post, you’ll learn how to combine coefficients effectively while maintaining the integrity of your algebraic expression. Finally, we will provide some practice problems designed to reinforce your understanding of combining like terms and prepare you for tackling complex mathematical challenges.

Table of Contents

Identifying Like Terms

In algebra, it is essential to recognize and understand the concept of like terms in order to simplify expressions and solve equations effectively. This section will give you a thorough comprehension of like terms, their identification and the significance they have in algebra.

Like terms are algebraic expressions that have the same variables raised to the same powers. For example, 3x2y and -5x2y are considered like terms because both contain x2y as their variable part. On the other hand, 4xy2 and 7x3y would not be classified as like terms since their variable parts (xy2 and x3y) differ from each other.

A few key points for identifying like terms

  • The coefficients (the numerical factors) can be different; only the variables need to match.
  • If there is more than one variable present in a term, all variables must appear with identical exponents for those expressions to qualify as “like.”
  • Numerical constants without any attached variables can also be combined together when simplifying an expression or equation.

Identifying like terms is an important step in simplifying expressions. Let’s proceed to the next area of Simplifying Expressions and discover how we can benefit from this understanding.

Simplifying Expressions

Algebraic simplification is a key ability which can help you to resolve issues with greater ease and accuracy. One of the primary methods for simplification is combining like terms. In this section, we will explore the process of combining like terms in order to simplify algebraic expressions.

To begin with, let’s understand what it means to combine like terms. Like terms are those which have the same variables raised to the same powers; they only differ by their coefficients (the numbers multiplied by these variables). For example, 3x2y and -5x2y are like terms because both have x2y as their variable part.

The process of combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged. Here’s a step-by-step guide on how to do this:

  1. Identify all pairs of like terms: Scan through your expression and find any groups of two or more similar-looking items that can be combined together.
  2. Add or subtract coefficients: Once you’ve identified all pairs/groups of like terms, add/subtract their respective numerical parts (coefficients) accordingly based on whether they’re being added (+) or subtracted (-).
  3. Rewrite expression with simplified term(s): Replace each group/pair with its simplified version – i.e., one term whose coefficient equals the sum/difference calculated in Step #2 above – leaving everything else unchanged within your original equation/expression.

Note: It’s important not just to identify but also arrange these matched sets properly so as not to accidentally mix up unrelated elements during subsequent steps.

Let’s look at an example to better understand the process:

Example: Simplify 5x + 3y – 2x + y

  1. Identify like terms: We have two sets of like terms here (5x – 2x) and (3y + y).
  2. Add/subtract coefficients: For the x-terms, we get (5 – 2)x = 3x. For the y terms, we get (3 + 1)y = 4y.
  3. Rewrite expression with simplified term(s): Our final simplified expression is thus: 3x + 4y.

In some cases, you may need to use the distributive property before combining like terms. The distributive property states that a(b+c)=ab+ac for any numbers a,b,c. Applying this rule can help reveal hidden like terms within expressions containing parentheses or other grouping symbols.

Example: Simplify x(2+x) – x(x+1)

  1. Distribute first: We get (2x+x2) – (x2 + x).
  2. Rewrite subtraction as addition of negative term(s): This gives us [(2x + x2) + (-x2 – x)].
  3. Combine like terms: We get (2x – x) + (x2 – x2) = x.
  4. Rewrite expression with simplified term(s): Our final simplified expression is thus: x.

Combining like terms is a fundamental concept in algebra that can help simplify complex expressions. By following the steps outlined above, you can easily identify and combine like terms to make solving algebraic expressions more manageable.

Simplifying expressions is a fundamental skill for any math student, and mastering it can open up many opportunities in mathematics. Moving on to the next topic of combining coefficients will further build upon this foundation by introducing students to more complex operations with variables.

Important Lesson:

The process of combining like terms in algebra involves identifying pairs or groups of similar-looking items and adding/subtracting their coefficients while keeping the variable part unchanged. This helps simplify complex expressions, making it easier to solve problems more efficiently and effectively. The distributive property can also be used to reveal hidden like terms within expressions containing parentheses or other grouping symbols.

Combining Coefficients

In this section, we will explore how to combine coefficients when combining like terms in algebraic expressions. The process of combining coefficients is crucial for simplifying expressions and solving equations effectively.

To start, let us define a coefficient as the numerical factor that multiplies a variable or variables in an algebraic expression. In an algebraic expression, the coefficient is the numerical factor that multiplies a variable or variables. For example, in the term (5x + 5) is the coefficient of x. When you encounter similar terms with different coefficients (like terms), you can add or subtract their coefficients while keeping the variable part unchanged.

Addition and Subtraction of Coefficients

The first step in combining like terms involves adding or subtracting their respective coefficients based on whether they have positive (+) or negative (-) signs:

  1. If both terms are positive (or both are negative), add their absolute values and keep the same sign as before.
  2. If one term has a positive sign and another has a negative sign, find the difference between their absolute values and use whichever sign corresponds to the larger value.

Example:

If we have the expression 3x + 7x – 2x, we can combine the like terms 3x, 7x, and -2x by adding their coefficients. This gives us 8x.

Combining Coefficients in Polynomials

In a polynomial with multiple terms and variables, it’s essential to identify like terms first before combining their coefficients. Remember that like terms have the same variable(s) raised to the same power.

Example:

If we have the polynomial 3x2 + 2xy + 5x2 – 4xy, we can combine the like terms 3x2 and 5x2 by adding their coefficients to get 8x2. We can also combine the like terms 2xy and -4xy by subtracting their coefficients to get -2xy.

Combining coefficients is an important skill to master in order to understand and solve more complex equations. Let’s now proceed to tackling some sample exercises in order to hone our aptitude for this concept.

Practice Problems

Remember that the key to success is identifying like terms and then combining their coefficients.

Problem 1

Simplify the expression: 5x + 7y – 3x + y

To combine like terms, first identify them: (5x – 3x) + (7y + y). Now, combine the coefficients of each set of like terms: (5-3)x + (7+1)y = 2x + 8y.

Problem 2

Simplify the expression: -6a²b³ – a²b³ + ab² – ab²

The given expression has two sets of like terms:

  1. -6a²b³ and -a²b³
  2. +ab² and -ab²

Combine their respective coefficients:

(-6 – 1)a²b³ + (1 – 1)ab² = -7a²b³

Note on Exponents:

Remember that when dealing with exponents, only variables with matching exponents are considered as “like” for our purposes.

Note on Subtraction:

When subtracting a term from another term or adding a negative term to another term, consider it as adding its opposite value.

For example: 5x – 3x = 5x + (-3x).

This makes it easier to combine the coefficients.

Problem 3

Simplify the expression: 4x²y – 2xy² + x²y – xy²

The given expression has two sets of like terms:

  1. 4x²y and x²y
  2. -2xy² and -xy²

Combine their respective coefficients:

(4 + 1)x²y + (-2 – 1)xy² = 5x²y – 3xy²

FAQs in Relation to Combining Like Terms

What is the rule for combining like terms?

The rule for combining like terms states that you can only add or subtract terms with the same variable and exponent. To combine them, simply add or subtract their coefficients while keeping the variable and exponent unchanged.

What is important to know about combining like terms?

It’s essential to understand that only similar variables with identical exponents can be combined. Also, remember to follow the proper order of operations (PEMDAS) when simplifying expressions involving multiple mathematical operations.

What is combining like terms and examples?

Combining like terms means adding or subtracting algebraic expressions with the same variables and exponents. For example, in 2x + 5y – x + 4y, we have two sets of like terms: ‘2x’ and ‘-x’, ‘5y’ and ‘4y’. Combining these gives us a simplified expression: x + 9y.

How do you know which terms to combine when combining like terms?

To identify which terms to combine, look for those having the same variable raised to an identical power. Group these together by either adding or subtracting their coefficients as required by the given expression.

Conclusion

Combining Like Terms

In this article, we learned how to simplify algebraic expressions by combining like terms. When we talk about like terms, we refer to terms that have the same variables raised to the same power. For example, 3x and 5x are like terms because they both have x raised to the first power. However, 3x and 3y are unlike terms because they have different variables.

When we combine like terms, we add or subtract their coefficients while keeping the variables the same. Let’s look at an example:

2x + 3y – 5x – 2y

First, we identify the like terms: 2x and -5x, and 3y and -2y. We can then combine them:

2x – 5x + 3y – 2y = -3x + y

It’s important to note that we cannot combine unlike terms. For example, we cannot simplify 3x + 2y because x and y are unlike terms.

By practicing problems involving combining like terms, math students can improve their skills in simplifying algebraic expressions. Remember to always look for similar variables with matching exponents and add or subtract their coefficients accordingly.

If you want to learn more about any other math-related topic, visit The Math Index!

Combining Like Terms - The Math Index (2024)

FAQs

How do you combine like terms in math? ›

When combining like terms, such as 2x and 3x, we add their coefficients. For example, 2x + 3x = (2+3)x = 5x.

What is an example of a like term in math? ›

Examples of like terms in math are x, 4x, -2x, and 7x. These are like terms because they all contain the same variable, x. The terms 8y2, y2, and -2y2 are like terms as well. These all contain the same variable, y, raised to the second power.

What grade is combining like terms? ›

Students will first learn about combining like terms as part of expressions and equations in 6th grade.

Are 2x and 3x like terms? ›

Summary. Like terms are terms that have exactly the same variable and power in them—whether that's x, x3, y, or even no variable! So, for example, 2x and 3x would be like terms since they both have the variable x and they're both to the first power.

Are 3x and 3y like terms? ›

Marcus daims that in the expression 3x + 3y, the terms 3x and 3y are like terms because both are variable terms and both have the same coefficient.

How do you solve a combining equation? ›

To combine two equations, add the left sides together, and add the right sides together. If you set your equation up right, one of the variables should cancel.

What is the step after combining like terms in a two step equation? ›

Step 2: Simplify each side of the equation by combining like terms. Step 3: Isolate the term. Use the Addition Property of Equality to get the variable on one side of the equal sign and the numerical values on the other. Step 4: Isolate the variable.

What are like terms in math 6th grade? ›

Like terms are terms that have the same variable raised to the same exponent. Expressions can be simplified by combining like terms. Both the original expression and the simplified expression are equivalent.

What do we mean when we say combining like terms? ›

Like terms are mathematical terms that have the exact same variables and exponents, but they can have different coefficients. Combining like terms will simplify a math problem and is also the proper form for writing a polynomial. To combine like terms, just add the coefficients of each like term.

Is combining like terms pre algebra? ›

Adding like terms is a fundamental concept in algebra. Coefficients are the numbers in front of variables, and they can be added when the variables are the same.

How do you know which terms to combine when combining like terms? ›

To combine like terms, you want to first reorder the terms so the powers are from highest to lowest. The like terms can be combined. That is, the terms with the same power can be combined.

How do you combine like terms facing math? ›

Like Terms: Terms that have identical variable parts (same variable(s) and same exponent(s)). When simplifying using addition and subtraction, you combine “like terms” by keeping the "like term" and adding or subtracting the numerical coefficients.

How do you combine like terms in multi-step equations? ›

Multi-step Equations with Like Terms
  1. Procedure to Solve Equations:
  2. Step 1: Remove any parentheses by using the Distributive Property or the Multiplication Property of Equality.
  3. Step 2: Simplify each side of the equation by combining like terms.
  4. Step 3: Isolate the term. ...
  5. Step 4: Isolate the variable.
Nov 29, 2023

How do you combine like terms with variables on both sides? ›

Combine like Terms (add things that have the same variable) Distribute when needed (multiply each of the things inside the parentheses) Add the additive inverse of terms to both sides. Multiply by the multiplicative inverse to both sides.

How to factor out like terms? ›

If the terms are “like terms,” you can use the distributive property to “factor out” the common variable part. a) Factor out the common variable part x2. −5x2−9x2=(−5−9)x2 Use the distributive property. =−14x2 Simplify: −5−9=−5+(−9)=−14.

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