Combining Like Terms In Algebra A Step By Step Guide

In the realm of mathematics, particularly algebra, the ability to simplify expressions is a fundamental skill. Simplifying algebraic expressions often involves combining like terms, which are terms that share the same variable raised to the same power. This article delves into the concept of like terms, how to identify them, and the process of combining them to simplify expressions. We will use the expression $6 + 3.2m + \frac{2}{5}n - \frac{m}{5}$ as a case study to illustrate these concepts. Understanding like terms is crucial for solving equations, graphing functions, and performing various mathematical operations. It's a building block for more advanced algebraic concepts, making it essential for students and anyone working with mathematical expressions. This article aims to provide a comprehensive guide to combining like terms, ensuring a solid understanding of this fundamental algebraic principle. The concept of like terms is not limited to simple expressions; it extends to more complex algebraic structures, including polynomials and rational expressions. Mastering the ability to identify and combine like terms will significantly enhance your problem-solving skills in mathematics. Let's explore the intricacies of like terms and how they play a vital role in simplifying algebraic expressions. We'll break down the expression step by step, identifying like terms and demonstrating the process of combining them. This will provide a clear understanding of the underlying principles and practical application of this algebraic technique.

Identifying Like Terms

What are Like Terms?

Like terms are terms that have the same variable(s) raised to the same power. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical for terms to be considered "like." For example, $3x$ and $-5x$ are like terms because they both have the variable $x$ raised to the power of 1. Similarly, $2y^2$ and $7y^2$ are like terms because they both have the variable $y$ raised to the power of 2. However, $3x$ and $3x^2$ are not like terms because the variable $x$ is raised to different powers (1 and 2, respectively). Understanding this distinction is crucial for accurately combining terms in an expression. It's not just about having the same variable; the exponent associated with the variable must also match. This subtle difference is often a source of errors, so paying close attention to the exponents is essential. Recognizing like terms is the first step in simplifying algebraic expressions, and it's a skill that improves with practice. The more you work with algebraic expressions, the more easily you'll be able to identify like terms and combine them efficiently. This foundational knowledge will serve you well as you progress to more advanced topics in algebra. Let's now apply this concept to our specific expression, $6 + 3.2m + \frac{2}{5}n - \frac{m}{5}$, and identify the like terms present.

Applying the Definition to the Expression

In the given expression, $6 + 3.2m + \frac{2}{5}n - \frac{m}{5}$, we need to identify the terms that share the same variable raised to the same power. Let's break down each term:

• 6: This is a constant term (a number without a variable).
• 3.2m: This term has the variable m raised to the power of 1.
• $\frac{2}{5}n$: This term has the variable n raised to the power of 1.
• $-\frac{m}{5}$: This term also has the variable m raised to the power of 1. We can rewrite this term as $-\frac{1}{5}m$ to make the coefficient more explicit.

By examining these terms, we can see that 3.2m and $-\frac{m}{5}$ (or $-\frac{1}{5}m$) are like terms because they both have the variable m raised to the power of 1. The terms 6 and $\frac{2}{5}n$ are not like terms with any other terms in the expression, as 6 is a constant and $\frac{2}{5}n$ has the variable n. The ability to dissect an expression in this manner is key to simplifying it correctly. It's about carefully analyzing each term and comparing its variable part with the variable parts of other terms. This methodical approach ensures that you don't inadvertently combine terms that are not alike. Now that we've identified the like terms in our expression, let's move on to the next step: combining them.

Combining Like Terms

The Process of Combining

To combine like terms, we simply add or subtract their coefficients while keeping the variable part the same. This is based on the distributive property of multiplication over addition and subtraction. For example, to combine $3x + 5x$, we add the coefficients 3 and 5, resulting in $8x$. The variable part, $x$, remains unchanged. Similarly, to combine $7y^2 - 2y^2$, we subtract the coefficients 2 from 7, resulting in $5y^2$. Again, the variable part, $y^2$, remains the same. This process is straightforward once you've identified the like terms. It's a matter of performing the arithmetic operation on the coefficients and then writing the result with the common variable part. Remember, you can only combine terms that are alike. Attempting to combine unlike terms will lead to an incorrect simplification. The distributive property is the mathematical justification for this process. It allows us to factor out the common variable part, making the addition or subtraction of coefficients more apparent. Let's now apply this process to the like terms we identified in our expression.

Combining the Like Terms in the Expression

In our expression, $6 + 3.2m + \frac{2}{5}n - \frac{m}{5}$, we identified 3.2m and $-\frac{m}{5}$ as like terms. To combine these terms, we need to add their coefficients. First, let's rewrite $-\frac{m}{5}$ as $-\frac{1}{5}m$. Now we have $3.2m - \frac{1}{5}m$. To add or subtract fractions and decimals, it's often helpful to convert them to a common format. Let's convert 3.2 to a fraction and $-\frac{1}{5}$ to a decimal.

• 3. 2 as a fraction is $\frac{32}{10}$ which simplifies to $\frac{16}{5}$.
• $-\frac{1}{5}$ as a decimal is -0.2.

Now we can rewrite the expression as $\frac{16}{5}m - \frac{1}{5}m$ or $3.2m - 0.2m$. Adding the coefficients, we have:

• $\frac{16}{5} - \frac{1}{5} = \frac{15}{5} = 3$
• 3. 2 - 0.2 = 3

So, combining the like terms 3.2m and $-\frac{m}{5}$ results in 3m. The simplified expression is now $6 + 3m + \frac{2}{5}n$. This demonstrates the power of combining like terms to reduce the complexity of an expression. By performing this simplification, we've made the expression easier to understand and work with. This skill is essential for solving equations and performing other algebraic manipulations. Let's summarize our findings and provide a clear answer to the question.

Solution and Conclusion

Identifying the Correct Terms

Based on our analysis, the two terms that can be combined in the expression $6 + 3.2m + \frac{2}{5}n - \frac{m}{5}$ are 3.2m and $-\frac{m}{5}$. These terms are like terms because they both have the variable m raised to the power of 1. Combining these terms simplifies the expression and makes it easier to work with. This process of identifying and combining like terms is a fundamental skill in algebra and is essential for simplifying expressions and solving equations. Mastering this skill will significantly enhance your ability to tackle more complex mathematical problems. It's not just about finding the correct answer; it's about understanding the underlying principles and applying them effectively. The ability to break down an expression, identify its components, and combine like terms is a testament to your algebraic proficiency. This skill will serve you well as you continue your mathematical journey.

Final Answer

Therefore, the correct answer is:

B. $3.2m$ and $-\frac{m}{5}$

By understanding the principles of like terms and practicing the process of combining them, you can confidently simplify algebraic expressions and solve a wide range of mathematical problems. Remember, the key is to identify terms with the same variable raised to the same power and then add or subtract their coefficients. This article has provided a comprehensive guide to this essential algebraic skill, equipping you with the knowledge and confidence to succeed in your mathematical endeavors.