Combining Like Terms: Which Term Combines With 5a?

Oct 15, 2025 by ADMIN 51 views

Hey guys! Let's dive into some algebra and figure out how to combine like terms. It’s a fundamental concept in mathematics, and once you grasp it, you’ll be simplifying expressions like a pro. We're going to break down an expression and identify which terms can be combined. So, let's get started!

Understanding Like Terms

Before we tackle the expression, let's define what like terms actually are. Like terms are terms that have the same variable 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 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 $y^2$ . However, $4x$ and $4x^2$ are not like terms because the exponents of $x$ are different. One has an exponent of 1, and the other has an exponent of 2. Understanding this basic principle is crucial for simplifying algebraic expressions.

Now, let’s consider constants. Constants are numbers without any variables (like -2, 5, or 10). Constants are also considered like terms because they can be combined with each other. Think of it this way: they're just numbers, and you can always add or subtract numbers. For example, in the expression $3 + 4x - 2$ , the constants 3 and -2 are like terms and can be combined to give 1. So the simplified expression becomes $1 + 4x$ .

Why is this important? Because combining like terms helps us simplify complex expressions, making them easier to work with. It’s like tidying up a messy room – you group similar items together to make everything more organized and manageable. In algebra, this means grouping terms with the same variable and exponent, and then performing the indicated operations (addition or subtraction) on their coefficients. This process is a cornerstone of algebraic manipulation and is used extensively in solving equations, graphing functions, and more. Mastering this skill will significantly boost your confidence and ability in handling mathematical problems.

Analyzing the Expression: $-2 + 5a + b + 2a - 5b$

Okay, guys, let's take a close look at the expression we're dealing with: $-2 + 5a + b + 2a - 5b$ . Our goal is to figure out which term can be combined with $5a$ because they are like terms. Remember, like terms have the same variable raised to the same power. Let’s break down each term in the expression:

-2: This is a constant term. It doesn't have any variables, so it can only be combined with other constants.
5a: This term has the variable $a$ raised to the power of 1. We’re looking for other terms that also have $a$ raised to the power of 1.
b: This term has the variable $b$ raised to the power of 1. It’s a different variable than $5a$ , so it’s not a like term.
2a: Aha! This term also has the variable $a$ raised to the power of 1. So, it looks like we might have a match here. Let’s keep going.
-5b: This term has the variable $b$ raised to the power of 1. Again, this is a different variable than $5a$ , so it’s not a like term.

From our analysis, we can see that the term 2a is the one that matches the criteria for being a like term with 5a. Both terms have the same variable, $a$ , raised to the same power, which is 1. This means we can combine these terms by adding their coefficients. The other terms, -2, b, and -5b, have either different variables or no variables at all, so they cannot be combined directly with 5a.

Understanding how to break down an expression like this is a key skill in algebra. It allows you to identify the individual components and determine how they relate to each other. By systematically examining each term, you can confidently identify like terms and simplify the expression. This methodical approach will serve you well as you tackle more complex algebraic problems.

Identifying the Like Term

So, after carefully examining the expression $-2 + 5a + b + 2a - 5b$ , we’ve pinpointed the term that can be combined with $5a$ . Remember, the key to combining terms is that they must be like terms, meaning they have the same variable raised to the same power. Let's recap the options:

A. $-2$ : This is a constant. Constants can only be combined with other constants.
B. $b$ : This term has the variable $b$ . It’s not the same variable as $5a$ .
C. $2a$ : This term has the variable $a$ . This looks promising!
D. $-5b$ : This term has the variable $b$ . Again, not the same variable as $5a$ .

Clearly, the term that can be combined with $5a$ is 2a. Both terms have the variable $a$ raised to the power of 1. This means we can add their coefficients together. In this case, we would add 5 (from 5a) and 2 (from 2a) to get 7a. So, the combined term would be 7a. Recognizing these like terms allows us to simplify the expression further.

To drive the point home, let's think about what combining these terms actually means. Imagine you have 5 apples (represented by 5a) and you add 2 more apples (represented by 2a). How many apples do you have in total? You have 7 apples (represented by 7a). This simple analogy helps to illustrate the concept of combining like terms in a real-world context. It's all about grouping similar items together to make things simpler and clearer. And guys, you've nailed it!

Conclusion

Alright, guys! We've successfully identified that the term 2a can be combined with $5a$ in the expression $-2 + 5a + b + 2a - 5b$ . This is because they are like terms, sharing the same variable $a$ raised to the same power. Understanding how to identify and combine like terms is a fundamental skill in algebra, and you've taken a big step in mastering it today. Keep practicing, and you'll become even more confident in simplifying algebraic expressions! Remember, math isn't about memorization; it's about understanding the concepts and applying them. You got this!