Problem 54 - Prealgebra

Question

Combine like terms. $$3 a b+7 a b$$

Step-by-Step Solution

Answer

10ab

Step 1: Identify Like Terms

To combine like terms, first identify them within the expression. Here, both terms \(3ab\) and \(7ab\) have the common variable components \(ab\), which means they are like terms.

Step 2: Combine the Coefficients

After identifying the like terms, combine them by adding their coefficients. The coefficients of the terms are 3 and 7. Add these together: \(3 + 7 = 10\).

Step 3: Write the Combined Term

Attach the sum of the coefficients to the common variable part, which is \(ab\). So, the expression becomes \(10ab\).

Key Concepts

Understanding Like Terms

In algebra, a crucial concept to grasp is that of "like terms." Like terms are terms within an expression that have exactly the same variable components. This means that the variables are identical in type and degree. For example, in the expression \(3ab + 7ab\), both terms share the same variables, "a" and "b," which makes them like terms.
Identifying like terms is essential for simplifying expressions. When terms have the same variable components, they can be combined by manipulating their coefficients. This simplification is one of the fundamental techniques in algebra, allowing for a cleaner and more concise representation of expressions. Remember, it's the similarity in the variable part that categorizes terms as "like."

The Role of Coefficients

Coefficients are the numerical parts that multiply the variable components of an expression. They are the numbers placed directly in front of the variables. In an expression like \(3ab + 7ab\), the coefficients are "3" and "7."
When combining like terms, these coefficients play a pivotal role. We add or subtract the coefficients while the variable part remains unchanged. In our example, we combine the coefficients as follows: \(3 + 7 = 10\).
Understanding the role of coefficients is vital for executing operations with algebraic expressions correctly. This knowledge ensures that when simplifying terms, the operations only affect the numbers, leaving the variable part the same.

Exploring Variable Components

Variable components refer to the variable part of a term, which includes all the variables and their respective exponents. These are the symbols often seen as letters like \(a\), \(b\), \(x\), and \(y\), which substitute unknown quantities or numbers in expressions.
In the expression \(3ab + 7ab\), the variable components are \(ab\). Here, both terms have identical variable components, which is necessary for them to be considered like terms. It's the consistent variable part that allows terms to be combined through their coefficients.
Recognizing and correctly handling variable components is crucial for successful algebraic manipulation. It allows you to spot like terms quickly and combine them efficiently, simplifying the expression while maintaining its original meaning.