Problem 108 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 7 w+w-2 w $$

Step-by-Step Solution

Answer

The simplified expression is \(6w\).

Step 1: Identify Like Terms

The expression given is \(7w + w - 2w\). Begin by identifying the like terms, which in this case are all the terms containing the variable \(w\). These include \(7w\), \(w\), and \(-2w\).

Step 2: Combine Like Terms

Add or subtract the coefficients of the like terms identified. Combine \(7w\), \(w\), and \(-2w\) by adding their coefficients: \(7 + 1 - 2\).

Step 3: Simplify the Coefficients

Calculate the result of the expression in coefficients: \(7 + 1 - 2 = 6\). Therefore, the combined expression is \(6w\).

Key Concepts

Like Terms

In algebra, like terms are essential when simplifying expressions. Like terms are terms that contain the same variable raised to the same power. For example, in the expression \(7w + w - 2w\), all terms are like terms because they share the variable \(w\). Identifying like terms is the first step in simplifying an algebraic expression. It allows you to see which terms can be combined together.
When the terms have different variables or powers, they are not like terms, and you cannot combine them directly through addition or subtraction. Always focus on identifying your like terms correctly to simplify the expression accurately.

Coefficients

Coefficients are the numbers that are multiplied by the variables in an algebraic expression. They represent how many times the variable is being counted. For instance, in the term \(7w\), the number 7 is the coefficient. Coefficients show the measure or scale of the term they accompany.
In the expression \(7w + w - 2w\), the coefficients are 7, 1, and -2 respectively. Each coefficient tells you how many times \(w\) is being added or subtracted. The coefficient of \(w\) (without a number in front of it) is understood to be 1. Understanding coefficients is crucial as they help determine the result when you combine like terms.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions. Once you identify the like terms, the next step is to combine them by adding or subtracting their coefficients. This process helps in reducing the expression to its simplest form.
For example, with the expression \(7w + w - 2w\), you first determine that \(w\) appears in each term, making them like terms. Then, you combine them by adding the coefficients: \(7 + 1 - 2 = 6\). This results in the simpler expression \(6w\).
Simplifying expressions through combining like terms is a foundational skill in algebra and helps make complex expressions much more manageable. Always perform each step carefully to ensure accuracy in your results.