Problem 14 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 7 c-8-c $$

Step-by-Step Solution

Answer

The expression simplifies to \(6c - 8\).

Step 1: Identify Like Terms

In the expression \(7c - 8 - c\), the like terms are those that have the same variable part. Here, \(7c\) and \(-c\) are like terms. The constant \(-8\) is a separate term.

Step 2: Combine Like Terms

To simplify the expression, add the coefficients of the like terms. The coefficients of \(7c\) and \(-c\) are 7 and -1 respectively. Combining these amounts, we have \(7c - c = (7 - 1)c = 6c\).

Step 3: Rewrite the Simplified Expression

After combining like terms, the expression simplifies to \(6c - 8\). This expression has no further like terms to combine.

Key Concepts

Like Terms

When working with algebraic expressions, identifying like terms is crucial to simplify an expression. Like terms are terms that have the same variable raised to the same power. They only differ in their coefficients, which are the numerical parts of the terms. For example, in the expression \(7c - 8 - c\), the terms \(7c\) and \(-c\) are like terms because they share the same variable \(c\) with an implicit power of 1. The constant term, \(-8\), is not a like term as it does not have a variable attached to it. Identifying like terms is the first step in combining terms to simplify expressions. This step is essential for accurate algebraic manipulation.

Variable Terms

Variable terms in an algebraic expression are those parts of the expression that include variables (such as \(c\) in our example). These terms are key components because they define the expression's algebraic nature. In \(7c - 8 - c\), the variable terms are \(7c\) and \(-c\). These terms' coefficients are what you will combine when simplifying the expression. By focusing on terms that include the same variable, you can efficiently simplify expressions by combining like coefficients. Understanding the connection between variables and their coefficients is essential for successful expression manipulation.

Combining Coefficients

After identifying like terms, the next step is to combine their coefficients. Coefficients are the numbers in front of the variables in each term. In \(7c - 8 - c\), the coefficients are 7 for \(7c\) and -1 for \(-c\). By adding these coefficients, \(7 + (-1)\), you receive the simplified coefficient for the variable term, \(c\), which results in \(6c\).
Here's a breakdown:

• Identify coefficients: 7 and -1.
• Combine: 7 + (-1) = 6.

The constant term \(-8\) remains unchanged, as it has no like terms to combine with. The final simplified expression becomes \(6c - 8\). Combining coefficients correctly is crucial for accurate simplification of algebraic expressions.