Problem 39 - Prealgebra

Question

Simplify expression. \(6 m+2 n+10 m\)

Step-by-Step Solution

Answer

The simplified expression is \(16m + 2n\).

Step 1: Identify Like Terms

The expression we need to simplify is \(6m + 2n + 10m\). The first step is to identify like terms. Like terms are terms that have the same variable raised to the same power. Here, \(6m\) and \(10m\) are like terms because they both contain the variable \(m\). The term \(2n\) is a separate term because it involves a different variable, \(n\).

Step 2: Combine Like Terms

Now, add the coefficients of the like terms \(6m\) and \(10m\) together. The coefficients are the numbers in front of the variables: 6 for \(6m\) and 10 for \(10m\). Adding these coefficients gives us \(6 + 10 = 16\). So, \(6m + 10m = 16m\).

Step 3: Write the Simplified Expression

Combine the simplified like terms \(16m\) with the other term \(2n\) that we identified as not like earlier. The fully simplified expression is \(16m + 2n\).

Key Concepts

Understanding 'Like Terms'

In algebra, like terms are terms that contain the same variable raised to the same power. These are the building blocks when it comes to simplifying expressions. To identify like terms, look at the variable part of each term. If they have the same variable and exponent, they can be considered like terms.
For example, in the expression \(6m + 2n + 10m\), the terms \(6m\) and \(10m\) share the same variable \(m\), making them like terms. However, the term \(2n\) is different because it contains the variable \(n\), so it is not a like term with the \(m\) terms. Recognizing like terms is essential as it enables the next steps in simplification.

Breaking Down 'Coefficients'

The coefficients in an algebraic expression are the numerical parts that are multiplied by the variables. They tell us how many of each variable, or how much of each term, we have. Recognizing coefficients is important because they need to be combined appropriately when simplifying expressions with like terms.
For instance, consider the terms \(6m\) and \(10m\) in our expression. Here, 6 and 10 are the coefficients. They are indicating how many times the variable \(m\) is being counted.

• \(6m\) has a coefficient of 6.
• \(10m\) has a coefficient of 10.

When simplifying, we focus on adding these coefficients together: \(6 + 10 = 16\). This is how we simplify the expression to \(16m\).

The Art of 'Combining Like Terms'

Combining like terms is the process of simplifying expressions by adding or subtracting coefficients of like terms. This is a fundamental skill in algebra and is crucial for solving equations and simplifying expressions quickly.
When we combine like terms, we focus only on those terms that have the same variables. In the expression \(6m + 2n + 10m\), we already identified \(6m\) and \(10m\) as like terms.

• Start by identifying the like terms.
• Add the coefficients of these like terms: \(6 + 10\) results in \(16m\).
• Finally, write the expression in its simplest form: \(16m + 2n\).

This process reduces the expression to its simplest form, making it easier to manage in further calculations or problem-solving scenarios.