Problem 41 - Algebra A Combined Function

Question

Simplify the expressions. $$ (3 n+6 m)+(2 n-6 m) $$

Step-by-Step Solution

Answer

The simplified expression is \(5n\).

Step 1: Identify Like Terms

First, we need to identify the like terms in the expression \((3n + 6m) + (2n - 6m)\). Like terms are terms that have the same variables raised to the same power. In this expression, \(3n\) and \(2n\) are like terms, and \(6m\) and \(-6m\) are also like terms.

Step 2: Combine Like Terms

Now, let's combine the like terms. Start with the \(n\) terms:\(3n + 2n = 5n\).Then, combine the \(m\) terms:\(6m - 6m = 0\).So the expression simplifies to \(5n + 0\).

Step 3: Simplify the Expression

Simplify \(5n + 0\) by removing the zero term, which does not affect the expression. Thus, the simplified expression is \(5n\).

Key Concepts

Like Terms

In algebra, understanding the concept of "like terms" is fundamental. Like terms are terms in an algebraic expression that include exactly the same variables, each raised to the same power. For example, in the expression \((3n + 6m) + (2n - 6m)\), both \(3n\) and \(2n\) are considered like terms because they share the variable \(n\). Likewise, \(6m\) and \(-6m\) are like terms because they both involve the variable \(m\). This similarity allows them to be combined and simplified.

When identifying like terms, it's crucial to ignore the coefficients (the numbers in front of the variables) and focus solely on the variables themselves. Variables and their exponents (if any) must match perfectly for terms to be considered "like". In our example, since both terms with \(n\) and \(m\) fit that criteria, they can be easily combined in the next steps of the simplification process.

Combining Terms

Once you have identified like terms in an expression, the next step is to combine them. This process involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged. In the expression \((3n + 6m) + (2n - 6m)\), combining terms means:

• Add the coefficients of the \(n\) terms: \(3n + 2n = 5n\).
• Combine the coefficients of the \(m\) terms: \(6m - 6m = 0\).

It's as simple as basic arithmetic! The variable part remains the same, so you only focus on the numbers (coefficients).

This combination reduces the complexity of the algebraic expression, paving the way for further simplification. Always double-check to make sure all possible like terms have been successfully and accurately combined.

Simplification Steps

Once the like terms are combined, the expression enters the simplification phase. Simplification is all about making the expression as compact and straightforward as possible, eliminating any unnecessary parts. From our combined expression \(5n + 0\), you can see that the \(0\) term doesn't contribute anything valuable.

In algebra, adding zero does not change the value of an expression. Therefore, it can be removed without affecting the equation's balance, simplifying \(5n + 0\) to just \(5n\).
These steps ensure the expression is reduced to its simplest form, enhancing readability and making further calculations easier.

Thus, the final result in our original problem becomes simply \(5n\), showcasing the power and utility of the simplification process.