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Chapter 15: Problem 84

Remove parentheses and simplify. $$ 14 m-6(2 n-3 m)+n $$

Short Answer

Expert verified
The simplified expression is \(32m - 11n\).

Step by step solution

01

- Identify the Terms Inside the Parentheses

First, identify the terms inside the parentheses: \(2n - 3m\).
02

- Distribute the Negative Sign

Distribute the \(-6\) (negative 6) to each term inside the parentheses: \[-6(2n - 3m) = -6 \times 2n - 6 \times (-3m) = -12n + 18m\].
03

- Rewrite the Expression Without Parentheses

Substitute the simplified expression back into the original equation: \[14m - 12n + 18m + n\].
04

- Combine Like Terms

Combine the like terms: \[(14m + 18m) - 12n + n\]. This simplifies to \[32m - 11n\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The distributive property is a fundamental concept in algebra. It allows you to multiply a single term by each term within a parenthesis. By doing this, you can simplify expressions and solve equations more easily. For example, in our exercise, we have the expression: \[ 14m - 6(2n - 3m) + n \]When we apply the distributive property, we distribute \( -6 \) to each term inside the parenthesis: \[ -6 \times 2n - 6 \times (-3m) = -12n + 18m \]This step transforms the expression inside the parentheses, making it easier to handle. Remember:
  • Always multiply the term outside the parentheses by each term inside.
  • Don't forget to apply the negative sign when necessary.
Understanding the distributive property is crucial because it sets the foundation for simplifying more complex algebraic expressions.
simplification
Simplification in algebra involves making an expression easier to work with or understand. By breaking down complex expressions into simpler ones, you can solve problems more efficiently. In our exercise, after distributing \( -6 \) to \( 2n - 3m \), we get:\[ 14m - 12n + 18m + n \]The next step is to simplify the expression by removing the parentheses, which gives us the same expression since we have already accounted for the negative sign. Simplifying expressions can include:
  • Removing parentheses.
  • Applying the distributive property.
  • Combining like terms.
Understanding how to simplify is essential because it transforms complex problems into simpler forms, making them more manageable and easier to solve.
combining like terms
Combining like terms involves adding or subtracting terms that have the same variable components. This step is crucial for further simplification. In our example, the expression we have after distributing is:\[ 14m - 12n + 18m + n \]To combine like terms, group the terms with the same variables together:
  • Combine the \( m \)-terms: \(14m + 18m = 32m \).
  • Combine the \( n \)-terms: \(-12n + n = -11n \).
Our final expression is:\[ 32m - 11n \]Combining like terms simplifies the expression, making it easier to understand and solve. Remember, only terms with the same variable and exponent can be combined.

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Most popular questions from this chapter

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