Problem 10 Simplify each expression. $$2 ... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 3: Problem 10

Simplify each expression. $$2 m-4(m-n)+2 n$$

Short Answer

Expert verified

The simplified form of the given expression is $ -2m +6n $

Step by step solution

Distribute -4

One can start by applying the distributive property of multiplicative operation over addition/subtraction, which means that each term inside the parentheses gets multiplied by the term outside the parentheses, that is $ -4 \times m $ and $ -4 \times -n $. As a result, we obtain $ 2m -4m +4n +2n $

Combine Like Terms

In the equation obtained above, one will then proceed to combine like terms. For that, find terms of $ m $ and terms of $ n $. We have $ 2m -4m $ and $ 4n +2n $. Doing this, we get the expression as $ -2m +6n $

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Key Concepts

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

Understanding Distributive Property in Algebra

The distributive property is a fundamental concept in algebra that helps in simplifying expressions. It allows you to multiply a single term by multiple terms inside a set of parentheses. Think of it as distributing the multiplication over addition or subtraction within the parentheses.
For example, in the expression $-4(m-n)$:

Multiply $-4$ by $m$, which gives you $-4m$.
Then, multiply $-4$ by $-n$, and this becomes $+4n$ because multiplying two negatives results in a positive.

Using this property effectively can help simplify complex algebraic expressions by eliminating parentheses, thus making it easier to handle subsequent steps like combining like terms. The distributive property is not just a rule but a powerful tool in helping to understand and manipulate algebraic expressions.

The Art of Combining Like Terms

After applying the distributive property, the next step in simplifying an expression is often to combine like terms. Like terms in algebra are terms that have the same variable raised to the same power.
In the expression obtained from the previous step $2m - 4m + 4n + 2n$:

Look for terms containing $m$. These are $2m$ and $-4m$. Adding them together gives $-2m$.
Then, identify terms with $n$, which are $4n$ and $2n$. Combining them yields $6n$.

Combining like terms simplifies the expression further and makes it cleaner and easier to understand. It is crucial to carefully add and subtract coefficients of the same variable. Always pay attention to the signs in front of the terms, as they determine whether you add or subtract the coefficients.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can contain constants, coefficients, and variables. For students, understanding these components is key as they allow you to perform operations like simplification using the distributive property or combining like terms.
In the given expression $2m-4(m-n)+2n$, several elements are at play:

$2m$ and $2n$ are simple products of variables and coefficients.
$-4(m-n)$ contains both multiplication and subtraction within the parentheses.

Every part of an algebraic expression has a role, and knowing how to manipulate these parts is essential for solving equations. Simplifying algebraic expressions helps establish the groundwork for more complex algebraic operations and provides a deepened understanding of mathematical structures.

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91影视

Simplify each expression. $$2 m-4(m-n)+2 n$$

Short Answer

Step by step solution

Distribute -4

Combine Like Terms

Key Concepts

Understanding Distributive Property in Algebra

The Art of Combining Like Terms

Exploring Algebraic Expressions

One App. One Place for Learning.

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