Problem 98 Simplify. $$ 4 m-9 n+3(2 m-n... [FREE SOLUTION]

Chapter 1: Problem 98

Simplify. $$ 4 m-9 n+3(2 m-n) $$

Short Answer

Expert verified

The simplified expression is $10m - 12n$.

Step by step solution

Distribute the 3

In the expression $4m - 9n + 3(2m - n)$, distribute the 3 across the terms inside the parenthesis. This means multiplying 3 by each term in the parentheses: $3 \cdot 2m$ and $3 \cdot -n$.This gives: \[ (4m - 9n) + (6m - 3n) \]

Combine like terms

Group and add the coefficients of the like terms. Combine the terms with $m$ and the terms with $n$: $4m + 6m - 9n - 3n$ becomes $ (4 + 6)m $ and $ (-9 - 3)n $.

Simplify

Simplify the combined terms by performing the additions and subtractions:$ (4+6)m = 10m $ and $ (-9 - 3)n = -12n $.So, the expression simplifies to: \[ 10m - 12n \]

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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 an essential concept in algebra and is used to remove parentheses by distributing a multiplier across the terms inside. In our example, we start with the equation: $$4m - 9n + 3(2m - n)$$. Here, we need to distribute the 3 to both terms inside the parenthesis. This means multiplying 3 by each term within the parentheses:
3 times 2m and 3 times -n. This leads to:
$$(4m - 9n) + (6m - 3n)$$. By applying this property, we create separate, simplified terms that make it easier to combine like terms in the next step.

Combining Like Terms

Combining like terms is the process of merging terms with the same variable to simplify the expression further. In the expression that we reached through the distributive property:
$$(4m - 9n) + (6m - 3n)$$, we identify similar terms and combine them. For the m terms:
$$4m + 6m$$
and for the n terms, we have:
$$-9n - 3n$$.
Next, we add and subtract the coefficients:
- For m terms:
$(4 + 6)m = 10m$
- For n terms:
$(-9 - 3)n = -12n$.
This step ensures all similar variable terms are together, making the final simplification easier.

Algebraic Simplification

Algebraic simplification brings together the previous steps to achieve the simplest form of an algebraic expression. From the like terms combination:$$10m - 12n,$$ we can see the variables m and n have been paired with their coefficients, without any further operations required. Simplified expressions are crucial as they:

Make algebraic manipulation easier.
Help in solving equations faster.
Provide clearer insight into variable relationships.

The final simplified form, 10m - 12n, is now much easier to work with in further calculations or evaluations. Hence, mastering these steps not only helps in simplifying expressions but also forms the foundation for solving more complex algebraic problems.

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91影视

Simplify. $$ 4 m-9 n+3(2 m-n) $$

Short Answer

Step by step solution

Distribute the 3

Combine like terms

Simplify

Key Concepts

Distributive Property

Combining Like Terms

Algebraic Simplification

One App. One Place for Learning.

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