Problem 69 - Algebra 2

Question

Simplify each expression. $$ -2(m-4 n)-3(5 n+6) $$

Step-by-Step Solution

Answer

The simplified expression is \(-2m - 7n - 18\).

Step 1: Distribute Each Term

First, distribute the constants outside each parentheses to every term inside the parentheses. For the first term, distribute -2 to both \(m\) and \(-4n\). For the second part, distribute -3 to both \(5n\) and \(6\). This gives: \[-2(m) + 2(4n) - 3(5n) - 3(6)\] Simplifying gives: \[-2m + 8n - 15n - 18\].

Step 2: Combine Like Terms

Now, combine the like terms from the expression \(-2m + 8n - 15n - 18\). Focus on terms involving \(n\): Combine \(8n\) and \(-15n\), which are the like terms: \(8n - 15n = -7n\). So, you have the expression: \[-2m - 7n - 18\].

Step 3: Conclusion: Simplified Expression

The expression is simplified by distributing the multiplication to eliminate the parentheses and then combining the like terms to reach the final simplified expression: \(-2m - 7n - 18\).

Key Concepts

Distributive Property

The distributive property is a useful and fundamental tool in algebra. It allows us to remove parentheses by distributing a multiplier outside the parentheses across the terms within the parentheses. This operation ensures we correctly apply multiplication over addition or subtraction. Imagine you have an expression like \[a(b+c)\],using the distributive property, it becomes:\[a \cdot b + a \cdot c\].
In our exercise, we started with \[-2(m-4n)-3(5n+6)\].
To distribute, multiply \(-2\) by both \(m\) and \(-4n\), resulting in:\[-2m + 8n\].
Next, multiply \(-3\) by \(5n\) and by \(6\), which produces:\[-15n - 18\].
By distributing correctly, we eliminate the parentheses and get:\[-2m + 8n - 15n - 18\].
This step lays the foundation for simplifying further by setting up the equation with individual terms.

Combining Like Terms

Combining like terms is a process used to simplify expressions by merging terms that have the same variable raised to the same power. This acts like tidying up your workspace - you gather similar items together for a neat, organized expression. For instance, in the simplified expression achieved from the distribution step:\[-2m + 8n - 15n - 18\],
look for terms that 'alike'. Here, \[8n\] and \[-15n\] are like terms because they both contain the same variable \(n\).
Combine them by performing the arithmetic operation:\[8n - 15n = -7n\].
Doing this refines the expression to:\[-2m - 7n - 18\].
Once like terms are combined, the expression becomes much simpler and easier to interpret.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. It involves applying various techniques like the distributive property and combining like terms to achieve it. This step is crucial for solving equations as it makes them more manageable and straightforward. In our exercise, after distributing and combining like terms, the final expression:\[-2m - 7n - 18\]
is the simplified form.
Simplifying helps in ensuring that equations and expressions are concise, which aids in interpretation and problem-solving. It highlights the essential parts of an expression and removes unnecessary complexity, making it easier for further mathematical processes or interpretations. It also often reveals patterns or insights that were not obvious before the expression was simplified. This skill not only applies to algebra but extends to many areas of mathematics. By practicing, one gains efficiency and clarity in mathematical thinking.