Problem 12 - Elementary Algebra

Question

Simplify by removing the parentheses. $$ -(5 m-2 n) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $-5m + 2n$.

Step 1: Distribute the negative sign

To remove the parentheses, distribute the negative sign to both terms within the parentheses. This means multiplying -1 with each term: $$ -1(5m) + -1(-2n) $$

Step 2: Simplify

Now, simplify each term resulting from the multiplication: $$ -5m + 2n $$ The expression is now simplified and the parentheses have been removed. The final answer is: $$ -5m + 2n $$

Key Concepts

Simplification

Simplification is an essential skill when working with algebraic expressions. It involves narrowing down expressions to their simplest form to make equations easier to interpret and solve. When simplifying, you aim to

• remove any unnecessary parentheses,
• combine like terms, and
• perform arithmetic to condense the expression.

In the example given, $$ -(5m - 2n), $$ we first use the Distributive Property to get rid of the parentheses and simplify. After applying the property, the resulting expression $$ -5m + 2n $$ is the simplified form, as there are no like terms left to combine, and all operations are complete.
Simplification helps to reveal the core structure of algebraic expressions, making them easier to work with in larger equations or problem-solving scenarios.

Distributive Property

The distributive property is a fundamental aspect of algebra that allows us to eliminate parentheses in expressions. It states that \( a(b + c) = ab + ac \),meaning you multiply each term inside the parentheses by the term outside.
In the original exercise, we deal with a negative sign outside the parentheses, which is equivalent to \(-1\).Using the distributive property here involves multiplying \(-1\)with every term inside the parentheses:

• \(-1\) multiplied by \(5m\) yields \(-5m\),
• \(-1\) multiplied by \(-2n\) yields \(2n\) (because a negative times a negative is a positive).

After applying the distributive property, the expression without parentheses reveals the operations clearly and provides an opportunity to simplify further if needed. Mastering this property is crucial, as it is a frequent operation in algebra, allowing for transformations that lead to solutions.

Negative Sign

In algebra, understanding the role of a negative sign is important for simplifying and solving expressions correctly. A negative sign affects everything within its scope by changing the signs of the terms it precedes.
In the expression \(-(5m - 2n)\),the negative sign is outside the parentheses. It's equivalent to multiplying the whole expression by \(-1\).
This turns \(5m\) into \(-5m\)and \(-2n\)into \(2n\).Key points when dealing with negative signs:

• A negative sign changes the sign of positive terms to negative and vice versa.
• A double negative, i.e., applying a negative to an already negative term, results in a positive term.

Grasping this concept helps prevent mistakes, especially in complex problems where multiple operations and terms are involved. Remember that the negative sign's primary job is to swap signs, and it can dramatically change the expression's overall form.