Problem 11 - Elementary Algebra

Question

Simplify by removing the parentheses. $$ -(x+2 y) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(-(x + 2y)\) Answer: \(-x - 2y\)

Step 1: Distribute the negative sign

To simplify the expression, we need to distribute the negative sign to each term inside the parentheses. So, we will multiply both \(x\) and \(2y\) by \(-1\). $$ -(x + 2y) = -1(x) + (-1)(2y) $$

Step 2: Simplify terms

Now, let's simplify the terms inside the expression by performing the multiplication between the numbers and the variables. $$ -1(x) + (-1)(2y) = -x - 2y $$

Step 3: Final form

Now that we have removed the parentheses and distributed the negative sign, the simplified expression is: $$ -x - 2y $$

Key Concepts

Simplification

Simplification in algebra refers to the process of making an expression as concise and straightforward as possible. This often involves reducing expressions by combining like terms, removing unnecessary parentheses, and simplifying mathematical operations. In the exercise above, the goal is to break down the expression into its simplest form by handling the elements carefully.

The process of simplification can be a series of steps that involve rules and properties of algebra. For example, one would often start by distributing to remove parentheses, then proceed to combine like terms. The final expression should be both correct and as concise as possible. For a beginner, recognizing which steps to take is an important skill to develop.

Distributive Property

The distributive property is a key concept in algebra that allows us to simplify and calculate more easily. It states that multiplying a factor across an addition or subtraction inside parentheses is the same as doing each multiplication separately and then adding (or subtracting) those results together.

In mathematical terms:

• \( a(b + c) = ab + ac \)
• \( a(b - c) = ab - ac \)

In our original exercise, the negative sign (which we can think of as \(-1\)) was distributed to each term within the parentheses:

• \(-(x + 2y) = -x - 2y\)

This is the distributive property in action, allowing us to rewrite the expression without parentheses.

Negative Sign

Handling the negative sign correctly is crucial in algebra, as an error can change the entire meaning of an expression. When you see a negative sign in front of parentheses, it affects everything inside them.

Essentially, it is as if you are multiplying each term inside the parentheses by \(-1\), inverting their signs:

• Positive becomes negative.
• Negative becomes positive.

For example:

• \[-(x + 2y)\]

Becomes:

• \(-x - 2y\)

Correctly applying the negative sign in this way is necessary for accurate simplification.

Parentheses in Expressions

Parentheses in algebraic expressions indicate grouping and should be handled with care. They show which operations should be performed first, according to the rules of order of operations.

In most cases, you will want to eliminate parentheses by simplifying what's inside them or using algebraic properties, like the distributive property. Always be mindful of any operation, such as a negative sign before the parentheses, which indicates that multiplication or distribution is required.

For example, in the expression \(- (x + 2y)\), the parentheses show that \(x\) and \(2y\) are grouped together as a single unit to be multiplied by \(-1\). The parentheses guide how the expression is simplified, showing which numbers and variables must be combined or distributed together. Proper attention to parentheses ensures the correct order and outcome in algebraic problem-solving.