Problem 9 - Elementary Algebra

Question

Simplify by removing the parentheses. $$ (2 a+3 b) $$

Step-by-Step Solution

Answer

A: The simplified algebraic expression is 2a+3b.

Step 1: Identify the terms inside the parentheses

In this case, we have two terms inside the parentheses, which are \(2a\) and \(3b\).

Step 2: Remove the parentheses

Since there are no terms to distribute, we can simply remove the parentheses and write the expression as: $$ 2a+3b $$ The simplified expression is \(2a+3b\).

Key Concepts

Simplifying Algebraic Expressions

In algebra, simplification involves making an expression as straightforward as possible. This process often includes basic arithmetic operations and organizing similar terms, ensuring calculations or further algebraic manipulations become easier. Simplification is essentially about fewer steps and less confusion. In the expression \((2a + 3b)\), simplification means ensuring everything is as easy to work with as possible.

• Identify and eliminate unnecessary operations.
• Organize similar terms or factors.
• Ensure the expression remains equivalent to its original form.

The goal is to make sure the expression is in its simplest form, which limits errors and improves understanding.

Removing Parentheses

When simplifying algebraic expressions, parentheses signal parts of an expression that need special attention. They are often used to group terms together that will interact with each other, especially in operations like distribution. Removing parentheses effectively repositions the terms within them into the wider expression.
Here's the basic idea of removing parentheses:

• If there is nothing to distribute outside the parentheses (like in \((2a + 3b)\)), you can simply drop the parentheses and rewrite the terms as they are.
• If there is a number or variable outside the parentheses, each term inside must be multiplied by that number. This step is called distribution.

In expressions like \((2a + 3b)\), where no distribution is needed, the terms inside remain unchanged, and the parentheses can be removed without altering the expression.

Understanding Terms in Algebra

The term "terms" in algebra refers to the individual components of an expression that are separated by addition or subtraction. Each term is usually made up of numbers and variables multiplied together. Recognizing different terms within an expression is crucial for both simplification and successful manipulation.

Key Points about Terms:

• Types of Terms: Terms can include constants like numbers, variables like "a" or "b", or combinations like "2a", "3b".
• Like Terms: Terms that have the same variable and power, such as "2a" and "5a". These can be combined in simplification processes.
• Unlike Terms: Terms that have different variables or different powers, such as "2a" and "3b", and are often left separated.

Understanding the structure of terms within an expression, like identifying "2a" and "3b" in \((2a + 3b)\), helps in deciding how to simplify or rearrange the expression efficiently.