Simplifying Algebraic Expressions: A Comprehensive Guide with Examples

When dealing with algebraic expressions, simplification is a fundamental skill that helps in understanding and solving complex problems. This guide provides a detailed, step-by-step approach to simplifying algebraic expressions, with a focus on understanding the underlying logical flow. Let's dive into the process, illustrated with examples.

Algebraic Expression Simplification: A Step-by-Step Approach

Expression: x - y y xy - x - 2y - 4xy - 2x

Let's simplify the algebraic expression x - y y xy - x - 2y - 4xy - 2x step-by-step, making use of basic algebraic rules and operations.

Step 1: Remove Parentheses

First, we remove the parentheses where necessary and identify the terms in the expression:

x - y y xy - x - 2y - 4xy - 2x

Step 2: Identify Negative Signs in Front of Parentheses

If there were any negative signs in front of parentheses, we would place a 1 next to them and distribute the sign to all terms within the parentheses. In this case, there are no parentheses to consider, so we move on to the next step.

Step 3: Distribute and Cancel Terms

We distribute and cancel out any like-terms. Here, we cancel out the y terms and the x terms:

x y xy - x - 2y - 4xy - 2x x - x y - 2y xy - 4xy - 2x -x - y - 3xy

Step 4: Combine Like-Terms and Arrange in Logical Order

Next, we combine the like-terms and arrange the remaining terms in a logical order:

-x - y - 3xy

However, the final simplified form of the expression is:

2x - 2y 5xy

Let's break down how this simplification was achieved:

Example Simplification: 2x - 2y 5xy

1. Initial Expression

Starting from the initial expression x - y y xy - x - 2y - 4xy - 2x, we proceed to simplify:

x - y y xy - x - 2y - 4xy - 2x

2. Simplification Steps

Step A: Remove Parentheses x - y y xy - x - 2y - 4xy - 2x Step B: Cancel Like-Terms x - x y - 2y xy - 4xy - 2x -x - y - 3xy

Making the expression more concise: