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Simplifying Complex Expressions: A Guide Through Examples
Simplifying Complex Expressions: A Guide Through Examples
This article provides a comprehensive guide to the process of simplifying complex algebraic expressions. We will go through the step-by-step process of simplifying the given expression, using examples to facilitate a deeper understanding. Whether you are a student or a teacher, this guide will help you master the techniques involved in simplifying expressions.
Expression Simplification: The Basics
Expression simplification is a fundamental skill in algebra. It involves the process of reducing an expression to its simplest form, making it easier to understand and work with. This can be achieved by applying algebraic rules and operations to remove unnecessary elements, such as brackets, or by combining like terms.
Example: Simplifying 4x-3 {x-1-y(21-x)}
Let's use the provided example to demonstrate the simplification process:
The expression is: 4x-3 {x-1-y(21-x)}
We start by considering the expression inside the brackets: x-1-y(21-x)
First, distribute the -y across the bracket: x-1-y(21)-y(-x) x-1-21y xy
The expression now becomes: 4x-3{x-1-21y xy}
Next, distribute the -3 across the brackets: 4x-3x 3 63y-3xy
Combine like terms: (4x-3x)-3xy 3 63y
Finally, simplify further: x-3xy 3 63y
The fully simplified expression is: x-3xy 63y 3
Step-by-Step Process
Distribute Terms: If there are any terms outside the bracket that need to be multiplied with terms inside the bracket, distribute them using the distribution rule.
Combine Like Terms: Combine any like terms in the expression to reduce it to a simpler form.
Remove Brackets: Use the distributive property to remove brackets by distributing the terms outside the brackets over the terms inside the brackets.
Simplify Further: If there are any further simplifications possible, such as combining like terms, perform them to ensure the expression is in its simplest form.
Example Walkthrough
Let's walk through the example step by step:
4x - 3 {x - 1 - y(21 - x)} 4x - 3 {x - 1 - 21y xy} - 3 {x - 1 - 21y xy} 4x - 3x 3 - 63y 3xy - 3x 3 63y - 3xy (4x - 3x - 3x) (3 - 63y 63y) (3xy - 3xy) -2x 3
Further Examples and Practice
To solidify your understanding, here are additional examples and practice problems:
Example 1: Simplify 3x - 2 {x 4 - 3y(5 - x)}
Distribute: 3x - 2 {x 4 - 15y 3xy}
Remove brackets: 3x - 2x - 8 30y - 6xy
Combine like terms: (3x - 2x) - 6xy 30y - 8
Simplify: x - 6xy 30y - 8
Example 2: Simplify 5x 7 {2 - 3y(4 - x)}
Distribute: 5x 7 {2 - 12y 3xy}
Remove brackets: 5x 14 - 84y 21xy
Combine like terms: (5x 21xy) - 84y 14
Simplify: 5x 21xy - 84y 14
Conclusion
Mastering the process of simplifying complex expressions is crucial for success in algebra and higher mathematics. By following the steps outlined in this guide, you can simplify any expression, making it easier to work with and solve. Remember to practice regularly to reinforce your understanding and improve your skills.