Simplifying Algebraic Expressions A Step-by-Step Guide To 2(6x - 2y) + (3x - 5y)

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex mathematical statements and reduce them to their most basic, understandable form. This process often involves combining like terms, applying the distributive property, and performing basic arithmetic operations. In this article, we will delve into the step-by-step simplification of the algebraic expression 2(6x - 2y) + (3x - 5y). This exercise will not only demonstrate the application of key algebraic principles but also highlight the importance of precision and methodical execution in mathematical problem-solving. Let’s embark on this journey of simplification and uncover the elegance hidden within algebraic expressions.

Understanding the Importance of Simplification

Before we dive into the specifics of simplifying the given expression, it's crucial to understand why simplification is such a vital aspect of mathematics. Simplification serves several key purposes:

• Clarity and Understanding: A simplified expression is easier to understand and interpret. It presents the underlying relationship between variables and constants in a clear and concise manner. This clarity is essential for problem-solving and decision-making.
• Efficiency in Calculations: Simplified expressions require fewer steps and calculations to evaluate. This is particularly important when dealing with complex expressions or when performing repeated calculations.
• Problem-Solving: Simplification is often a crucial step in solving equations and inequalities. By simplifying expressions, we can isolate variables and find solutions more easily.
• Generalization: Simplified expressions often reveal patterns and relationships that might not be immediately apparent in the original form. This can lead to broader generalizations and a deeper understanding of mathematical concepts.

In essence, simplification is about making mathematics more accessible and manageable. It allows us to focus on the core ideas and relationships without getting bogged down in unnecessary complexity. In the context of our expression, simplifying 2(6x - 2y) + (3x - 5y) will help us understand the relationship between the variables x and y and express it in its most fundamental form.

Step-by-Step Simplification of 2(6x - 2y) + (3x - 5y)

Now, let's embark on the journey of simplifying the expression 2(6x - 2y) + (3x - 5y). We will proceed step-by-step, explaining each action and the underlying mathematical principle. This approach will ensure clarity and understanding at every stage of the process.

Step 1: Applying the Distributive Property

The first step in simplifying the expression is to apply the distributive property. The distributive property states that a(b + c) = ab + ac. In our expression, we have the term 2(6x - 2y). We need to distribute the 2 to both terms inside the parentheses:

2 * (6x - 2y) = (2 * 6x) - (2 * 2y)

Performing the multiplications, we get:

12x - 4y

So, the expression now becomes:

12x - 4y + (3x - 5y)

The distributive property is a cornerstone of algebra, allowing us to eliminate parentheses and combine terms effectively. It's a fundamental tool in simplifying expressions and solving equations.

Step 2: Identifying Like Terms

In the next step, we need to identify like terms. Like terms are terms that have the same variable raised to the same power. In our expression, 12x and 3x are like terms because they both have the variable x raised to the power of 1. Similarly, -4y and -5y are like terms because they both have the variable y raised to the power of 1.

Identifying like terms is crucial for combining them and simplifying the expression. It allows us to consolidate terms with the same variable and reduce the overall complexity of the expression.

Step 3: Combining Like Terms

Now that we have identified the like terms, we can combine them. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). Let's start with the x terms:

12x + 3x = (12 + 3)x = 15x

Next, let's combine the y terms:

-4y - 5y = (-4 - 5)y = -9y

Combining like terms is a fundamental operation in algebra. It allows us to reduce the number of terms in an expression and express it in a more concise form.

Step 4: Writing the Simplified Expression

After combining the like terms, we can now write the simplified expression. We have 15x and -9y. Combining these terms, we get:

15x - 9y

This is the simplified form of the original expression, 2(6x - 2y) + (3x - 5y). We have successfully applied the distributive property and combined like terms to arrive at a more concise and understandable expression.

Significance of the Simplified Form

The simplified form, 15x - 9y, is much easier to work with than the original expression. It clearly shows the relationship between the variables x and y. This simplified form can be used for various purposes, such as:

• Evaluating the Expression: To evaluate the expression for specific values of x and y, we can simply substitute the values into the simplified form. This is much easier than substituting the values into the original expression.
• Graphing the Expression: If we set the expression equal to a constant, we can graph the resulting equation. The simplified form makes it easier to identify the slope and intercepts of the line.
• Solving Equations: The simplified form can be used to solve equations involving the expression. By manipulating the simplified form, we can isolate variables and find solutions.

In essence, the simplified form provides a more efficient and accessible representation of the original expression, making it easier to use in various mathematical contexts.

Common Mistakes to Avoid

When simplifying expressions, it's essential to avoid common mistakes that can lead to incorrect results. Here are some common pitfalls to watch out for:

• Incorrectly Applying the Distributive Property: Ensure that you distribute the term outside the parentheses to all terms inside the parentheses. For example, in 2(6x - 2y), the 2 must be multiplied by both 6x and -2y.
• Combining Unlike Terms: Only combine terms that have the same variable raised to the same power. For example, you cannot combine 12x and -4y because they have different variables.
• Sign Errors: Pay close attention to the signs of the terms when combining like terms. For example, -4y - 5y = -9y, not -1y.
• Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

By being mindful of these common mistakes, you can ensure accuracy and confidence in your simplification efforts.

Practice Problems

To solidify your understanding of simplifying expressions, let's work through a few practice problems:

1. Simplify: 3(2a + 5b) - (a - 2b)
2. Simplify: 4(x - 3y) + 2(2x + y)
3. Simplify: 5(m - n) - 3(2m - 4n)

Working through these practice problems will help you develop your skills and build confidence in simplifying expressions. Remember to follow the steps we discussed earlier: apply the distributive property, identify like terms, combine like terms, and write the simplified expression.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics. It involves applying the distributive property, identifying like terms, and combining them to arrive at a more concise and understandable form. We have successfully simplified the expression 2(6x - 2y) + (3x - 5y) to 15x - 9y. This simplified form is easier to work with and provides a clearer understanding of the relationship between the variables x and y. By avoiding common mistakes and practicing regularly, you can master the art of simplifying expressions and excel in your mathematical journey. Remember, simplification is not just about finding the answer; it's about understanding the underlying principles and developing a logical approach to problem-solving.