Simplifying (a-5b)-(6a-2b)+(2a-3b) A Step-by-Step Guide

In the realm of mathematics, simplifying algebraic expressions is a foundational skill that paves the way for tackling more complex equations and problems. This article delves into the process of simplifying the expression $(a - 5b) - (6a - 2b) + (2a - 3b)$, providing a step-by-step guide and valuable insights into the underlying principles. Whether you're a student grappling with algebra for the first time or a seasoned mathematician looking for a refresher, this comprehensive guide will equip you with the knowledge and techniques to confidently simplify algebraic expressions.

Understanding the Basics of Algebraic Expressions

Before we dive into the specifics of simplifying the given expression, it's crucial to understand the fundamental components of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. The coefficients are the numbers that multiply the variables.

In the expression $(a - 5b) - (6a - 2b) + (2a - 3b)$, the variables are a and b, and the coefficients are 1 (for a in the first term), -5 (for b in the first term), -6 (for a in the second term), 2 (for b in the second term), 2 (for a in the third term), and -3 (for b in the third term). Understanding these components is essential for effectively simplifying algebraic expressions.

To simplify an algebraic expression, our goal is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, 3x and -5x are like terms because they both have the variable x raised to the power of 1. Similarly, 2y² and 7y² are like terms because they both have the variable y raised to the power of 2. However, 4x and 9x² are not like terms because the variable x is raised to different powers. Constants, such as 5 and -2, are also considered like terms.

When simplifying expressions, we apply the distributive property, which states that a( b + c ) = ab + ac. This property is crucial for removing parentheses and combining like terms. Additionally, we adhere to the order of operations (PEMDAS/BODMAS), which dictates the sequence in which mathematical operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Step-by-Step Simplification of $(a - 5b) - (6a - 2b) + (2a - 3b)$

Let's now embark on a step-by-step simplification of the expression $(a - 5b) - (6a - 2b) + (2a - 3b)$. This process will illustrate the application of the concepts we've discussed and provide a clear methodology for simplifying similar expressions.

Step 1: Distribute the Negative Signs

The first step in simplifying this expression involves distributing the negative signs in front of the parentheses. Remember that subtracting a quantity is the same as adding the negative of that quantity. Therefore, we need to distribute the negative sign in the second term, $-(6a - 2b)$. This means multiplying each term inside the parentheses by -1.

Applying the distributive property, we get:

$(a - 5b) - (6a - 2b) + (2a - 3b) = (a - 5b) + (-1)(6a - 2b) + (2a - 3b)$

Now, distribute the -1 to both terms inside the second set of parentheses:

$(a - 5b) + (-6a + 2b) + (2a - 3b)$

The expression now becomes:

$a - 5b - 6a + 2b + 2a - 3b$

Step 2: Identify Like Terms

The next step is to identify the like terms in the expression. As we discussed earlier, like terms are those that have the same variables raised to the same powers. In this expression, we have terms with the variable a, terms with the variable b, and no constant terms. Let's group the like terms together:

(a terms): a, -6a, 2a

(b terms): -5b, 2b, -3b

Step 3: Combine Like Terms

Now, we combine the like terms by adding their coefficients. This involves adding the coefficients of the a terms and the coefficients of the b terms separately.

Combining the a terms:

a - 6a + 2a = (1 - 6 + 2)a = -3a

Combining the b terms:

-5b + 2b - 3b = (-5 + 2 - 3)b = -6b

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the results from the previous step:

-3a - 6b

Therefore, the simplified form of the expression $(a - 5b) - (6a - 2b) + (2a - 3b)$ is -3a - 6b. This final expression is much more concise and easier to work with than the original expression.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

• Incorrectly Distributing Negative Signs: A frequent error is failing to distribute the negative sign to all terms inside the parentheses. Remember that when subtracting a quantity, you are subtracting each term within that quantity.
• Combining Unlike Terms: Only like terms can be combined. Avoid adding or subtracting terms with different variables or different powers of the same variable.
• Ignoring the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) to ensure accurate simplification. Parentheses/Brackets should be addressed first, followed by exponents/orders, multiplication and division, and finally, addition and subtraction.
• Sign Errors: Be mindful of the signs of the coefficients. A simple sign error can lead to an incorrect result.
• Forgetting Coefficients: Remember that a variable without a visible coefficient has a coefficient of 1. For example, a is the same as 1a.

By being aware of these common mistakes, you can significantly reduce the chances of making errors in your simplifications.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, try working through these practice problems:

1. Simplify: $2(x + 3y) - (4x - y)$
2. Simplify: $5a - 3b + 2(a - 4b)$
3. Simplify: $(3m - 2n) - (m + 5n) + (4m - n)$
4. Simplify: $-4(p - 2q) + 3(2p + q)$
5. Simplify: $7x² + 2x - (3x² - 5x) + 2$

Working through these problems will provide valuable practice and reinforce the concepts we've discussed. Remember to follow the steps outlined in this article and pay close attention to detail.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics. By understanding the basic components of expressions, applying the distributive property, combining like terms, and avoiding common mistakes, you can confidently simplify a wide range of algebraic expressions. The step-by-step guide provided in this article offers a clear methodology for simplifying expressions like $(a - 5b) - (6a - 2b) + (2a - 3b)$, and the practice problems provide an opportunity to hone your skills. With practice and attention to detail, you can master the art of simplifying algebraic expressions and unlock the door to more advanced mathematical concepts.