Simplifying Algebraic Expressions Finding The Simplest Form Of (5(2x-3)-3(x+4))/9

In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to manipulate complex equations into more manageable forms, making them easier to understand and solve. This article delves into the process of simplifying a specific algebraic expression, providing a step-by-step guide to arrive at the simplest form. We'll break down the expression (5(2x-3)-3(x+4))/9, exploring the underlying mathematical principles and techniques involved. Understanding these concepts is crucial for success in various mathematical disciplines, from basic algebra to advanced calculus.

Understanding the Importance of Simplifying Expressions

Before we dive into the simplification process, it's essential to grasp why simplifying expressions is so important. Simplified expressions are not just aesthetically pleasing; they offer several practical advantages. A simplified expression is easier to evaluate. When we need to substitute a value for a variable, a simplified expression reduces the number of calculations required, minimizing the chances of errors. Simplified expressions reveal the underlying structure. By stripping away the clutter, we can often see the relationships between variables and constants more clearly. This can be invaluable in problem-solving and mathematical reasoning. Simplifying expressions is a key step in solving equations. Whether we're dealing with linear equations, quadratic equations, or more complex equations, simplification is often necessary to isolate the variable and find its value. In many real-world applications, mathematical models are used to represent various phenomena. These models often involve complex expressions, and simplifying them is crucial for making predictions and drawing conclusions. Consider the expression (5(2x-3)-3(x+4))/9 which is the focus of this article. At first glance, it appears somewhat convoluted. However, by applying the rules of algebra, we can systematically transform it into a much simpler form, revealing its true essence.

Step-by-Step Simplification of (5(2x-3)-3(x+4))/9

Now, let's embark on the journey of simplifying the expression (5(2x-3)-3(x+4))/9. We'll proceed step by step, explaining each operation and the rationale behind it. This methodical approach will not only lead us to the solution but also reinforce our understanding of algebraic principles.

Step 1: Distribute the Constants

The first step in simplifying this expression involves distributing the constants outside the parentheses to the terms inside. This is based on the distributive property of multiplication over addition and subtraction, which states that a(b + c) = ab + ac and a(b - c) = ab - ac. Applying this property to our expression, we get: 5(2x - 3) = 5 * 2x - 5 * 3 = 10x - 15 and -3(x + 4) = -3 * x - 3 * 4 = -3x - 12. Therefore, the expression becomes: (10x - 15 - 3x - 12)/9. This step effectively eliminates the parentheses, making the expression easier to work with. The distributive property is a cornerstone of algebra, and mastering it is crucial for simplifying expressions and solving equations.

Step 2: Combine Like Terms

After distributing the constants, we now have an expression with several terms. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, 10x and -3x are like terms, and -15 and -12 are like terms. To combine like terms, we simply add or subtract their coefficients. Combining the x terms, we have 10x - 3x = (10 - 3)x = 7x. Combining the constant terms, we have -15 - 12 = -27. Therefore, the expression becomes: (7x - 27)/9. This step consolidates the expression, reducing the number of terms and making it more concise. Combining like terms is a fundamental technique in algebra, and it's essential for simplifying expressions and solving equations efficiently.

Step 3: Express the Simplified Form

Now that we've combined like terms, we have the expression (7x - 27)/9. This expression is already in a simplified form, but we can also express it as a sum of two fractions. This can be useful in certain contexts, such as when we need to perform further operations on the expression. To do this, we simply divide each term in the numerator by the denominator: (7x - 27)/9 = 7x/9 - 27/9. We can further simplify the second fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9: 27/9 = 3. Therefore, the expression can also be written as: 7x/9 - 3. While both (7x - 27)/9 and 7x/9 - 3 are equivalent simplified forms, the former is generally considered the simplest form as it involves fewer terms and operations.

Identifying the Simplest Form Among the Options

Having simplified the expression (5(2x-3)-3(x+4))/9, we can now compare our result with the given options to identify the simplest form. Our simplified expression is (7x - 27)/9. Let's examine the options:

A. (7x - 3)/9 B. (13x - 27)/9 C. 7x - 3 D. (7x - 27)/9

By comparing our simplified expression with the options, we can clearly see that option D, (7x - 27)/9, matches our result. Therefore, the simplest form of the given expression is (7x - 27)/9. The other options are incorrect because they either involve errors in the simplification process or represent different expressions altogether. For instance, option A has an incorrect constant term in the numerator, option B has an incorrect coefficient for the x term, and option C is missing the denominator.

Common Mistakes to Avoid When Simplifying Expressions

Simplifying expressions is a skill that requires practice and attention to detail. There are several common mistakes that students often make, and being aware of these pitfalls can help you avoid them. One common mistake is incorrect distribution. When distributing a constant over a sum or difference, it's crucial to multiply the constant by each term inside the parentheses. For example, in the expression 5(2x - 3), you must multiply both 2x and -3 by 5. Forgetting to multiply all the terms can lead to errors. Another common mistake is combining unlike terms. Only terms with the same variable raised to the same power can be combined. For example, 7x and -27 are unlike terms and cannot be combined. Trying to combine them will result in an incorrect simplification. Sign errors are also a frequent source of mistakes. Pay close attention to the signs of the terms when distributing constants and combining like terms. A misplaced negative sign can completely change the result. For example, -3(x + 4) is equal to -3x - 12, not -3x + 12. Another mistake is incorrect 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), and Addition and Subtraction (from left to right). Ignoring the order of operations can lead to incorrect results. Finally, not simplifying completely is a common mistake. Make sure to simplify the expression as much as possible by combining like terms and reducing fractions. Leaving the expression partially simplified may not be the final answer.

Conclusion: Mastering Simplification for Algebraic Success

Simplifying algebraic expressions is a fundamental skill that forms the foundation for more advanced mathematical concepts. By following a systematic approach, such as distributing constants, combining like terms, and expressing the result in its simplest form, we can effectively manipulate complex expressions into more manageable forms. In this article, we've demonstrated the process of simplifying the expression (5(2x-3)-3(x+4))/9, arriving at the simplest form of (7x - 27)/9. We've also highlighted the importance of simplification, common mistakes to avoid, and the broader implications for algebraic success. Mastering simplification techniques not only enhances your ability to solve equations and tackle mathematical problems but also cultivates a deeper understanding of the underlying structure and relationships within mathematical expressions. So, continue to practice and refine your simplification skills, and you'll be well-equipped to excel in your algebraic endeavors. The ability to simplify expressions efficiently and accurately is a valuable asset in mathematics and beyond.