Simplifying Algebraic Expressions A Step By Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to rewrite complex mathematical statements into more manageable and understandable forms. This article delves into the process of simplifying the expression $30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)$, providing a step-by-step guide to arrive at the equivalent expression. Whether you're a student grappling with algebraic concepts or simply seeking to refresh your mathematical prowess, this comprehensive guide will equip you with the knowledge and techniques to confidently tackle such problems.

Understanding the Expression

The given expression, $30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)$, involves variables, coefficients, and constants. To simplify it effectively, we'll employ the distributive property and combine like terms. The distributive property is a cornerstone of algebra, allowing us to multiply a single term by each term within a set of parentheses. This property is crucial for expanding expressions and setting the stage for further simplification. In this particular expression, we see the application of the distributive property in two distinct parts, each involving a different set of terms within the parentheses. By carefully applying this property, we can break down the complexity of the expression into smaller, more manageable components, paving the way for a clearer understanding and a more streamlined simplification process. Understanding the properties and principles that govern algebraic manipulations is key to not only solving this specific problem but also to building a solid foundation for tackling more complex mathematical challenges in the future.

Step 1 Applying the Distributive Property

The first step in simplifying this expression is to apply the distributive property. This property states that a(b + c) = ab + ac. We'll apply this to both parts of the expression.

For the first part, $30(\frac{1}{2}x - 2)$, we distribute the 30 to both terms inside the parentheses:

$30 * (\frac{1}{2}x) - 30 * 2 = 15x - 60$

Similarly, for the second part, $40(\frac{3}{4}y - 4)$, we distribute the 40:

$40 * (\frac{3}{4}y) - 40 * 4 = 30y - 160$

By applying the distributive property, we've effectively eliminated the parentheses, transforming the original expression into a sum of individual terms. This is a crucial step in simplifying algebraic expressions, as it allows us to identify and combine like terms, which is the next key step in our simplification journey. The ability to confidently apply the distributive property is a fundamental skill in algebra, and mastering it will significantly enhance your ability to manipulate and simplify a wide range of algebraic expressions. This step not only simplifies the expression visually but also lays the groundwork for further algebraic manipulations, ultimately leading us to the most simplified form of the expression.

Step 2 Combining Like Terms

Now that we've applied the distributive property, our expression looks like this:

$15x - 60 + 30y - 160$

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have constant terms (-60 and -160) that can be combined.

Combining the constants, we get:

-60 - 160 = -220

So, our expression now becomes:

$15x + 30y - 220$

The process of combining like terms is a crucial step in simplifying algebraic expressions. It involves identifying terms that share the same variable and exponent, and then adding or subtracting their coefficients. This step not only reduces the number of terms in the expression but also makes it easier to understand and work with. In our example, the constant terms -60 and -160 were like terms because they both lacked a variable. By combining them, we consolidated them into a single constant term, -220. This simplification process is essential for arriving at the most concise and manageable form of the expression. The ability to accurately identify and combine like terms is a fundamental skill in algebra and is essential for solving equations, simplifying expressions, and performing other algebraic manipulations.

Identifying the Equivalent Expression

After simplifying the expression, we arrive at $15x + 30y - 220$. Comparing this with the given options, we can see that it matches option C.

Therefore, the equivalent expression is:

C. $15x + 30y - 220$

The process of identifying the equivalent expression involves comparing the simplified form of the expression with the provided options. This step requires careful attention to detail and a thorough understanding of the simplification process. In our case, after applying the distributive property and combining like terms, we arrived at the expression $15x + 30y - 220$. By meticulously comparing this simplified expression with the given options, we were able to confidently identify option C as the correct equivalent expression. This final step underscores the importance of accuracy and precision in mathematical problem-solving. It also highlights the value of a systematic approach, where each step is carefully executed and the final result is thoroughly verified against the available options. The ability to accurately identify equivalent expressions is a critical skill in algebra and is essential for solving equations, simplifying complex expressions, and performing various other mathematical operations.

Why Other Options Are Incorrect

To further solidify our understanding, let's examine why the other options are incorrect:

• A. $45xy - 220$: This option incorrectly multiplies the x and y terms, which is not part of the correct simplification process.
• B. $15x - 30y - 220$: This option has the wrong sign for the y term. It should be +30y, not -30y.
• D. $15x + 30y - 64$: This option has an incorrect constant term. The correct constant term is -220, not -64.

Understanding why the incorrect options are wrong is just as important as knowing why the correct option is right. This process of elimination not only reinforces the correct solution but also deepens our understanding of the underlying mathematical principles. In this case, by analyzing each incorrect option, we can pinpoint the specific errors in their simplification processes. Option A incorrectly multiplies the x and y terms, a common mistake that violates the rules of algebraic manipulation. Option B demonstrates an error in the sign of the y term, highlighting the importance of paying close attention to positive and negative signs throughout the simplification process. Option D presents an incorrect constant term, indicating a potential mistake in combining the constant terms during the simplification process. By carefully examining these errors, we can learn valuable lessons about common pitfalls in algebraic simplification and develop strategies to avoid them in the future. This process of error analysis is a powerful tool for enhancing our mathematical understanding and problem-solving skills.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By applying the distributive property and combining like terms, we can rewrite complex expressions into simpler, equivalent forms. In this case, we successfully simplified the expression $30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)$ to $15x + 30y - 220$. This step-by-step guide provides a clear and concise approach to solving similar problems, empowering you to confidently tackle algebraic challenges. Remember, practice is key to mastering these skills, so continue to explore and simplify various expressions to strengthen your mathematical foundation.

This comprehensive guide has not only demonstrated the process of simplifying a specific algebraic expression but has also emphasized the importance of understanding the underlying principles and techniques. The distributive property and the process of combining like terms are fundamental concepts in algebra, and mastering them will significantly enhance your ability to manipulate and simplify a wide range of mathematical expressions. By understanding why the incorrect options are wrong, we have further solidified our understanding of the correct solution and the principles that govern algebraic simplification. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the reasoning behind it. By consistently practicing and applying these principles, you can develop a strong foundation in algebra and confidently tackle more complex mathematical challenges.