Simplifying Algebraic Expressions A Step-by-Step Guide

In the realm of mathematics, particularly in algebra, simplifying expressions is a fundamental skill. It allows us to manipulate equations and inequalities, solve for unknowns, and gain a deeper understanding of mathematical relationships. This article delves into the simplification of a specific algebraic expression: $3(\frac{7}{5} x+4)-2(\frac{3}{2}-\frac{5}{4} x)$. We will break down the process step-by-step, ensuring clarity and comprehension for learners of all levels. This detailed explanation aims to provide not just the answer, but also the underlying principles and techniques involved in algebraic simplification.

Understanding the Expression

Before we embark on the simplification journey, let's first dissect the expression itself. We have $3(\frac{7}{5} x+4)-2(\frac{3}{2}-\frac{5}{4} x)$. This expression involves several key components:

• Variables: The variable in this expression is 'x', representing an unknown quantity.
• Coefficients: These are the numerical values multiplying the variable 'x', such as $\frac{7}{5}$ and $-\frac{5}{4}$.
• Constants: These are the numerical terms without any variable, such as 4 and $\frac{3}{2}$.
• Parentheses: These indicate the order of operations, telling us which operations to perform first.
• Distributive Property: This property will be crucial in simplifying the expression, as it allows us to multiply a number by a sum or difference within parentheses.
• Arithmetic Operations: The expression involves addition, subtraction, and multiplication.

To simplify this expression effectively, we will utilize the order of operations (PEMDAS/BODMAS) and the distributive property. Understanding these basic components is key to successfully navigating the simplification process. Grasping these elements lays the groundwork for a smooth and accurate simplification process, ensuring we handle each term and operation correctly. This initial understanding is critical for avoiding common mistakes and building confidence in algebraic manipulations.

Step 1: Applying the Distributive Property

The distributive property is the cornerstone of simplifying expressions involving parentheses. It states that $a(b + c) = ab + ac$. In our expression, we have two instances where we need to apply the distributive property:

1. Distributing the 3: $3(\frac{7}{5} x+4)$ becomes $3 * \frac{7}{5} x + 3 * 4$.
2. Distributing the -2: $-2(\frac{3}{2}-\frac{5}{4} x)$ becomes $-2 * \frac{3}{2} + (-2) * (-\frac{5}{4} x)$.

Let's perform these multiplications:

• $3 * \frac{7}{5} x = \frac{21}{5} x$
• $3 * 4 = 12$
• $-2 * \frac{3}{2} = -3$
• $(-2) * (-\frac{5}{4} x) = \frac{10}{4} x = \frac{5}{2} x$

Now, our expression looks like this: $\frac{21}{5} x + 12 - 3 + \frac{5}{2} x$. This step effectively removes the parentheses, allowing us to combine like terms in the subsequent steps. The distributive property is a powerful tool in algebra, and mastering its application is crucial for simplifying complex expressions. Careful attention to signs and fractions is essential during this step to ensure accuracy.

Step 2: Combining Like Terms

In this step, we identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, $\frac{21}{5} x + 12 - 3 + \frac{5}{2} x$, we have two terms with 'x' and two constant terms.

Let's group them together:

• Terms with 'x': $\frac{21}{5} x + \frac{5}{2} x$
• Constant terms: $12 - 3$

Now, we need to add the coefficients of the 'x' terms. To do this, we need a common denominator for the fractions $\frac{21}{5}$ and $\frac{5}{2}$. The least common denominator (LCD) of 5 and 2 is 10. So, we convert the fractions:

• $\frac{21}{5} x = \frac{21 * 2}{5 * 2} x = \frac{42}{10} x$
• $\frac{5}{2} x = \frac{5 * 5}{2 * 5} x = \frac{25}{10} x$

Now we can add them: $\frac{42}{10} x + \frac{25}{10} x = \frac{67}{10} x$.

Next, we combine the constant terms: $12 - 3 = 9$.

Our expression now simplifies to $\frac{67}{10} x + 9$. Combining like terms is a crucial step in simplifying algebraic expressions, as it reduces the number of terms and makes the expression more manageable. The process involves identifying terms with the same variable and exponent, finding a common denominator for fractions, and then adding or subtracting the coefficients.

Step 3: Final Simplified Form

After applying the distributive property and combining like terms, we have arrived at the simplified form of the expression. Our expression, $\frac{67}{10} x + 9$, cannot be simplified further because there are no more like terms to combine. This is the final answer.

Therefore, the simplified form of $3(\frac{7}{5} x+4)-2(\frac{3}{2}-\frac{5}{4} x)$ is $\frac{67}{10} x + 9$.

Comparing this result with the given options:

• A. $-\frac{39}{5} x-\frac{11}{2}$ (Incorrect)
• B. $\frac{67}{10} x+9$ (Correct)
• C. $\frac{3}{10} x+\frac{5}{2}$ (Incorrect)

Thus, the correct answer is B. $\frac{67}{10} x+9$. This final step reinforces the importance of accurate calculations and careful attention to detail throughout the simplification process. By systematically applying the distributive property and combining like terms, we can confidently arrive at the correct simplified form of the expression. The ability to simplify algebraic expressions is a fundamental skill in mathematics, with applications in various fields, including science, engineering, and economics.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and certain common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them:

1. Incorrectly Applying the Distributive Property: A common error is forgetting to distribute the number outside the parentheses to every term inside. For instance, in $3(\frac{7}{5} x+4)$, you must multiply both $\frac{7}{5} x$ and 4 by 3.
2. Sign Errors: Pay close attention to signs, especially when distributing a negative number. For example, when distributing -2 in $-2(\frac{3}{2}-\frac{5}{4} x)$, remember that $-2 * -\frac{5}{4} x$ results in a positive term.
3. Combining Unlike Terms: Only like terms (terms with the same variable and exponent) can be combined. Do not add or subtract terms like $\frac{67}{10} x$ and 9, as they are not like terms.
4. Fraction Arithmetic Errors: When adding or subtracting fractions, ensure they have a common denominator. Mistakes in finding the least common denominator or in adding/subtracting the numerators can lead to incorrect results.
5. Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Failing to follow this order can lead to significant errors.

By understanding and avoiding these common mistakes, you can improve your accuracy and confidence in simplifying algebraic expressions. Regular practice and careful attention to detail are key to mastering this skill. Recognizing these common pitfalls is an essential part of the learning process, allowing you to develop strategies for error prevention and correction. Consistent practice and a methodical approach will help you build a solid foundation in algebraic manipulation.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, here are some practice problems:

1. Simplify: $4(2x - 3) + 5(x + 1)$
2. Simplify: $-2(3y + 5) - (4y - 2)$
3. Simplify: $\frac{1}{2}(4a - 6) + \frac{2}{3}(9a + 3)$
4. Simplify: $5(\frac{2}{5}b + 1) - 3(\frac{1}{3}b - 2)$
5. Simplify: $2.5(x + 4) - 1.5(2x - 1)$

Working through these problems will give you valuable experience in applying the distributive property, combining like terms, and avoiding common mistakes. Remember to show your steps clearly and double-check your work. The more you practice, the more comfortable and proficient you will become at simplifying algebraic expressions. Solving these problems will not only reinforce the concepts discussed but also help you develop problem-solving skills applicable to various mathematical contexts. Take your time, work through each step carefully, and don't hesitate to review the previous sections if you encounter any difficulties.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics. In this article, we have walked through the process of simplifying the expression $3(\frac{7}{5} x+4)-2(\frac{3}{2}-\frac{5}{4} x)$ step-by-step, emphasizing the importance of the distributive property, combining like terms, and avoiding common mistakes. By understanding these concepts and practicing regularly, you can develop a strong foundation in algebra and confidently tackle more complex mathematical problems. The ability to simplify expressions is not just a mathematical exercise; it is a powerful tool for problem-solving and critical thinking in various fields. Mastering this skill will open doors to further mathematical studies and applications, empowering you to approach challenges with confidence and clarity.