Simplify Algebraic Expression Step-by-Step Solution

In the realm of mathematics, particularly in algebra, simplifying expressions is a fundamental skill. It involves manipulating an expression to put it in its most concise and manageable form, making it easier to understand and work with. This article delves into the process of simplifying the algebraic expression $3\left(\frac{7}{5} x+4\right)-2\left(\frac{3}{2}-\frac{5}{4} x\right)$, providing a step-by-step guide to arrive at the correct simplified form. We will explore the underlying principles and techniques involved, ensuring a clear understanding of the simplification process. Mastering these techniques is crucial for success in algebra and higher-level mathematics.

Breaking Down the Expression

To effectively simplify the expression, we need to follow the correct order of operations and apply the distributive property. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform mathematical operations. The distributive property, on the other hand, allows us to multiply a number or variable by each term inside a set of parentheses. Before we dive into the step-by-step simplification, let's take a closer look at the expression:

$3\left(\frac{7}{5} x+4\right)-2\left(\frac{3}{2}-\frac{5}{4} x\right)$

This expression involves fractions, variables, and parentheses, making it a good example to illustrate the simplification process. By carefully applying the distributive property and combining like terms, we can transform this complex expression into a simpler, more manageable form.

Step-by-Step Simplification

Let's embark on the journey of simplifying the given expression step by step:

Step 1: Distribute the Constants

The first step in simplifying the expression is to distribute the constants outside the parentheses to the terms inside. This involves multiplying the constant by each term within the parentheses. For the first set of parentheses, we multiply 3 by both $\frac{7}{5}x$ and 4. For the second set of parentheses, we multiply -2 by both $\frac{3}{2}$ and $-\frac{5}{4}x$. This gives us:

$3 \cdot \frac{7}{5}x + 3 \cdot 4 - 2 \cdot \frac{3}{2} + 2 \cdot \frac{5}{4}x$

Step 2: Perform the Multiplication

Next, we perform the multiplication operations. Multiplying the constants by the terms involving $x$ and the constant terms, we get:

$\frac{21}{5}x + 12 - 3 + \frac{10}{4}x$

Step 3: Simplify Fractions

Now, let's simplify the fraction $\frac{10}{4}$ to its simplest form, which is $\frac{5}{2}$. The expression now becomes:

$\frac{21}{5}x + 12 - 3 + \frac{5}{2}x$

Step 4: Combine Like Terms

Like terms are terms that have the same variable raised to the same power. In this expression, we have two terms involving $x$ ($\frac{21}{5}x$ and $\frac{5}{2}x$) and two constant terms (12 and -3). We can combine these like terms by adding their coefficients. To add the terms involving $x$, we need to find a common denominator for the fractions $\frac{21}{5}$ and $\frac{5}{2}$. The least common denominator is 10. So, we rewrite the fractions with a denominator of 10:

$\frac{21}{5}x = \frac{42}{10}x$

$\frac{5}{2}x = \frac{25}{10}x$

Now we can add the terms involving $x$:

$\frac{42}{10}x + \frac{25}{10}x = \frac{67}{10}x$

We can also combine the constant terms:

$12 - 3 = 9$

Step 5: Write the Simplified Expression

Finally, we combine the simplified terms to obtain the simplified expression:

$\frac{67}{10}x + 9$

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

The Correct Answer and Why

Based on our step-by-step simplification, the correct answer is:

B. $\frac{67}{10} x+9$

This answer aligns perfectly with the result we obtained through our detailed simplification process. The other options are incorrect due to errors in the distribution, multiplication, or combination of like terms. Understanding the correct application of these principles is key to arriving at the accurate simplified form.

Common Mistakes to Avoid

Simplifying algebraic expressions can sometimes be tricky, and it's easy to make mistakes if you're not careful. Here are some common errors to watch out for:

• Incorrectly Distributing: A frequent mistake is not distributing the constant to every term inside the parentheses. Remember, the distributive property applies to all terms within the parentheses.
• Sign Errors: Pay close attention to the signs (positive and negative) when distributing and combining terms. A simple sign error can lead to a completely different answer.
• Combining Unlike Terms: Only combine terms that have the same variable raised to the same power. For example, you cannot combine $x$ and $x^2$ or a constant term with a term involving a variable.
• Fraction Arithmetic Errors: Mistakes in adding, subtracting, multiplying, or dividing fractions are common. Ensure you have a solid understanding of fraction arithmetic before tackling complex expressions.
• Order of Operations: Always follow the order of operations (PEMDAS) to ensure you perform operations in the correct sequence.

By being mindful of these common mistakes, you can significantly improve your accuracy in simplifying algebraic expressions.

Importance of Simplification

Simplifying algebraic expressions is not just an exercise in mathematical manipulation; it's a crucial skill with far-reaching applications. Here are some reasons why simplification is so important:

• Making Expressions Easier to Understand: Simplified expressions are more concise and easier to grasp. This makes it easier to identify patterns, relationships, and key features within the expression.
• Solving Equations and Inequalities: Simplification is often a necessary step in solving equations and inequalities. By simplifying both sides of an equation or inequality, you can isolate the variable and find its value.
• Graphing Functions: To graph a function, it's often helpful to simplify its expression first. This can make it easier to identify key features of the graph, such as intercepts and asymptotes.
• Calculus and Higher-Level Math: Simplification is a fundamental skill in calculus and other advanced mathematical topics. Many calculus problems involve simplifying complex expressions before applying calculus techniques.
• Real-World Applications: Algebraic expressions are used to model real-world phenomena in various fields, including physics, engineering, economics, and computer science. Simplifying these expressions can make it easier to analyze and interpret these models.

In summary, mastering the art of simplifying algebraic expressions is an invaluable asset in mathematics and beyond. It empowers you to tackle complex problems, gain deeper insights, and apply mathematical concepts effectively.

Practice Makes Perfect

The best way to master simplifying algebraic expressions is through practice. Work through a variety of examples, starting with simpler expressions and gradually progressing to more complex ones. Pay close attention to the steps involved, and don't hesitate to review the concepts and techniques discussed in this article. Remember, consistent practice is the key to building confidence and proficiency in simplifying algebraic expressions. By diligently practicing, you can hone your skills and unlock the power of simplification in your mathematical journey.

Conclusion

In conclusion, simplifying the algebraic expression $3\left(\frac{7}{5} x+4\right)-2\left(\frac{3}{2}-\frac{5}{4} x\right)$ involves a systematic application of the distributive property and the combination of like terms. By following the step-by-step process outlined in this article, we arrive at the simplified form $\frac{67}{10} x+9$. This process underscores the importance of understanding the fundamental principles of algebra and the need for careful attention to detail. Mastering the simplification of algebraic expressions is a crucial skill for success in mathematics and its applications in various fields. Embrace the challenge, practice diligently, and unlock the power of simplified expressions! By understanding the process and avoiding common mistakes, you can confidently simplify a wide range of algebraic expressions and excel in your mathematical endeavors.