Simplify Algebraic Expressions A Step-by-Step Guide

Jul 11, 2025 by Admin 52 views

$Simplifying Algebraic Expressions: A Step-by-Step Guide to Solving $2(\frac{3}{5} x+2 \frac{3}{4} y-\frac{1}{4} x-1 \frac{1}{2} y+3)$$

In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to rewrite complex expressions in a more manageable and understandable form. This article delves into the process of simplifying the algebraic expression $2(\frac{3}{5} x+2 \frac{3}{4} y-\frac{1}{4} x-1 \frac{1}{2} y+3)$ , providing a step-by-step guide to arrive at the solution. We will explore the underlying principles of combining like terms and the distributive property, ensuring a clear understanding of the simplification process. This comprehensive guide will not only demonstrate the solution but also elucidate the mathematical concepts involved, making it a valuable resource for students and anyone looking to enhance their algebraic proficiency.

Understanding the Problem: A Detailed Breakdown

The problem at hand requires us to simplify the algebraic expression $2(\frac{3}{5} x+2 \frac{3}{4} y-\frac{1}{4} x-1 \frac{1}{2} y+3)$ . This expression involves variables ( $x$ and $y$ ), fractions, and a constant term. To simplify it effectively, we need to follow a systematic approach, utilizing the principles of algebra. The first crucial step is to address the mixed numbers within the expression. Converting mixed numbers into improper fractions will streamline the subsequent calculations. For instance, $2 \frac{3}{4}$ can be converted to $\frac{11}{4}$ and $1 \frac{1}{2}$ to $\frac{3}{2}$ . This conversion allows for easier manipulation of the fractions during the simplification process. Next, we need to identify and combine the like terms. Like terms are terms that contain the same variable raised to the same power. In our expression, we have terms with $x$ , terms with $y$ , and a constant term. The coefficients of the like terms can be combined through addition or subtraction. Finally, we will apply the distributive property, which involves multiplying the constant outside the parentheses (in this case, 2) by each term inside the parentheses. This step is crucial for eliminating the parentheses and further simplifying the expression. By carefully executing these steps, we can transform the given expression into its simplest form, making it easier to interpret and use in further calculations. This initial breakdown sets the stage for a detailed step-by-step simplification process, ensuring clarity and accuracy in our solution.

Step-by-Step Simplification: Unraveling the Solution

To simplify the expression $2(\frac{3}{5} x+2 \frac{3}{4} y-\frac{1}{4} x-1 \frac{1}{2} y+3)$ , we will follow a meticulous step-by-step approach, ensuring clarity and accuracy at each stage. This process involves several key algebraic principles, including converting mixed numbers to improper fractions, combining like terms, and applying the distributive property. By breaking down the problem into manageable steps, we can effectively navigate the complexities of the expression and arrive at the simplified form. Each step will be explained in detail, providing a clear understanding of the mathematical operations involved. This methodical approach not only leads to the correct solution but also enhances our understanding of algebraic simplification techniques. Let's embark on this step-by-step journey to unravel the solution and master the art of simplifying algebraic expressions.

Step 1: Converting Mixed Numbers to Improper Fractions

The initial step in simplifying the expression is to convert the mixed numbers into improper fractions. This conversion makes it easier to perform arithmetic operations, especially when dealing with addition and subtraction of fractions. We have two mixed numbers in the expression: $2 \frac{3}{4}$ and $1 \frac{1}{2}$ . To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, while the denominator remains the same. For $2 \frac{3}{4}$ , we multiply 2 by 4 (which equals 8) and add 3, resulting in 11. So, $2 \frac{3}{4}$ becomes $\frac{11}{4}$ . Similarly, for $1 \frac{1}{2}$ , we multiply 1 by 2 (which equals 2) and add 1, resulting in 3. Thus, $1 \frac{1}{2}$ becomes $\frac{3}{2}$ . Replacing the mixed numbers with their improper fraction equivalents, the expression now becomes $2(\frac{3}{5} x+\frac{11}{4} y-\frac{1}{4} x-\frac{3}{2} y+3)$ . This conversion is a crucial foundation for the subsequent steps, as it allows us to work with fractions in a more streamlined manner. By ensuring accuracy in this initial conversion, we set the stage for a successful simplification process.

Step 2: Distributing the Constant

Having converted the mixed numbers to improper fractions, the next step is to apply the distributive property. This property states that $a(b + c) = ab + ac$ . In our expression, the constant 2 is multiplied by the entire expression inside the parentheses: $2(\frac{3}{5} x+\frac{11}{4} y-\frac{1}{4} x-\frac{3}{2} y+3)$ . To distribute the 2, we multiply each term inside the parentheses by 2. This gives us $2 * \frac{3}{5} x + 2 * \frac{11}{4} y - 2 * \frac{1}{4} x - 2 * \frac{3}{2} y + 2 * 3$ . Performing these multiplications, we get $\frac{6}{5} x + \frac{22}{4} y - \frac{2}{4} x - \frac{6}{2} y + 6$ . Now, we can simplify the fractions. $\frac{22}{4}$ simplifies to $\frac{11}{2}$ , $\frac{2}{4}$ simplifies to $\frac{1}{2}$ , and $\frac{6}{2}$ simplifies to 3. The expression now looks like this: $\frac{6}{5} x + \frac{11}{2} y - \frac{1}{2} x - 3y + 6$ . Distributing the constant is a pivotal step in eliminating the parentheses and preparing the expression for combining like terms. By carefully applying the distributive property, we ensure that each term is correctly multiplied, setting the stage for further simplification.

Step 3: Combining Like Terms: Grouping Similar Elements

After distributing the constant, the expression now stands as $\frac{6}{5} x + \frac{11}{2} y - \frac{1}{2} x - 3y + 6$ . The next crucial step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have terms with $x$ and terms with $y$ , along with a constant term. To combine like terms, we group them together and then add or subtract their coefficients. Let's first group the $x$ terms: $\frac{6}{5} x$ and $-\frac{1}{2} x$ . Then, we group the $y$ terms: $\frac{11}{2} y$ and $-3y$ . The constant term is 6. Now, we will combine the $x$ terms: $\frac{6}{5} x - \frac{1}{2} x$ . To do this, we need a common denominator, which is 10. So, we convert $\frac{6}{5}$ to $\frac{12}{10}$ and $\frac{1}{2}$ to $\frac{5}{10}$ . Thus, $\frac{12}{10} x - \frac{5}{10} x = \frac{7}{10} x$ . Next, we combine the $y$ terms: $\frac{11}{2} y - 3y$ . We can rewrite 3y as $\frac{6}{2} y$ . So, $\frac{11}{2} y - \frac{6}{2} y = \frac{5}{2} y$ . Now, we have simplified the $x$ and $y$ terms. The expression now becomes $\frac{7}{10} x + \frac{5}{2} y + 6$ . Combining like terms is a fundamental step in simplifying algebraic expressions, allowing us to consolidate similar elements and arrive at a more concise form. By accurately grouping and combining these terms, we move closer to the final simplified expression.

Step 4: Expressing the Result in the Required Format

Having combined the like terms, our simplified expression is now $\frac{7}{10} x + \frac{5}{2} y + 6$ . The final step is to express this result in the format specified in the options. We have a fractional coefficient for $x$ , a fractional coefficient for $y$ , and a constant term. The coefficient of $x$ is $\frac{7}{10}$ , which is already in its simplest form. The coefficient of $y$ is $\frac{5}{2}$ . We can express this improper fraction as a mixed number. To do this, we divide 5 by 2, which gives us 2 with a remainder of 1. So, $\frac{5}{2}$ is equivalent to $2 \frac{1}{2}$ . The constant term is 6, which is already in the desired format. Therefore, the final simplified expression is $\frac{7}{10} x + 2 \frac{1}{2} y + 6$ . This matches option B in the given choices. Expressing the result in the required format ensures that our solution aligns with the expected answer and demonstrates a thorough understanding of the simplification process. By carefully reviewing the steps and ensuring accuracy in each operation, we arrive at the correct simplified expression.

Final Answer and Conclusion: Presenting the Simplified Expression

After meticulously following the step-by-step simplification process, we have arrived at the final answer. The simplified form of the expression $2(\frac{3}{5} x+2 \frac{3}{4} y-\frac{1}{4} x-1 \frac{1}{2} y+3)$ is $\frac{7}{10} x + 2 \frac{1}{2} y + 6$ . This result was achieved by first converting mixed numbers to improper fractions, then distributing the constant, combining like terms, and finally, expressing the result in the required format. Each step was carefully executed to ensure accuracy and clarity. This exercise demonstrates the importance of understanding and applying the fundamental principles of algebra, such as the distributive property and combining like terms. These skills are essential for simplifying complex expressions and solving algebraic equations. The ability to simplify expressions not only makes mathematical problems more manageable but also enhances our overall understanding of algebraic concepts. In conclusion, the simplified expression $\frac{7}{10} x + 2 \frac{1}{2} y + 6$ is the solution to the given problem, showcasing the power and elegance of algebraic simplification techniques. This comprehensive guide has provided a clear and concise roadmap for tackling similar problems, empowering readers to confidently simplify algebraic expressions.

Final Answer: The final answer is $\frac{7}{10} x+2 \frac{1}{2} y+6$