Solving The Equation 1/2 - 1/3(5x - 3) = 13/2 A Step-by-Step Guide

Introduction

In this article, we will embark on a mathematical journey to dissect and understand how Fiona tackled the equation 1/2 - 1/3(5x - 3) = 13/2. This equation, a blend of fractions and algebraic expressions, presents a common challenge in algebra. Our goal is not just to find the solution, but to meticulously break down each step, offering insights and clarity along the way. We will delve into the fundamental principles of algebra, such as the order of operations, distribution, and combining like terms, to provide a comprehensive understanding of the problem-solving process. Whether you're a student grappling with similar equations or a math enthusiast seeking to sharpen your skills, this detailed explanation will serve as a valuable guide. By the end of this exploration, you'll not only grasp the solution but also the underlying logic that makes it all come together.

Unraveling the Equation: Step-by-Step Solution

To begin, let's revisit the equation Fiona aimed to solve: 1/2 - 1/3(5x - 3) = 13/2. The first crucial step in tackling this equation is to address the distribution. The term -1/3 needs to be multiplied across the terms inside the parenthesis, namely 5x and -3. This process is based on the distributive property of multiplication over subtraction. When we multiply -1/3 by 5x, we obtain -5x/3. Similarly, multiplying -1/3 by -3 yields +1. The equation now transforms into a more expanded form: 1/2 - 5x/3 + 1 = 13/2. This expansion is a pivotal moment, as it simplifies the equation and prepares it for further manipulation. The next phase involves consolidating the constant terms on the left side of the equation. We have 1/2 and +1, which can be combined. To do this effectively, we need a common denominator. We can rewrite 1 as 2/2, making it easier to add to 1/2. The sum of 1/2 and 2/2 is 3/2. Thus, our equation now looks like this: 3/2 - 5x/3 = 13/2. This step showcases the importance of understanding fraction arithmetic in solving algebraic equations. By combining the constants, we've streamlined the equation, bringing us closer to isolating the variable x. The initial distribution and the subsequent combination of constants lay the groundwork for the subsequent steps in solving the equation. They demonstrate the fundamental algebraic principle of simplifying expressions to reveal the underlying structure and make the equation more tractable.

Isolating the Variable: A Journey to Find 'x'

Having simplified the equation to 3/2 - 5x/3 = 13/2, our primary objective now shifts to isolating the variable x. This is a fundamental technique in algebra, allowing us to determine the value of the unknown. The first step in this isolation process is to eliminate the constant term on the left side of the equation, which is 3/2. To achieve this, we subtract 3/2 from both sides of the equation. This maintains the balance of the equation, a critical principle in algebraic manipulation. Subtracting 3/2 from both sides gives us -5x/3 = 13/2 - 3/2. On the right side, we now have a simple subtraction of fractions with a common denominator. 13/2 minus 3/2 equals 10/2, which simplifies to 5. Our equation now stands as -5x/3 = 5. This transformation is a significant step forward, as it isolates the term containing x on one side of the equation. The next task is to eliminate the fraction associated with x, which is -5/3. To do this, we multiply both sides of the equation by the reciprocal of -5/3, which is -3/5. Multiplying both sides by -3/5 gives us (-3/5) * (-5x/3) = 5 * (-3/5). On the left side, the fractions cancel out, leaving us with just x. On the right side, we multiply 5 by -3/5. The 5 in the numerator and the 5 in the denominator cancel out, leaving us with -3. Thus, we arrive at the solution: x = -3. This final step demonstrates the power of using inverse operations to isolate and solve for a variable. By systematically eliminating the constants and coefficients surrounding x, we successfully determined its value. The process of isolating the variable involves careful application of algebraic principles, ensuring that each operation maintains the equation's balance and brings us closer to the solution.

Verification: Ensuring the Accuracy of Our Solution

With the solution x = -3 in hand, the critical next step is verification. In mathematics, verifying a solution is akin to proofreading an essay; it ensures accuracy and completeness. To verify our solution, we substitute x = -3 back into the original equation, which was 1/2 - 1/3(5x - 3) = 13/2. This process is not merely a formality; it's a robust check that confirms whether our algebraic manipulations were correct and that the solution holds true for the original equation. Substituting x = -3 into the equation, we get 1/2 - 1/3(5(-3) - 3) = 13/2. The next step is to simplify the expression inside the parenthesis. 5 times -3 equals -15, so the expression becomes 1/2 - 1/3(-15 - 3) = 13/2. Further simplifying inside the parenthesis, -15 - 3 equals -18. Our equation now looks like this: 1/2 - 1/3(-18) = 13/2. The next operation is to multiply -1/3 by -18. A negative times a negative results in a positive, and 1/3 of 18 is 6. So, the equation transforms to 1/2 + 6 = 13/2. Now, we need to add 1/2 and 6. To do this, we convert 6 into a fraction with a denominator of 2. 6 is equivalent to 12/2. Adding 1/2 and 12/2 gives us 13/2. Thus, the left side of the equation simplifies to 13/2. Our equation now reads 13/2 = 13/2, which is a true statement. This equality confirms that our solution, x = -3, is indeed correct. The verification process underscores the importance of precision in algebra. It demonstrates that each step, from the initial distribution to the final simplification, must be executed with care to arrive at the correct solution. By substituting the solution back into the original equation and confirming its validity, we solidify our understanding of the problem-solving process and gain confidence in our mathematical abilities.

Common Pitfalls and How to Avoid Them

Solving equations like 1/2 - 1/3(5x - 3) = 13/2 can be tricky, and it's easy to stumble upon common pitfalls if one isn't careful. Recognizing these potential errors and learning how to avoid them is crucial for mastering algebra. One of the most frequent mistakes occurs during the distribution phase. When multiplying -1/3 by (5x - 3), it's essential to distribute the negative sign correctly. A common error is to multiply -1/3 by 5x but forget to multiply it by -3. This oversight can lead to an incorrect equation and, consequently, a wrong solution. To avoid this, always double-check that the term outside the parenthesis is multiplied by every term inside, paying close attention to signs. Another pitfall lies in the arithmetic of fractions. When adding or subtracting fractions, a common denominator is a must. For instance, in our equation, we needed to combine 1/2 and 1. A mistake here would be to add the numerators without finding a common denominator first. The correct approach is to convert 1 to 2/2 and then add it to 1/2. To prevent such errors, always ensure that fractions have a common denominator before performing addition or subtraction. Sign errors are another significant source of mistakes in algebra. A misplaced negative sign can completely alter the outcome of a problem. For example, when moving terms across the equals sign, remember to change their signs. If you're subtracting a term on one side, you must add it to the other side, and vice versa. Keeping track of signs meticulously is key to avoiding these errors. Verification, as we discussed earlier, is a powerful tool for catching mistakes. By substituting the solution back into the original equation, you can quickly identify whether an error has been made. If the equation doesn't balance, there's likely a mistake in your calculations. In conclusion, while solving algebraic equations, be vigilant about distribution, fraction arithmetic, and sign conventions. Regular practice and careful checking of each step are the best ways to steer clear of these common pitfalls and enhance your problem-solving skills.

Alternative Approaches to Solving the Equation

While we've meticulously dissected Fiona's equation using a step-by-step method, it's worth noting that there are often multiple paths to arrive at the same destination in mathematics. Exploring alternative approaches can deepen our understanding and provide us with a more versatile problem-solving toolkit. One such alternative approach to solving 1/2 - 1/3(5x - 3) = 13/2 involves eliminating the fractions early on in the process. This can simplify the equation and make it easier to work with, especially for those who find fractions cumbersome. To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which in this case are 2 and 3. The LCM of 2 and 3 is 6. Multiplying both sides of the equation by 6, we get 6 * [1/2 - 1/3(5x - 3)] = 6 * [13/2]. Now, we distribute the 6 on both sides. On the left side, 6 * 1/2 equals 3, and 6 * [-1/3(5x - 3)] equals -2(5x - 3). On the right side, 6 * 13/2 equals 39. The equation now transforms into 3 - 2(5x - 3) = 39. This equation, devoid of fractions, might appear less daunting to some. The next step is to distribute the -2 across the terms inside the parenthesis, which gives us 3 - 10x + 6 = 39. Combining like terms on the left side, 3 and 6 add up to 9, so the equation becomes 9 - 10x = 39. Now, we subtract 9 from both sides to isolate the term with x, resulting in -10x = 30. Finally, we divide both sides by -10 to solve for x, which yields x = -3. This alternative approach demonstrates that clearing fractions at the outset can be a viable strategy. It's essential to recognize that different methods may resonate differently with individuals, and having a repertoire of techniques can make problem-solving more adaptable and efficient. Each approach offers a unique perspective on the equation, and mastering multiple methods can enhance one's mathematical intuition and flexibility.

Conclusion

In conclusion, the journey through Fiona's equation, 1/2 - 1/3(5x - 3) = 13/2, has been a comprehensive exploration of algebraic problem-solving. We began by dissecting the equation step-by-step, emphasizing the importance of distribution, combining like terms, and isolating the variable. The solution, x = -3, was not just an end result but a testament to the methodical application of algebraic principles. Verification played a crucial role, underscoring the necessity of ensuring accuracy in mathematical calculations. By substituting the solution back into the original equation, we confirmed its validity, reinforcing our confidence in the problem-solving process. Furthermore, we delved into common pitfalls that often plague students tackling similar equations. Errors in distribution, fraction arithmetic, and sign conventions can lead to incorrect solutions. Recognizing these potential stumbling blocks and implementing strategies to avoid them is essential for mathematical proficiency. We also explored an alternative approach to solving the equation, highlighting the versatility of algebraic techniques. Eliminating fractions early on provided a different perspective and demonstrated that multiple paths can lead to the same correct answer. This adaptability is a hallmark of strong problem-solving skills. Ultimately, mastering algebra is not just about finding solutions; it's about understanding the underlying logic and principles that govern mathematical operations. Fiona's equation served as a microcosm of the broader algebraic landscape, offering valuable lessons in precision, attention to detail, and the power of systematic thinking. As we conclude this exploration, we hope that the detailed breakdown, the emphasis on verification, the awareness of common pitfalls, and the exploration of alternative approaches have equipped you with a deeper understanding of algebra and enhanced your problem-solving capabilities. The journey of mathematical discovery is ongoing, and each equation solved is a step forward in that enriching experience.