Solving The Equation 15 = (1/2)(x + 10) A Step-by-Step Guide

Introduction

In the realm of mathematics, equations serve as fundamental tools for problem-solving and understanding relationships between variables. This comprehensive guide delves into the equation $15=\frac{1}{2}(x+10)$, providing a step-by-step solution and exploring the underlying mathematical principles. Our focus will be on solving the equation, understanding the order of operations, and applying the distributive property. We'll also discuss how to verify the solution and explore real-world applications of such equations. By the end of this guide, you will have a solid understanding of how to tackle similar algebraic problems and appreciate their significance in various fields. This detailed exploration aims to enhance your mathematical skills and confidence in equation solving. Let's embark on this journey to decode the equation and reveal its solution.

Breaking Down the Equation: Understanding the Components

Before we dive into the solution, let's dissect the equation $15=\frac{1}{2}(x+10)$ and understand its components. The equation involves a variable, constants, and mathematical operations. The variable, denoted by 'x', represents an unknown value we aim to find. Constants are numerical values, such as 15 and 10, that remain fixed. The mathematical operations include addition, multiplication, and division. The fraction ${\frac{1}{2}}$ represents division by 2, and the parentheses indicate that the addition operation should be performed before multiplication. Understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for solving equations correctly. This order dictates the sequence in which operations should be performed to arrive at the correct answer. In our equation, we first address the expression within the parentheses, then handle the multiplication by ${\frac{1}{2}}$, and finally, isolate the variable 'x' to find its value. By carefully analyzing each component and the operations involved, we set the stage for a systematic approach to solving the equation. Recognizing the interplay between variables, constants, and operations is key to mastering algebraic problem-solving. Let's now move on to the first step in isolating the variable.

Step-by-Step Solution: Isolating the Variable

To solve the equation $15 = \frac{1}{2}(x + 10)$, we need to isolate the variable 'x'. This involves a series of algebraic manipulations that maintain the equality of the equation. Our first step is to eliminate the fraction by multiplying both sides of the equation by 2. This gives us: 2 * 15 = 2 * ${\frac{1}{2}}$(x + 10), which simplifies to 30 = x + 10. Now, we have a simpler equation to work with. The next step is to isolate 'x' by subtracting 10 from both sides of the equation. This gives us: 30 - 10 = x + 10 - 10, which simplifies to 20 = x. Therefore, the solution to the equation is x = 20. Each step in this process is crucial for maintaining the balance of the equation and arriving at the correct solution. By carefully applying algebraic principles, we have successfully isolated the variable and determined its value. This step-by-step approach is fundamental to equation solving and can be applied to a wide range of algebraic problems. Let's now verify our solution to ensure its accuracy.

Verification: Ensuring the Accuracy of the Solution

After solving the equation $15 = \frac{1}{2}(x + 10)$ and finding x = 20, it's crucial to verify our solution to ensure its accuracy. Verification involves substituting the value we found for 'x' back into the original equation and checking if the equation holds true. Substituting x = 20 into the equation, we get: 15 = ${\frac{1}{2}}$(20 + 10). Simplifying the expression inside the parentheses, we have: 15 = ${\frac{1}{2}}$(30). Now, multiplying ${\frac{1}{2}}$ by 30, we get: 15 = 15. Since the left side of the equation equals the right side, our solution x = 20 is indeed correct. This process of verification is an essential step in equation solving, as it helps to identify any potential errors made during the solution process. By confirming the accuracy of our solution, we can be confident in our answer and the methods we used to obtain it. Verification not only ensures the correctness of the solution but also reinforces our understanding of the equation and the algebraic principles involved. Now that we've verified our solution, let's explore some real-world applications of this type of equation.

Real-World Applications: Connecting Equations to Everyday Scenarios

Equations like $15 = \frac{1}{2}(x + 10)$ might seem abstract, but they have numerous real-world applications. These equations can be used to model various situations, from calculating financial investments to determining measurements in construction projects. For instance, imagine a scenario where a store is offering a 50% discount on an item, and the discounted price is $15. If the original price is represented by 'x', and an additional $10 is added to the price for some reason (like a processing fee), the equation can help determine the original price of the item. Another application could be in geometry, where 'x' might represent the length of a side of a figure, and the equation helps to find the unknown length based on given conditions. In finance, such equations can be used to calculate simple interest or to determine the principal amount needed to achieve a certain return. The ability to translate real-world problems into mathematical equations and solve for unknown variables is a crucial skill in many fields. By understanding how equations model real-world scenarios, we can appreciate their practical significance and develop our problem-solving abilities. Let's consider another example in physics, where similar equations might be used to calculate distances or velocities. The possibilities are vast, making the mastery of equation solving a valuable asset.

Advanced Techniques: Exploring Alternative Solution Methods

While we've solved the equation $15 = \frac{1}{2}(x + 10)$ using a straightforward algebraic approach, it's beneficial to explore alternative solution methods. Understanding different techniques can enhance our problem-solving skills and provide flexibility in tackling more complex equations. One alternative method is to distribute the ${\frac{1}{2}}$ across the parentheses before isolating the variable. This would involve multiplying ${\frac{1}{2}}$ by both 'x' and 10, resulting in the equation 15 = ${\frac{1}{2}}$x + 5. From here, we can subtract 5 from both sides to get 10 = ${\frac{1}{2}}$x, and then multiply both sides by 2 to find x = 20. This method, while slightly different, arrives at the same solution and demonstrates the distributive property in action. Another approach might involve rearranging the equation in a different order, but the fundamental principles of maintaining equality remain the same. By exploring these alternative methods, we gain a deeper understanding of the equation's structure and the flexibility of algebraic manipulations. This knowledge is particularly useful when dealing with more complex equations that may require creative problem-solving strategies. The ability to approach a problem from different angles is a hallmark of a skilled mathematician. Let's now summarize the key concepts and takeaways from this guide.

Conclusion: Key Takeaways and Further Exploration

In this comprehensive guide, we've meticulously dissected the equation $15 = \frac{1}{2}(x + 10)$, providing a step-by-step solution, verification, real-world applications, and exploring alternative solution methods. The key takeaway is the importance of understanding the order of operations, applying algebraic principles to isolate the variable, and verifying the solution to ensure accuracy. We've also seen how this seemingly simple equation can model various real-world scenarios, highlighting the practical significance of equation solving. Furthermore, exploring alternative solution methods enhances our problem-solving skills and provides flexibility in tackling more complex equations. To further your exploration, consider practicing similar equations with varying levels of complexity. You can also delve into more advanced algebraic concepts, such as systems of equations or quadratic equations. Numerous online resources and textbooks offer a wealth of information and practice problems to expand your mathematical knowledge. Remember, the key to mastering algebra is consistent practice and a solid understanding of the fundamental principles. By continuing to explore and practice, you'll develop the skills and confidence to tackle a wide range of mathematical challenges. This journey into equation solving is just the beginning, and the possibilities for further learning are endless.