Solve The Equation 74.23 = (73.95 + X) / 2 For X.

Introduction

In this article, we will explore how to solve for x in the equation 74.23 = (73.95 + x) / 2. This is a fundamental algebraic problem that involves isolating the variable x to find its value. Understanding how to solve such equations is crucial for various mathematical and real-world applications. Whether you're a student learning algebra or someone looking to refresh your math skills, this comprehensive guide will provide you with a step-by-step approach to solving this equation. We will break down each step in detail, ensuring clarity and understanding. By the end of this article, you will not only be able to solve this particular equation but also gain confidence in tackling similar algebraic problems. Solving for variables is a cornerstone of algebra, and mastering this skill can open doors to more advanced mathematical concepts and problem-solving scenarios. The ability to manipulate equations and isolate variables is essential in fields such as physics, engineering, economics, and computer science. Therefore, a solid understanding of these principles is invaluable for academic and professional success. In the following sections, we will delve into the specific steps required to solve the equation 74.23 = (73.95 + x) / 2, providing clear explanations and examples along the way. We will also discuss common mistakes to avoid and offer tips for verifying your solution to ensure accuracy. So, let's embark on this mathematical journey and unravel the value of x together.

Step 1: Understanding the Equation

Before we dive into solving the equation, it's essential to understand its structure. The equation 74.23 = (73.95 + x) / 2 is a linear equation in one variable, x. Linear equations are characterized by having a variable raised to the first power, and they can be represented graphically as a straight line. In this particular equation, x is the unknown value we need to determine. The equation states that the quantity (73.95 + x), when divided by 2, equals 74.23. To solve for x, our goal is to isolate x on one side of the equation. This means we need to perform algebraic operations to eliminate any other terms or coefficients affecting x. The order of operations is crucial in solving equations. We must follow the reverse order of operations (PEMDAS/BODMAS) to correctly isolate the variable. This means we address any addition or subtraction operations before multiplication or division. Understanding the structure of the equation and the principles of algebraic manipulation is the foundation for solving it. In this step, we are setting the stage for the subsequent steps by recognizing the type of equation we are dealing with and the overall strategy for finding the value of x. The ability to identify the type of equation and the steps needed to solve it is a fundamental skill in algebra and will serve as a valuable tool in your mathematical journey. Let's move on to the next step where we will begin the process of isolating x by applying algebraic operations.

Step 2: Multiplying Both Sides by 2

The first step in isolating x is to eliminate the division by 2. To do this, we multiply both sides of the equation by 2. This operation maintains the equality of the equation because we are performing the same operation on both sides. The equation is 74.23 = (73.95 + x) / 2. Multiplying both sides by 2 gives us: 2 * 74.23 = 2 * [(73.95 + x) / 2]. On the left side, 2 multiplied by 74.23 equals 148.46. On the right side, the multiplication by 2 cancels out the division by 2, leaving us with 73.95 + x. So, our equation now becomes 148.46 = 73.95 + x. This step is crucial because it simplifies the equation by removing the fraction, making it easier to isolate x. By multiplying both sides by the denominator, we effectively undo the division operation. This is a common technique used in solving algebraic equations. It's important to remember that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain balance. This principle is fundamental to solving equations and ensures that the solution remains valid. In this step, we have successfully eliminated the division and simplified the equation. Now, we are closer to isolating x and finding its value. Let's proceed to the next step where we will further isolate x by removing the addition term.

Step 3: Subtracting 73.95 from Both Sides

Now that we have the equation 148.46 = 73.95 + x, the next step is to isolate x by removing the addition of 73.95. To do this, we subtract 73.95 from both sides of the equation. This operation maintains the equality, just like in the previous step. Subtracting 73.95 from both sides gives us: 148.46 - 73.95 = 73.95 + x - 73.95. On the left side, 148.46 minus 73.95 equals 74.51. On the right side, the addition and subtraction of 73.95 cancel each other out, leaving us with x. So, our equation now simplifies to 74.51 = x. This step is a critical part of the process because it directly isolates x, allowing us to determine its value. By subtracting the constant term from both sides, we effectively undo the addition operation and bring x to one side of the equation. It's essential to perform this step accurately to arrive at the correct solution. Subtraction is the inverse operation of addition, and by applying it appropriately, we can isolate the variable and solve for its value. In this step, we have successfully isolated x and found its value. Now, we have a potential solution for the equation. Let's move on to the next step where we will verify our solution to ensure its accuracy.

Step 4: Verifying the Solution

After solving for x, it's crucial to verify the solution to ensure it is correct. This step helps to catch any potential errors made during the solving process and provides confidence in the final answer. We found that x = 74.51. To verify this solution, we substitute this value back into the original equation: 74.23 = (73.95 + x) / 2. Substituting x = 74.51 into the equation, we get: 74.23 = (73.95 + 74.51) / 2. Now, we simplify the right side of the equation. First, we add 73.95 and 74.51, which equals 148.46. So, the equation becomes: 74.23 = 148.46 / 2. Next, we divide 148.46 by 2, which equals 74.23. So, the equation further simplifies to: 74.23 = 74.23. Since both sides of the equation are equal, our solution x = 74.51 is verified and correct. This verification step is a vital part of the problem-solving process. It ensures that the solution satisfies the original equation and that no errors were made during the algebraic manipulations. By substituting the solution back into the equation and confirming the equality, we can be confident in the accuracy of our answer. In this step, we have successfully verified our solution and confirmed that x = 74.51 is the correct value. This completes the process of solving for x in the equation. Let's summarize the steps we took to arrive at this solution.

Conclusion

In conclusion, we have successfully solved for x in the equation 74.23 = (73.95 + x) / 2. We followed a step-by-step approach, which included understanding the equation, multiplying both sides by 2, subtracting 73.95 from both sides, and finally, verifying the solution. Through these steps, we determined that x = 74.51. Solving algebraic equations is a fundamental skill in mathematics, and the process we followed in this article can be applied to a wide range of similar problems. By understanding the principles of algebraic manipulation and the importance of each step, you can confidently tackle equations and find solutions. Remember to always verify your solution to ensure accuracy and to catch any potential errors. The ability to solve equations is not only essential in academic settings but also in various real-world applications. Whether you're calculating finances, designing structures, or analyzing data, the skills you develop in algebra will serve you well. We hope this comprehensive guide has provided you with a clear understanding of how to solve for x in this equation and has enhanced your problem-solving abilities. Keep practicing and exploring different types of equations to further develop your mathematical skills. The journey of learning mathematics is ongoing, and each problem you solve contributes to your growth and understanding. So, continue to challenge yourself, and you will find that mathematics becomes an increasingly valuable tool in your life.