Solving Half A Number Decreased By 8 Equals Zero A Step-by-Step Guide

Introduction: Understanding the Problem

In this article, we will explore the process of solving a mathematical equation that involves finding a number where half of it, when decreased by 8, results in zero. This type of problem falls under the category of basic algebra, where we use equations to represent relationships between numbers and variables. Understanding how to solve such equations is a fundamental skill in mathematics, applicable in various fields, from everyday calculations to more complex scientific and engineering problems. The equation we are tackling is a linear equation, which is characterized by having a variable raised to the power of one. These equations are relatively straightforward to solve, involving basic arithmetic operations such as addition, subtraction, multiplication, and division. The key to solving this specific equation lies in carefully isolating the variable on one side of the equation, revealing its value. This involves reversing the operations performed on the variable, one step at a time, until we arrive at the solution. Let's delve into the step-by-step process of solving this equation, ensuring a clear understanding of each stage involved. This journey will not only provide the answer to the specific problem but also enhance our overall problem-solving skills in mathematics. The ability to manipulate equations and solve for unknowns is a powerful tool that can be applied in many different contexts. Whether it's balancing a budget, calculating a discount, or understanding scientific formulas, the principles of algebra are universally applicable.

Setting Up the Equation: Translating Words into Math

The initial step in tackling any mathematical problem, especially those presented in words, is to translate the given information into a mathematical equation. This process involves identifying the unknowns, assigning variables to them, and then expressing the relationships between these variables using mathematical symbols. In our case, the problem states "half a number decreased by 8 equals zero." The key here is to break down this statement into smaller parts and represent each part mathematically. First, we identify the unknown, which is the "number" we are trying to find. Let's represent this number with the variable 'x'. The phrase "half a number" translates directly to 'x/2' or '1/2 * x', as it represents the number divided by 2. Next, we encounter the phrase "decreased by 8," which indicates a subtraction operation. So, we subtract 8 from 'x/2', resulting in the expression 'x/2 - 8'. Finally, the problem states that this expression "equals zero," which means we set the entire expression equal to 0. This gives us the complete equation: x/2 - 8 = 0. This equation now represents the problem mathematically, allowing us to use algebraic techniques to solve for the unknown variable 'x'. The ability to translate word problems into mathematical equations is a crucial skill in algebra and problem-solving. It requires careful reading and understanding of the relationships between the quantities involved. By breaking down complex statements into smaller, manageable parts, we can effectively represent them using mathematical symbols and equations. This sets the stage for solving the problem and finding the unknown value.

Isolating the Variable: Step-by-Step Solution

Now that we have successfully set up the equation x/2 - 8 = 0, the next step is to isolate the variable 'x' to find its value. Isolating the variable means getting 'x' by itself on one side of the equation. This is achieved by performing a series of operations on both sides of the equation to undo the operations that are currently being applied to 'x'. The order of operations is crucial here. We need to reverse the order in which the operations were applied to 'x'. In our equation, 'x' is first divided by 2, and then 8 is subtracted from the result. Therefore, we need to reverse these operations in the opposite order. The first step is to undo the subtraction of 8. To do this, we add 8 to both sides of the equation. This maintains the equality and helps us move closer to isolating 'x'. Adding 8 to both sides gives us: x/2 - 8 + 8 = 0 + 8, which simplifies to x/2 = 8. Now, 'x' is being divided by 2. To undo this division, we multiply both sides of the equation by 2. This is the inverse operation of division and will effectively isolate 'x'. Multiplying both sides by 2 gives us: (x/2) * 2 = 8 * 2, which simplifies to x = 16. Therefore, the solution to the equation is x = 16. This means that the number we were looking for is 16. By carefully following these steps, we have successfully isolated the variable and found its value. This process of isolating the variable is a fundamental technique in algebra and is used to solve a wide variety of equations.

Verification: Checking the Solution

After finding a solution to an equation, it's crucial to verify the solution to ensure its accuracy. Verification involves substituting the value we found for the variable back into the original equation and checking if the equation holds true. This step helps to identify any errors made during the solving process and confirms that the solution is correct. In our case, we found that x = 16 is the solution to the equation x/2 - 8 = 0. To verify this, we substitute 16 for 'x' in the original equation: (16)/2 - 8 = 0. Now, we simplify the equation following the order of operations. First, we divide 16 by 2, which gives us 8. So, the equation becomes: 8 - 8 = 0. Next, we subtract 8 from 8, which indeed equals 0. Therefore, the equation holds true: 0 = 0. This confirms that our solution, x = 16, is correct. The verification step is an essential part of the problem-solving process. It provides a check and balance, ensuring that the solution we have obtained is accurate and satisfies the given conditions. This practice is particularly important in more complex equations and problems, where the chances of making an error are higher. By verifying our solutions, we can build confidence in our problem-solving abilities and ensure the correctness of our answers. In this case, the verification process has confirmed that 16 is indeed the correct solution to the equation, giving us assurance in our answer.

Conclusion: The Answer and Its Significance

In conclusion, we have successfully solved the equation "half a number decreased by 8 equals zero" by translating it into a mathematical equation, isolating the variable, and verifying the solution. The solution we found is x = 16. This means that the number which, when halved and then decreased by 8, results in zero is 16. This problem serves as a good example of how algebraic equations can be used to represent and solve real-world problems. The process of translating words into mathematical expressions, manipulating equations to isolate variables, and verifying solutions are fundamental skills in mathematics. These skills are not only important for academic pursuits but also have practical applications in various aspects of life, such as finance, engineering, and science. The ability to solve equations allows us to make informed decisions, solve complex problems, and understand the relationships between different quantities. The significance of this problem lies not just in the answer itself but in the process we followed to arrive at the answer. The step-by-step approach, involving setting up the equation, isolating the variable, and verifying the solution, is a valuable framework that can be applied to a wide range of mathematical problems. By mastering these techniques, we can enhance our problem-solving skills and develop a deeper understanding of mathematical concepts. Furthermore, the confidence gained from successfully solving such problems can encourage us to tackle more challenging mathematical tasks in the future. In essence, this simple equation serves as a stepping stone to more advanced mathematical concepts and applications.