Solving For X When H(x) = 0 Given H(x) = -x + 5

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article delves into solving for x when given a function h(x) and a specific value for that function. We will explore the process step-by-step, ensuring a clear understanding of the underlying concepts. Specifically, we will focus on the linear function h(x) = -x + 5 and determine the value of x that makes h(x) equal to 0. This exercise is a cornerstone of algebra, applicable in various fields such as physics, engineering, and economics. Understanding how to solve such equations is crucial for anyone seeking to grasp mathematical principles and their practical applications. The problem we are addressing is a classic example of finding the root or zero of a linear function, a concept that forms the basis for more complex mathematical problems. Mastering this skill provides a strong foundation for tackling advanced topics in algebra and calculus. So, let's embark on this mathematical journey and unravel the solution to this intriguing problem. Remember, the key to success in mathematics is not just memorization, but understanding the logic and reasoning behind each step. This article aims to provide that understanding, making the process of solving equations not just a task, but an enlightening experience.

Understanding the Problem

The problem at hand involves finding the value of x that satisfies the equation h(x) = 0, where h(x) is defined as -x + 5. This type of problem falls under the category of solving linear equations. Linear equations are algebraic expressions where the highest power of the variable (in this case, x) is 1. They are characterized by their straight-line graphs when plotted on a coordinate plane, hence the name 'linear'. To solve this equation, we need to isolate x on one side of the equation. This means performing algebraic operations that maintain the equality while gradually separating x from the other terms. The concept of isolating variables is fundamental in algebra and is used extensively in solving various types of equations. Understanding this concept is key to mastering more complex mathematical problems. In our specific problem, we start with the equation -x + 5 = 0. Our goal is to manipulate this equation using valid algebraic operations to get x by itself on one side. This process involves strategic steps that maintain the balance of the equation. Each step taken must be justified by mathematical principles, ensuring that the solution obtained is accurate and reliable. As we proceed, we will break down each step, explaining the reasoning behind it, and highlighting the mathematical principles at play. This approach aims to provide not just the solution, but also a deep understanding of the process involved.

Step-by-Step Solution

To solve for x when h(x) = 0, given that h(x) = -x + 5, we will follow a step-by-step approach:

  1. Set up the equation: The first step is to set the function h(x) equal to 0. This gives us the equation -x + 5 = 0. This equation represents the core of our problem. It states that the expression -x + 5 must equal zero. Solving this equation will reveal the value of x that satisfies this condition. This initial step is crucial as it translates the problem from a functional notation into a solvable algebraic equation. It's important to understand that setting h(x) to zero is equivalent to finding the point where the graph of the function intersects the x-axis. This point is often referred to as the root or zero of the function. The equation -x + 5 = 0 is a linear equation, meaning it represents a straight line when graphed. The solution to this equation will be the x-coordinate of the point where this line crosses the x-axis. This graphical interpretation provides a visual understanding of the problem and the solution we are seeking.

  2. Isolate the term with x: To isolate the term with x, we need to subtract 5 from both sides of the equation. This maintains the balance of the equation and moves the constant term to the other side. Subtracting 5 from both sides of -x + 5 = 0 gives us -x = -5. This step is based on the fundamental principle of algebraic manipulation: whatever operation is performed on one side of the equation must also be performed on the other side to maintain equality. Subtracting 5 effectively cancels out the +5 on the left side, leaving us with the term containing x. This process is a key step in isolating the variable we are trying to solve for. The equation -x = -5 is now closer to our desired form, where x is isolated. However, we still have a negative sign associated with x, which we need to address in the next step. This step highlights the importance of maintaining balance in an equation while performing operations to isolate the variable. The goal is to gradually simplify the equation until x stands alone on one side.

  3. Solve for x: To solve for x, we can multiply both sides of the equation by -1. This will eliminate the negative signs and give us the value of x. Multiplying both sides of -x = -5 by -1, we get x = 5. This final step completes the process of isolating x. By multiplying both sides by -1, we effectively change the sign of both sides, resulting in a positive x and a positive 5. This operation is valid because multiplying both sides of an equation by the same non-zero number maintains the equality. The solution x = 5 means that when x is 5, the function h(x) equals 0. This is the value we were seeking to find. The solution can be verified by substituting x = 5 back into the original equation h(x) = -x + 5. If the equation holds true, then our solution is correct. In this case, h(5) = -5 + 5 = 0, which confirms our solution. This step-by-step process demonstrates the systematic approach to solving linear equations. By carefully applying algebraic principles, we can isolate the variable and find its value.

Verification

To ensure the accuracy of our solution, we can verify it by substituting the value of x we found (which is 5) back into the original function h(x) = -x + 5. If h(5) equals 0, then our solution is correct. Substituting x = 5 into the function, we get:

h(5) = -(5) + 5

h(5) = -5 + 5

h(5) = 0

Since h(5) indeed equals 0, our solution x = 5 is verified. This verification step is a crucial part of the problem-solving process. It provides a check on our work and ensures that we have not made any errors in our calculations. By substituting the solution back into the original equation, we are essentially reversing the steps we took to solve for x. If the equation holds true, then we can be confident that our solution is correct. This process not only confirms the solution but also reinforces our understanding of the relationship between the function and the value of x that makes it equal to zero. Verification is a practice that should be applied to all mathematical problem-solving, especially in algebra where errors can easily occur during the manipulation of equations. It is a simple yet powerful tool for ensuring the accuracy of our work and building confidence in our mathematical abilities. In this case, the verification clearly shows that when x is 5, the function h(x) equals 0, thus validating our solution.

Conclusion

In conclusion, we have successfully solved for x when h(x) = 0, given the function h(x) = -x + 5. By following a clear and systematic approach, we determined that x = 5 is the solution. This process involved setting up the equation, isolating the term with x, solving for x, and verifying the solution. Each step was crucial in arriving at the correct answer. This exercise demonstrates the fundamental principles of solving linear equations, a skill that is essential in various areas of mathematics and its applications. Understanding how to isolate variables and manipulate equations is key to tackling more complex problems in algebra and beyond. The verification step further reinforces the importance of checking our work to ensure accuracy. By substituting the solution back into the original equation, we confirmed that our answer was indeed correct. This process not only validates the solution but also deepens our understanding of the relationship between the function and its variables. Mastering these basic algebraic techniques lays a strong foundation for further exploration in mathematics. The ability to solve equations is a valuable skill that empowers us to analyze and solve problems in various real-world scenarios. So, continue practicing and applying these principles to enhance your mathematical proficiency. Remember, mathematics is not just about finding the right answer, but also about understanding the process and the reasoning behind each step.