Solving For X When H(x) = 0 Detailed Explanation

Jul 8, 2025 by qnaunigon 49 views

Solving for x When h(x) = 0: A Comprehensive Guide

In mathematics, solving for variables is a fundamental skill. It allows us to understand the relationships between different quantities and find solutions to equations. In this article, we will delve into the process of solving for x when given a function h(x) = -x + 5 and the condition h(x) = 0. This is a common problem in algebra and is a crucial step in understanding the behavior of linear functions. We will break down the steps involved, explain the underlying concepts, and provide examples to solidify your understanding. Whether you're a student learning algebra for the first time or someone looking to brush up on your skills, this guide will provide you with a clear and concise explanation of how to solve this type of problem.

Understanding the Problem

Before diving into the solution, it’s crucial to understand what the problem is asking. We are given a function, h(x) = -x + 5. This function defines a relationship between the input x and the output h(x). The function tells us that for any value of x, we can find the corresponding value of h(x) by multiplying x by -1 and adding 5. The problem then asks us to find the value of x that makes h(x) equal to 0. In other words, we need to find the x-value that, when plugged into the function, results in an output of 0. This is a fundamental concept in solving equations and understanding the behavior of functions. It is often referred to as finding the root or zero of the function. Understanding the problem is the first and most important step in finding the solution. Without a clear understanding of what is being asked, it is difficult to proceed with the correct steps. So, before we jump into the algebraic manipulation, let's make sure we have a solid grasp of the problem statement. We are looking for the specific x value that satisfies the equation h(x) = 0, given the function h(x) = -x + 5. This means we are looking for the x-intercept of the line represented by the function. This is a core concept in algebra and will be used extensively in more advanced mathematical topics.

Step-by-Step Solution

Now that we understand the problem, let's walk through the step-by-step solution. This process involves using algebraic manipulation to isolate the variable x and find its value. Here are the steps involved:

Set up the equation: We start by setting the function h(x) equal to 0. This gives us the equation:
```
-x + 5 = 0
```
This equation represents the condition we need to satisfy. We are looking for the value of x that makes this equation true. Setting up the equation correctly is the foundation for solving the problem. It's important to ensure that you have accurately transcribed the function and the desired output. In this case, we have correctly set the expression for h(x) equal to 0, which is the target value we are trying to achieve.
Isolate the term with x: Our goal is to isolate x on one side of the equation. To do this, we can subtract 5 from both sides of the equation:
```
-x + 5 - 5 = 0 - 5
-x = -5
```
This step is based on the fundamental principle of equality: we can perform the same operation on both sides of an equation without changing its solution. Subtracting 5 from both sides cancels out the +5 on the left side, leaving us with just the term involving x. This is a critical step in isolating x and getting closer to the solution.
Solve for x: We now have -x = -5. To solve for x, we need to get rid of the negative sign. We can do this by multiplying both sides of the equation by -1:
```
(-1) * (-x) = (-1) * (-5)
x = 5
```
Multiplying by -1 changes the sign of both sides of the equation. This allows us to isolate x and find its value. This step completes the algebraic manipulation and gives us the solution to the problem. It's a simple but crucial step in solving for x.

Therefore, the solution to the equation h(x) = 0 when h(x) = -x + 5 is x = 5. This means that when we plug 5 into the function h(x), the output will be 0. This is the value of x where the line represented by the function intersects the x-axis.

Verifying the Solution

It’s always a good practice to verify your solution. This helps to ensure that you haven't made any mistakes in your calculations. To verify our solution, we can substitute x = 5 back into the original function h(x) = -x + 5 and see if we get 0:

h(5) = -(5) + 5
h(5) = -5 + 5
h(5) = 0

Since h(5) = 0, our solution is correct. Verification is a critical step in problem-solving. It helps to catch any errors that may have occurred during the process. By substituting the solution back into the original equation, we can confirm that it satisfies the given condition. This step provides confidence in the correctness of the answer and reinforces the understanding of the problem.

Graphical Interpretation

Understanding the graphical interpretation of the solution can provide further insight. The function h(x) = -x + 5 represents a straight line. The solution x = 5 represents the point where this line intersects the x-axis. The x-axis is the line where y = 0, and in our case, h(x) represents y. So, the solution x = 5 is the x-coordinate of the point where the line crosses the x-axis. This point is also known as the x-intercept of the line. Visualizing the graph of the function can help to understand the meaning of the solution. The graph of h(x) = -x + 5 is a line with a slope of -1 and a y-intercept of 5. The point where this line crosses the x-axis is at x = 5, which confirms our algebraic solution. This graphical representation provides a visual confirmation of the solution and helps to build a deeper understanding of the relationship between the function and its roots. It connects the algebraic solution to a geometric interpretation, making the concept more concrete.

Real-World Applications

While solving for x when h(x) = 0 might seem like an abstract mathematical exercise, it has numerous real-world applications. Understanding how to find the zeros of a function is crucial in many fields, including:

Physics: Finding the time when a projectile hits the ground involves solving for the zeros of a quadratic function that represents the projectile's trajectory.
Engineering: Determining the equilibrium points of a system often requires finding the roots of a function.
Economics: Analyzing supply and demand curves involves finding the point where they intersect, which is essentially solving for the zeros of the difference between the two functions.
Computer Science: In algorithm design, finding the roots of a function can be used to optimize performance or find stable states.

The ability to solve for the zeros of a function is a fundamental skill that is applied in a wide range of disciplines. It's a powerful tool for analyzing and understanding real-world phenomena. This simple example of solving for x when h(x) = 0 is a building block for more complex mathematical modeling and problem-solving in various fields.

Conclusion

In this article, we have explored the process of solving for x when h(x) = 0, given the function h(x) = -x + 5. We walked through the step-by-step solution, verified the result, discussed the graphical interpretation, and highlighted real-world applications. This seemingly simple problem illustrates fundamental algebraic principles that are essential for further mathematical studies and problem-solving in various fields. The ability to isolate variables, understand the relationship between functions and their graphs, and apply these concepts to real-world situations is crucial for success in mathematics and beyond. By mastering these basic skills, you can build a solid foundation for more advanced mathematical concepts and problem-solving techniques. Remember, practice is key to mastering these skills. Work through various examples and try applying these concepts to different problems to solidify your understanding. The more you practice, the more confident and proficient you will become in solving algebraic equations and understanding the behavior of functions.