Solving 162 = (1/2)(18)h Step-by-Step Guide

Jul 11, 2025 by ADMIN 44 views

Solving for h in the Equation 162 = (1/2)(18)h: A Comprehensive Guide

Introduction

In this detailed guide, we will explore how to solve for the variable h in the equation 162 = (1/2)(18)h. This is a fundamental algebraic problem that involves simplifying the equation and isolating the variable. Understanding how to solve such equations is crucial for various mathematical and real-world applications. We will break down the steps in a clear, step-by-step manner, ensuring that you grasp the underlying concepts and can confidently tackle similar problems in the future. Whether you are a student looking to improve your algebra skills or simply someone interested in refreshing your mathematical knowledge, this guide will provide you with the tools and understanding you need. Let's dive in and unravel the solution to this equation.

Understanding the Equation

Before we delve into the solution, it’s essential to understand the components of the equation 162 = (1/2)(18)h. This equation is a linear equation in one variable, h. Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. The variable in this case is h, which represents an unknown value that we aim to find. The equation states that 162 is equal to half of 18 multiplied by h. The left-hand side (LHS) of the equation is 162, a constant. The right-hand side (RHS) of the equation is (1/2)(18)h, which involves a fraction, a constant, and the variable h. To solve for h, our goal is to isolate h on one side of the equation. This involves performing algebraic operations that maintain the equality of the equation, such as multiplication, division, addition, and subtraction. By understanding the structure and components of the equation, we can approach the problem methodically and arrive at the correct solution. In the following sections, we will break down each step of the solution process, ensuring clarity and comprehension.

Step 1: Simplify the Right-Hand Side

The first step in solving the equation 162 = (1/2)(18)h is to simplify the right-hand side (RHS). The RHS of the equation is (1/2)(18)h, which involves a product of a fraction, a constant, and the variable h. To simplify this expression, we can first multiply the fraction (1/2) by the constant 18. Multiplying 1/2 by 18 is the same as dividing 18 by 2, which results in 9. So, (1/2)(18) equals 9. Now, the RHS of the equation becomes 9h. The equation is now simplified to 162 = 9h. This simplification makes the equation easier to work with, as we have reduced the number of operations involved. By simplifying the RHS, we have taken the first step towards isolating the variable h. This step is crucial because it transforms the equation into a more manageable form, allowing us to proceed with the next steps in the solution process. In the next section, we will focus on isolating h to find its value. Remember, simplifying expressions is a fundamental skill in algebra, and mastering it will help you solve more complex equations with ease.

Step 2: Isolate the Variable h

Now that we have simplified the equation to 162 = 9h, the next step is to isolate the variable h. Isolating a variable means getting it by itself on one side of the equation. In this case, h is being multiplied by 9 on the right-hand side. To isolate h, we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 9. This is a crucial step because it maintains the equality of the equation. Whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation balanced. Dividing both sides of the equation 162 = 9h by 9, we get 162 / 9 = (9h) / 9. On the left-hand side, 162 divided by 9 equals 18. On the right-hand side, 9h divided by 9 simplifies to h. Therefore, the equation becomes 18 = h. This means that the value of h that satisfies the equation is 18. By isolating h, we have successfully solved for the variable. In the next section, we will verify our solution to ensure that it is correct. Remember, isolating variables is a fundamental skill in algebra, and it is essential for solving a wide range of equations.

Step 3: Verify the Solution

After solving for a variable, it is always a good practice to verify the solution. This ensures that the value we found for the variable makes the original equation true. In our case, we found that h equals 18. To verify this solution, we will substitute h = 18 back into the original equation, 162 = (1/2)(18)h. Substituting h = 18, the equation becomes 162 = (1/2)(18)(18). Now, we simplify the right-hand side of the equation. First, we multiply (1/2) by 18, which equals 9. So, the equation becomes 162 = 9(18). Next, we multiply 9 by 18. 9 multiplied by 18 equals 162. Therefore, the equation becomes 162 = 162. This shows that the left-hand side (LHS) of the equation is equal to the right-hand side (RHS) of the equation when h = 18. Since the equation holds true, we have successfully verified our solution. This step is crucial because it confirms the accuracy of our calculations and ensures that we have found the correct value for h. By verifying the solution, we can be confident in our answer and proceed with applying this knowledge to other problems. In the final section, we will summarize the steps we took to solve the equation and highlight the key concepts involved.

Conclusion

In conclusion, we have successfully solved the equation 162 = (1/2)(18)h for the variable h. We began by understanding the structure of the equation, identifying it as a linear equation in one variable. The first step in solving the equation was to simplify the right-hand side, (1/2)(18)h, by multiplying 1/2 by 18, which resulted in 9h. This simplified the equation to 162 = 9h. Next, we isolated the variable h by dividing both sides of the equation by 9. This gave us 162 / 9 = 9h / 9, which simplifies to 18 = h. Therefore, the solution for h is 18. To ensure the accuracy of our solution, we verified it by substituting h = 18 back into the original equation. This resulted in 162 = (1/2)(18)(18), which simplifies to 162 = 162. This confirmed that our solution is correct. The key concepts involved in solving this equation include understanding linear equations, simplifying expressions, isolating variables, and verifying solutions. These are fundamental skills in algebra that are essential for solving a wide range of mathematical problems. By mastering these concepts, you can confidently tackle similar equations and apply them to real-world scenarios. We hope this comprehensive guide has provided you with a clear understanding of how to solve this type of equation and has enhanced your problem-solving skills in mathematics.