Solving The Equation Z/32 + 5/8 = 13/16 A Step-by-Step Guide
In the realm of mathematics, equations serve as fundamental tools for expressing relationships between variables and constants. Solving equations is a core skill that empowers us to determine the value of unknown variables. This comprehensive guide delves into the process of solving the equation z/32 + 5/8 = 13/16, providing a step-by-step approach that demystifies the solution. This article aims to provide a detailed explanation of how to solve the equation, ensuring a clear understanding for anyone looking to improve their algebra skills. Let’s dive into the step-by-step solution to this algebraic equation, breaking down each part to ensure clarity and understanding.
Understanding the Equation
At its essence, an equation is a mathematical statement that asserts the equality of two expressions. In our case, the equation z/32 + 5/8 = 13/16 presents a relationship between the variable 'z' and numerical constants. The goal is to isolate 'z' on one side of the equation, thereby revealing its value. Before we begin solving, it's important to understand the structure of our equation. We have a variable, z, divided by 32, then we add 5/8, and the result equals 13/16. The key to solving this equation is to isolate z by performing inverse operations.
Breaking Down the Components
- Variable: The variable in this equation is 'z', which represents the unknown value we aim to find.
- Constants: The constants are the numerical values in the equation: 32, 5/8, and 13/16. These values are fixed and do not change.
- Operations: The equation involves two primary operations: division (z/32) and addition (+ 5/8). Understanding these components is crucial for applying the correct steps in solving the equation. Each part plays a critical role in the equation's balance, and manipulating them correctly is essential for finding the correct value of z.
Step 1: Isolate the Term with the Variable
Our initial objective is to isolate the term containing the variable, which is z/32. To achieve this, we need to eliminate the constant term 5/8 from the left side of the equation. The principle we employ here is the concept of inverse operations. Since 5/8 is added to z/32, we perform the inverse operation, which is subtraction. We subtract 5/8 from both sides of the equation to maintain the balance. This is a fundamental rule in algebra: whatever operation you perform on one side of the equation, you must also perform on the other side to keep the equation balanced. This ensures that the equality remains valid throughout the solving process.
Subtracting 5/8 from Both Sides
Subtracting 5/8 from both sides of the equation, we get:
z/32 + 5/8 - 5/8 = 13/16 - 5/8
This simplifies to:
z/32 = 13/16 - 5/8
Now, we have successfully isolated the term with the variable, z/32, on one side of the equation. The next step involves dealing with the fractions on the right side of the equation. This step is crucial because it simplifies the equation and brings us closer to finding the value of z. By performing this subtraction, we ensure that only the term with z remains on the left side, making it easier to isolate z in the following steps. The isolation of this term is a key milestone in solving for z.
Step 2: Simplify the Right Side of the Equation
Now that we have isolated the term z/32, our next step involves simplifying the right side of the equation, which currently reads 13/16 - 5/8. To subtract fractions, they must have a common denominator. In this case, the least common denominator (LCD) for 16 and 8 is 16. Therefore, we need to convert 5/8 into an equivalent fraction with a denominator of 16. This process involves multiplying both the numerator and the denominator of 5/8 by 2. This is a standard procedure in fraction arithmetic and ensures that we can accurately perform the subtraction.
Finding a Common Denominator
Converting 5/8 to an equivalent fraction with a denominator of 16, we multiply both the numerator and the denominator by 2:
5/8 * (2/2) = 10/16
Now, our equation looks like this:
z/32 = 13/16 - 10/16
Subtracting the fractions on the right side, we get:
z/32 = 3/16
Simplifying the right side of the equation is a crucial step as it makes the equation easier to manage and solve. By finding a common denominator and performing the subtraction, we have reduced the complexity of the equation, bringing us closer to isolating z. This step demonstrates the importance of understanding fraction arithmetic in solving algebraic equations. The result, z/32 = 3/16, sets the stage for the final step in solving for z.
Step 3: Solve for z
With the equation simplified to z/32 = 3/16, our final task is to isolate 'z' completely. Currently, 'z' is being divided by 32. To undo this division, we perform the inverse operation, which is multiplication. We multiply both sides of the equation by 32. This ensures that the equation remains balanced, and we can isolate 'z' on one side. This is a fundamental principle in solving algebraic equations: using inverse operations to isolate the variable.
Multiplying Both Sides by 32
Multiplying both sides of the equation by 32, we get:
(z/32) * 32 = (3/16) * 32
On the left side, the multiplication by 32 cancels out the division by 32, leaving us with just 'z'. On the right side, we multiply 3/16 by 32. To simplify this, we can think of 32 as 32/1 and then multiply the fractions:
z = (3/16) * (32/1)
This simplifies to:
z = (3 * 32) / (16 * 1)
z = 96 / 16
Now, we divide 96 by 16:
z = 6
Thus, we have successfully solved for 'z'. The solution is z = 6. This final step demonstrates the power of using inverse operations to isolate the variable and find its value. By multiplying both sides of the equation by 32, we effectively undid the division and revealed the value of z. The solution, z = 6, is the answer to the original equation.
Final Answer
Therefore, the solution to the equation z/32 + 5/8 = 13/16 is z = 6. We arrived at this solution by systematically isolating the variable 'z' through the application of inverse operations and simplification of fractions. Each step in the process was crucial, from isolating the term with the variable to finding a common denominator and performing the necessary arithmetic. This comprehensive approach ensures accuracy and a clear understanding of the solution. Understanding these steps is essential for solving similar algebraic equations. This exercise highlights the importance of a methodical approach in mathematics, where each step builds upon the previous one to reach the final solution. The value z = 6 makes the original equation true, confirming our solution.
Conclusion
Solving algebraic equations is a fundamental skill in mathematics. In this guide, we have demonstrated a step-by-step approach to solving the equation z/32 + 5/8 = 13/16, arriving at the solution z = 6. The key to solving such equations lies in understanding the principles of inverse operations and maintaining the balance of the equation. By isolating the variable, simplifying fractions, and performing the necessary arithmetic, we can confidently solve for the unknown. This process not only provides the solution but also enhances our understanding of algebraic principles. Mastering these skills opens doors to more complex mathematical problems and applications. The ability to solve equations like this is crucial in various fields, from engineering to finance. We hope this guide has provided you with a clear and comprehensive understanding of how to solve this type of equation. Remember, practice is key to mastering these skills, so continue to challenge yourself with similar problems. With consistent effort and a solid understanding of the fundamentals, you can confidently tackle any algebraic equation that comes your way. This step-by-step method can be applied to numerous other equations, making it a valuable skill for anyone studying mathematics or related fields.
