Solving For D In D/3 - 1.1 = 3 A Step-by-Step Guide

Introduction: Unveiling the Value of 'd'

In the realm of mathematics, solving for variables is a fundamental skill. This article focuses on a specific algebraic equation: $\frac{d}{3}-1.1=3$. Our goal is to isolate 'd' and determine its value. We'll embark on a step-by-step journey, unraveling the equation's structure and applying essential algebraic principles. Understanding how to solve such equations is crucial for various mathematical applications, from basic arithmetic to advanced calculus. This problem serves as a great example to illustrate the core concepts of equation-solving, and by the end of this guide, you'll be equipped with the knowledge to tackle similar challenges with confidence. We will break down each step, providing clear explanations and justifications. This method will empower you to understand not just the how but also the why behind each algebraic manipulation. This comprehensive approach makes the process easier to understand and retain, leading to a stronger grasp of algebra. Let's dive in and discover the value of 'd'!

Understanding the Equation: $\frac{d}{3}-1.1=3$

Before diving into the solution, let's take a closer look at the equation $\frac{d}{3}-1.1=3$. At its core, this is an algebraic equation, expressing a relationship between an unknown variable ('d') and known numbers. The left-hand side, $\frac{d}{3}-1.1$, involves two operations: division by 3 and subtraction of 1.1. The right-hand side is simply the number 3. The equals sign signifies that the expression on the left has the same value as the expression on the right. In essence, we are trying to find the value of 'd' that, when divided by 3 and then reduced by 1.1, results in 3. To solve for 'd', we need to isolate it on one side of the equation. This involves performing inverse operations, carefully undoing the operations that are currently applied to 'd'. Understanding the structure of the equation is vital for choosing the correct steps to solve it. By recognizing the order of operations and the relationships between terms, we can develop a clear strategy for isolating 'd'. This analytical approach forms the foundation of algebraic problem-solving, enabling us to tackle more complex equations in the future. We'll meticulously examine each step in the solution process, ensuring that the reasoning behind each manipulation is clear and logical.

Step 1: Isolating the Term with 'd'

The first step in solving for 'd' in the equation $\frac{d}{3}-1.1=3$ is to isolate the term containing 'd'. In this case, the term is $\frac{d}{3}$. To isolate this term, we need to eliminate the -1.1 that is being subtracted from it. The fundamental principle here is to perform the inverse operation. The inverse of subtraction is addition. Therefore, we add 1.1 to both sides of the equation. This maintains the equality because we are performing the same operation on both sides. Adding 1.1 to both sides, we get: $\frac{d}{3}-1.1 + 1.1 = 3 + 1.1$. On the left side, -1.1 and +1.1 cancel each other out, leaving us with $\frac{d}{3}$. On the right side, 3 + 1.1 equals 4.1. So, the equation now becomes: $\frac{d}{3} = 4.1$. This step is crucial because it brings us closer to isolating 'd'. By adding 1.1 to both sides, we have effectively removed the constant term from the left side, leaving only the term with 'd'. This is a common strategy in solving algebraic equations: isolating the variable by performing inverse operations. The next step will focus on further isolating 'd' by dealing with the division by 3.

Step 2: Solving for 'd' by Undoing Division

Now that we have the equation $\frac{d}{3} = 4.1$, the next step is to isolate 'd' completely. Currently, 'd' is being divided by 3. To undo this division, we perform the inverse operation, which is multiplication. We multiply both sides of the equation by 3. Again, it is crucial to perform the same operation on both sides to maintain the equality. Multiplying both sides by 3, we get: 3 * $\frac{d}{3}$ = 3 * 4.1. On the left side, the multiplication by 3 cancels out the division by 3, leaving us with just 'd'. On the right side, 3 multiplied by 4.1 equals 12.3. So, the equation simplifies to: d = 12.3. This is the solution! We have successfully isolated 'd' and found its value. The process of multiplying both sides by 3 effectively reverses the division operation, allowing us to determine the value of 'd' directly. This step demonstrates the power of inverse operations in solving algebraic equations. By carefully applying these operations, we can systematically isolate the variable we are trying to find. This method is widely applicable to a variety of algebraic problems, making it a fundamental skill in mathematics.

Solution: d = 12.3

After performing the steps of isolating the variable 'd', we have arrived at the solution: d = 12.3. This means that the value of 'd' that satisfies the original equation $\frac{d}{3}-1.1=3$ is 12.3. To confirm our solution, we can substitute this value back into the original equation and check if it holds true. Substituting d = 12.3 into the equation, we get: $\frac{12.3}{3}-1.1=3$. First, we divide 12.3 by 3, which equals 4.1. So, the equation becomes: 4.1 - 1.1 = 3. Then, we subtract 1.1 from 4.1, which equals 3. Therefore, the equation simplifies to: 3 = 3. This confirms that our solution, d = 12.3, is correct. This verification step is an essential part of problem-solving. It provides a way to check our work and ensure that we have arrived at the correct answer. By substituting the solution back into the original equation, we can be confident that we have solved for 'd' accurately. The process of finding the solution and verifying it reinforces our understanding of the equation and the algebraic principles involved.

Conclusion: Mastering Algebraic Equations

In conclusion, we have successfully solved for 'd' in the equation $\frac{d}{3}-1.1=3$, arriving at the solution d = 12.3. This process involved two key steps: first, isolating the term containing 'd' by adding 1.1 to both sides, and second, isolating 'd' itself by multiplying both sides by 3. We also verified our solution by substituting it back into the original equation, confirming its accuracy. This exercise highlights the fundamental principles of solving algebraic equations, particularly the use of inverse operations to isolate the variable of interest. By understanding and applying these principles, you can confidently tackle a wide range of algebraic problems. The ability to solve equations is a cornerstone of mathematics, with applications in various fields, including science, engineering, and economics. This step-by-step guide has demonstrated not only the mechanics of solving the equation but also the underlying logic and reasoning. This deeper understanding will empower you to approach future mathematical challenges with greater confidence and skill. Remember, practice is key to mastering algebra. The more you solve equations, the more intuitive the process will become. So, keep exploring, keep practicing, and keep building your mathematical prowess!