Rahul's Equation Solving Identifying The Addition Property Of Equality

Introduction

In this article, we will delve into the step-by-step solution of an equation provided by Rahul and pinpoint exactly where he applied the addition property of equality. This property is a fundamental concept in algebra, allowing us to manipulate equations while maintaining their balance. By understanding this principle, we can solve a wide range of algebraic problems with confidence. Our main focus will be on analyzing each step of Rahul's solution to identify the precise moment the addition property was used. This involves examining how terms were moved across the equals sign and how operations were performed on both sides of the equation to isolate the variable x. In addition to identifying the specific step, we will also discuss the significance of the addition property in the broader context of equation solving. This includes how it relates to other properties of equality and how it contributes to the overall strategy of isolating the variable. Understanding the addition property is crucial for anyone studying algebra, as it forms the basis for many more advanced algebraic techniques. So, let's embark on this equation-solving journey and uncover the step where Rahul skillfully employed the addition property of equality to progress towards his solution.

The Problem: Solving the Equation

The equation Rahul solved is:

$2\left(x-\frac{1}{8}\right)-\frac{3}{5} x=\frac{55}{4}$.

To determine where Rahul used the addition property of equality, we need to meticulously analyze each step of his solution. The addition property of equality states that if you add the same number to both sides of an equation, the equation remains true. This property is crucial for isolating the variable and finding its value. By carefully examining each transformation in Rahul's solution, we can pinpoint exactly where he strategically applied this property. This involves looking for steps where a term was added to both sides of the equation, effectively moving it from one side to the other while maintaining the balance. Understanding how and when to use the addition property is a cornerstone of algebraic problem-solving, and by dissecting Rahul's method, we can gain a deeper appreciation for its practical application. Furthermore, identifying the use of the addition property in this specific context will help us reinforce our understanding of the broader principles of equation manipulation. This includes recognizing how the addition property works in conjunction with other properties, such as the distributive property and the multiplication property of equality, to achieve the ultimate goal of solving for the unknown variable. Therefore, our analysis will not only focus on the specific step but also on the underlying reasoning and strategies involved in using the addition property effectively.

Rahul's Solution: A Step-by-Step Analysis

Let's break down Rahul's solution step-by-step:

Now, let's identify the step where the addition property of equality was used. To effectively pinpoint the application of the addition property, we need to carefully compare each step with its predecessor. The addition property, in essence, allows us to move terms from one side of the equation to the other by adding their additive inverse to both sides. This maintains the equation's balance while simplifying it. In the context of Rahul's solution, we are looking for a step where a constant or a term involving x is effectively transferred from one side to the other through the act of addition. This means we need to examine how terms are grouped and rearranged in each step. By scrutinizing the transitions between the equations, we can identify the precise moment when the addition property of equality was strategically employed to advance the solution process. This analysis will not only highlight the specific step but also underscore the importance of understanding and applying this fundamental property in solving algebraic equations. Therefore, our detailed examination will provide a clear understanding of how the addition property contributes to the overall solution strategy.

Identifying the Addition Property of Equality Step

Looking at the steps, we can see that in Step 4, the term $-\frac{1}{4}$ was moved from the left side of the equation to the right side. This was achieved by adding $\frac{1}{4}$ to both sides of the equation.

This is a clear application of the addition property of equality. The addition property of equality is a fundamental principle in algebra that allows us to manipulate equations while preserving their balance. It states that if we add the same value to both sides of an equation, the equation remains true. This property is essential for isolating variables and solving for their values. In the context of Rahul's solution, the strategic use of the addition property in Step 4 enabled him to consolidate constant terms on one side of the equation, bringing him closer to the solution. By adding $\frac{1}{4}$ to both sides, Rahul effectively neutralized the $-\frac{1}{4}$ term on the left side, transferring its equivalent to the right side as $+\frac{1}{4}$. This maneuver is a classic example of how the addition property is used to rearrange and simplify equations. Recognizing this step as an application of the addition property reinforces our understanding of its importance in the broader landscape of algebraic problem-solving. Furthermore, it highlights the need for a clear grasp of the properties of equality to effectively solve equations and tackle more complex mathematical challenges. Therefore, identifying Step 4 as the point where the addition property was used is crucial for understanding Rahul's solution process and appreciating the role of this fundamental property in algebra.

Conclusion

Therefore, Rahul used the addition property of equality in Step 4 of his solution. This step was crucial in isolating the terms with x on one side and the constants on the other, paving the way for the final solution. Understanding the properties of equality is essential for solving algebraic equations effectively. The addition property of equality is just one piece of the puzzle, but it plays a vital role in simplifying and manipulating equations. By mastering this property, along with others such as the subtraction, multiplication, and division properties of equality, students can confidently tackle a wide range of algebraic problems. The ability to strategically apply these properties is what transforms a seemingly complex equation into a manageable and solvable one. In the context of Rahul's solution, the addition property was instrumental in streamlining the equation, allowing for the combination of like terms and the eventual isolation of the variable x. This highlights the importance of not only knowing the properties but also understanding when and how to apply them effectively. Therefore, recognizing the addition property in action, as we did in Step 4, is a key step in developing a robust understanding of algebraic equation solving and building a strong foundation for further mathematical studies.