Rahul's Equation Solution Identifying The Addition Property Of Equality
In which step of Rahul's solution to the equation 2(x - 1/8) - (3/5)x = 55/4 did he use the addition property of equality?
In the realm of mathematics, solving equations is a fundamental skill. Equations are the language of science, engineering, and economics, and the ability to manipulate and solve them is crucial for understanding and modeling the world around us. This article delves into the step-by-step solution of a linear equation, highlighting the application of the addition property of equality. Understanding how equations are solved not only strengthens mathematical proficiency but also enhances problem-solving abilities in various fields.
We will dissect Rahul's solution to the equation 2(x - 1/8) - (3/5)x = 55/4, meticulously examining each step to pinpoint where the addition property of equality was employed. By breaking down the process, we aim to provide a clear and comprehensive understanding of the techniques used in solving algebraic equations. This detailed analysis will be beneficial for students, educators, and anyone interested in reinforcing their algebraic skills.
The problem at hand is to solve the linear equation 2(x - 1/8) - (3/5)x = 55/4. Linear equations are algebraic expressions in which the highest power of the variable is one. They are the simplest type of equation to solve and form the basis for more complex mathematical models. In this specific equation, we are tasked with finding the value of 'x' that satisfies the equality.
The equation presents a combination of arithmetic operations, including multiplication, subtraction, and fractions, which requires a systematic approach to solve. Rahul's solution provides a structured method for isolating the variable 'x' and determining its value. By examining each step, we can appreciate the logical progression and the properties of equality that underpin the solution process. Understanding the structure of the equation and the goal of isolating 'x' is the first step in unraveling the problem.
Rahul's solution to the equation 2(x - 1/8) - (3/5)x = 55/4 is presented in a step-by-step format, allowing for a clear understanding of the process. Each step involves specific algebraic manipulations aimed at isolating the variable 'x'. Let's dissect each step to understand the underlying principles and identify the use of the addition property of equality.
To effectively analyze Rahul's solution, we need to consider the initial equation and the transformations applied in each subsequent step. This involves tracing the changes made to both sides of the equation and identifying the properties of equality that justify these changes. By carefully examining the transitions between steps, we can pinpoint the exact moment when the addition property of equality is employed.
The addition property of equality is a fundamental principle in algebra. It states that if you add the same value to both sides of an equation, the equation remains balanced. This property is crucial for manipulating equations and isolating the variable. In simpler terms, if a = b, then a + c = b + c.
The addition property of equality allows us to strategically add terms to both sides of an equation to eliminate unwanted terms and simplify the expression. This is particularly useful when dealing with equations involving subtraction or negative terms. By adding the appropriate value, we can effectively move terms from one side of the equation to the other while maintaining the equality. Identifying where this property is used in Rahul's solution is key to understanding his approach.
The critical part of our analysis involves identifying the exact step in Rahul's solution where the addition property of equality is applied. This requires careful comparison between consecutive steps. We need to look for instances where a term has been added to both sides of the equation to eliminate another term or simplify the expression.
To pinpoint the step, we will examine each transition in Rahul's solution, comparing the equations before and after the step. This comparative analysis will help us isolate the specific instance where the addition property was used. The identification of this step is the crux of the problem and demonstrates the practical application of a core algebraic principle.
To accurately determine the step where Rahul used the addition property of equality, we need a detailed breakdown of each step in his solution. Let's assume Rahul's solution unfolds as follows (note: since the actual steps are missing, this is a hypothetical solution for illustrative purposes):
- Original Equation: 2(x - 1/8) - (3/5)x = 55/4
- Step 1: Distribute the 2: 2x - 1/4 - (3/5)x = 55/4
- In this step, the distributive property is applied, not the addition property of equality.
- Step 2: Combine like terms: (10/5)x - (3/5)x - 1/4 = 55/4 => (7/5)x - 1/4 = 55/4
- This step involves combining like terms, not the addition property of equality.
- Step 3: Add 1/4 to both sides: (7/5)x - 1/4 + 1/4 = 55/4 + 1/4 => (7/5)x = 56/4
- Here, the addition property of equality is used. Adding 1/4 to both sides eliminates the -1/4 term on the left side.
- Step 4: Simplify: (7/5)x = 14
- This step simplifies the fraction on the right side.
- Step 5: Multiply both sides by 5/7: (5/7)(7/5)x = 14(5/7) => x = 10
- This step uses the multiplication property of equality to isolate x.
Therefore, in this hypothetical solution, the addition property of equality is used in Step 3. By adding 1/4 to both sides of the equation, Rahul maintained the balance while progressing towards isolating the variable 'x'. This detailed breakdown exemplifies how each step can be analyzed to pinpoint the application of specific algebraic properties.
The addition property of equality is not merely a theoretical concept; it is a practical tool used extensively in various real-world applications. This fundamental principle underpins countless calculations in fields ranging from engineering to economics.
In engineering, for example, structural engineers use equations to calculate the forces acting on a bridge or building. The addition property of equality is essential for manipulating these equations to determine unknown variables, such as the tension in a cable or the stress on a beam. Similarly, electrical engineers use it to analyze circuits and determine current, voltage, and resistance. The ability to balance equations by adding the same value to both sides is crucial for ensuring the safety and stability of structures and electrical systems.
In economics, the addition property of equality is used in supply and demand analysis, cost-benefit calculations, and macroeconomic modeling. Economists often need to solve equations to predict market trends, analyze the impact of government policies, or assess the profitability of investments. The addition property allows them to rearrange equations and isolate key variables, providing valuable insights for decision-making.
Beyond these technical fields, the addition property of equality is also used in everyday situations. For instance, when balancing a checkbook, you are essentially applying this principle to ensure that the debits and credits match. Similarly, when calculating expenses or budgeting for a trip, you are using the addition property to keep track of your finances and make informed decisions. The principle of maintaining balance by adding equal amounts is a cornerstone of financial management.
The ubiquity of the addition property of equality underscores its importance as a foundational concept in mathematics and its practical relevance in numerous disciplines. Whether it's designing a bridge, forecasting economic trends, or managing personal finances, this principle provides a powerful tool for solving problems and making informed decisions. Its applications highlight the interconnectedness of mathematics and the real world.
While the addition property of equality is straightforward, there are common mistakes that students and practitioners sometimes make. Recognizing these pitfalls can help in avoiding errors and ensuring accurate solutions.
One of the most frequent mistakes is forgetting to add the same value to both sides of the equation. The addition property of equality hinges on maintaining balance; if a value is added to one side but not the other, the equation becomes invalid. This can lead to incorrect solutions and flawed analyses. It's crucial to always perform the same operation on both sides to preserve the equality.
Another common mistake is incorrectly applying the addition property in conjunction with other operations. For instance, students might add a term before properly distributing or combining like terms. It's essential to follow the order of operations (PEMDAS/BODMAS) and ensure that the addition property is applied at the appropriate stage in the solution process. Failing to do so can complicate the equation and make it harder to solve.
Furthermore, misunderstanding the concept of inverse operations can lead to errors. The addition property is often used to eliminate terms by adding their additive inverse. For example, to remove a term that is being subtracted, you add its positive counterpart. Confusing inverse operations can result in the wrong term being added, hindering the solution process.
To avoid these mistakes, it's important to practice solving equations regularly and to check your work carefully. Pay close attention to each step, ensuring that the addition property is applied correctly and that the equation remains balanced. Seeking feedback from teachers or peers can also help identify and correct errors. By being mindful of these common pitfalls, you can strengthen your understanding of the addition property of equality and improve your problem-solving skills.
In conclusion, the addition property of equality is a cornerstone of algebraic problem-solving. Understanding and correctly applying this property is essential for solving equations and manipulating mathematical expressions. By carefully analyzing Rahul's solution, we can pinpoint the exact step where this property was used, gaining a deeper appreciation for its role in isolating variables and finding solutions.
The addition property of equality is not just a mathematical concept; it's a tool that empowers us to solve problems in various contexts. From engineering and economics to everyday financial management, the ability to maintain balance by adding equal amounts is invaluable. By avoiding common mistakes and practicing regularly, we can master this property and enhance our mathematical proficiency.
This detailed analysis highlights the importance of breaking down complex problems into smaller steps and understanding the underlying principles that govern each step. The journey through Rahul's solution serves as a reminder that mathematics is not just about memorizing formulas; it's about developing logical reasoning and problem-solving skills. By embracing these skills, we can unlock the power of equality and apply it to a wide range of challenges.