Solving For W In The Equation X = 3(y + W) - 1 A Step By Step Guide

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Introduction

In this article, we will delve into the process of solving for a specific variable within an equation. Our focus will be on the equation x = 3(y + w) - 1, where our objective is to isolate w and express it in terms of x and y. This type of algebraic manipulation is a fundamental skill in mathematics and is essential for various applications in science, engineering, and economics. Understanding how to rearrange equations allows us to solve for unknown quantities, analyze relationships between variables, and make predictions based on mathematical models. We will explore the step-by-step process of isolating w, providing a clear and concise explanation for each step involved. By the end of this article, you will have a solid understanding of how to solve this type of equation and will be able to apply these techniques to similar problems.

Solving for a specific variable in an equation is a fundamental concept in algebra. It involves using algebraic operations to isolate the desired variable on one side of the equation, expressing it in terms of other variables or constants. This process is crucial for various applications, such as solving for unknowns in formulas, manipulating equations in physics and engineering, and understanding relationships between different quantities. This skill is not just confined to the classroom; it's a practical tool used in everyday problem-solving and decision-making. For instance, rearranging a formula to calculate the required input for a desired output or determining the effect of changing one variable on another. The ability to confidently manipulate equations empowers individuals to tackle a wide range of problems, from simple calculations to complex mathematical modeling. This article aims to provide a comprehensive guide to solving for w in the given equation, equipping you with the knowledge and skills to tackle similar algebraic challenges.

The importance of algebraic manipulation cannot be overstated. It forms the backbone of many scientific and mathematical disciplines. Mastering this skill allows students and professionals to move beyond rote memorization of formulas and delve into a deeper understanding of the underlying relationships between variables. In fields such as physics, for example, solving for a variable in a formula might be necessary to determine the force required to achieve a certain acceleration, or to calculate the resistance in an electrical circuit. In economics, it might involve rearranging equations to model supply and demand, or to analyze the impact of government policies on market equilibrium. Moreover, the process of solving for a variable hones critical thinking and problem-solving skills. It requires a logical and systematic approach, where each step builds upon the previous one, leading to a final solution. This structured thinking process is transferable to many other areas of life, making algebraic manipulation a valuable tool beyond the confines of mathematics itself. By mastering this skill, individuals gain a powerful advantage in their academic pursuits, professional careers, and everyday lives.

Step-by-Step Solution

To find the value of w in terms of x and y, we need to isolate w on one side of the equation. Let's break down the process step by step:

  1. Start with the given equation:

    • We begin with the original equation: x = 3(y + w) - 1.

  2. Distribute the 3:

    • The first step in isolating w is to eliminate the parentheses. We do this by distributing the 3 across the terms inside the parentheses: x = 3y + 3w - 1.

  3. Isolate the term with w:

    • Our next goal is to get the term containing w (which is 3w) by itself on one side of the equation. To do this, we add 1 to both sides of the equation: x + 1 = 3y + 3w. Then, we subtract 3y from both sides to isolate the term with w: x + 1 - 3y = 3w.

  4. Solve for w:

    • Now that we have isolated the term 3w, we can solve for w by dividing both sides of the equation by 3: (x + 1 - 3y) / 3 = w. This gives us w = (x - 3y + 1) / 3.

Detailed Explanation of Each Step

Step 1: Start with the Given Equation

The foundation of any algebraic solution is the original equation. In our case, the given equation is x = 3(y + w) - 1. This equation establishes a relationship between the variables x, y, and w. Our aim is to manipulate this equation using valid algebraic operations to express w in terms of x and y. Starting with the correct equation is crucial, as any error at this stage will propagate through the subsequent steps, leading to an incorrect solution. It is always a good practice to double-check the original equation before proceeding with the solution. Understanding the equation's structure and the roles of different variables is the first step towards effectively solving for the desired variable. This foundational understanding will guide the subsequent algebraic manipulations and ensure a clear and logical progression towards the final solution.

Step 2: Distribute the 3

The second step in solving for w involves distributing the 3 across the terms inside the parentheses. This means multiplying the 3 by both y and w. Mathematically, this step transforms the equation from x = 3(y + w) - 1 to x = 3y + 3w - 1. The distributive property is a fundamental concept in algebra, which states that a(b + c) = ab + ac. Applying this property correctly is essential for simplifying equations and isolating the desired variable. By eliminating the parentheses, we make the equation more manageable and pave the way for further algebraic manipulations. This step is crucial because it separates the terms involving w from the other terms, bringing us closer to isolating w on one side of the equation. The correct application of the distributive property is a key skill in solving algebraic equations and forms the basis for many subsequent steps.

Step 3: Isolate the Term with w

After distributing the 3, we have the equation x = 3y + 3w - 1. The next step is to isolate the term containing w, which is 3w. To achieve this, we need to move all other terms to the opposite side of the equation. First, we add 1 to both sides of the equation, resulting in x + 1 = 3y + 3w. This eliminates the -1 term from the right side. Then, we subtract 3y from both sides to isolate the term with w: x + 1 - 3y = 3w. This step is crucial because it groups all the terms not containing w on one side of the equation, leaving only the term 3w on the other side. The principle behind this step is to maintain the balance of the equation by performing the same operation on both sides. Isolating the term with the desired variable is a critical step in solving equations and brings us closer to finding the solution.

Step 4: Solve for w

Having isolated the term 3w, we now have the equation x + 1 - 3y = 3w. The final step is to solve for w by dividing both sides of the equation by 3. This gives us (x + 1 - 3y) / 3 = w, which can be rewritten as w = (x - 3y + 1) / 3. This step effectively isolates w on one side of the equation, expressing it in terms of x and y. The division operation undoes the multiplication by 3, allowing us to find the value of w. This final step completes the solution, providing the value of w in terms of the other variables. The ability to confidently perform this step is a hallmark of algebraic proficiency and demonstrates a clear understanding of equation-solving principles.

Final Answer and Conclusion

The value of w in terms of x and y is w = (x - 3y + 1) / 3. Therefore, the correct answer is B. This exercise demonstrates the importance of applying algebraic principles systematically to isolate a variable. By following these steps, you can solve a wide range of algebraic equations.

In conclusion, solving for a specific variable in an equation is a crucial skill in mathematics and its applications. In this article, we have walked through the process of solving for w in the equation x = 3(y + w) - 1. By distributing, isolating terms, and dividing, we arrived at the solution w = (x - 3y + 1) / 3. This process highlights the importance of understanding algebraic principles and applying them methodically. The ability to rearrange equations is essential for solving problems in various fields, including science, engineering, and economics. By mastering these techniques, individuals can confidently tackle mathematical challenges and apply their knowledge to real-world scenarios. Algebraic manipulation is not just a theoretical exercise; it is a practical tool that empowers us to understand and solve problems in diverse contexts. The skills developed in solving equations form a foundation for more advanced mathematical concepts and are invaluable in both academic and professional pursuits.