Solving For W: A Step-by-Step Guide

Hey there, math enthusiasts! Ever found yourself staring at an equation, wondering how to isolate a specific variable? Well, you're in the right place! Today, we're diving into a common algebraic problem: solving for a variable. Specifically, we'll be tackling the equation x = 3(y + w) - 1 and figuring out how to express w in terms of x and y. This is a fundamental skill in algebra, and understanding it will open doors to solving more complex problems. So, buckle up, grab your pencils, and let's get started!

Understanding the Problem: Isolating the Variable

Our mission is clear: we want to rewrite the equation so that w is all by itself on one side, and everything else (x and y) is on the other side. This process is called isolating the variable. Think of it like a treasure hunt where w is the treasure, and our goal is to unearth it. To do this, we'll use the principles of algebra, which allow us to manipulate equations while maintaining their balance. These principles include adding, subtracting, multiplying, and dividing both sides of the equation by the same value. The key is to perform these operations in a way that gradually peels away all the terms and coefficients surrounding w, leaving it exposed. We have the following steps to take to achieve that.

Step-by-Step Solution: Unveiling W

Let's break down the process step by step, making it as clear and easy to follow as possible:

1. Original Equation: We begin with our starting point: x = 3(y + w) - 1. This equation tells us the relationship between x, y, and w. Our goal is to rearrange this equation to have w = .... We need to reverse the order of operations to isolate w. The first operation is the subtraction of 1, the next is the multiplication by 3, and the last is the addition of y.
2. Add 1 to Both Sides: To get rid of the '- 1', we add 1 to both sides of the equation. This maintains the equality. Doing this, we get: x + 1 = 3(y + w) - 1 + 1. This simplifies to x + 1 = 3(y + w). This is the first step toward isolating w.
3. Divide Both Sides by 3: Now, we want to eliminate the '3' that's multiplying the entire term (y + w). We do this by dividing both sides of the equation by 3. This gives us: (x + 1) / 3 = [3(y + w)] / 3. The equation simplifies to: (x + 1) / 3 = y + w. Now, w is almost isolated.
4. Subtract y from Both Sides: Finally, to isolate w, we need to remove the y that's added to it. We do this by subtracting y from both sides of the equation. This yields: (x + 1) / 3 - y = y + w - y. This simplifies to: (x + 1) / 3 - y = w. or w = (x + 1) / 3 - y. Now we have isolated w and expressed it in terms of x and y.

Explanation of Each Step

Each step in this process is carefully chosen to systematically unravel the equation and reveal the value of w. Remember that in algebra, we always aim to maintain the equation's balance. Each operation performed on one side must be mirrored on the other side to ensure the equality holds true. This is the cornerstone of solving algebraic equations. The addition and subtraction steps isolate the term containing w, and the division step gets rid of the coefficient multiplying that term. In essence, we're working backward through the order of operations, undoing the calculations that were initially performed on w. With consistent practice, these steps will become second nature, and you'll be solving similar equations with ease. If we have the initial equation, x = 3(y + w) - 1, and we want to determine w, we can follow the steps mentioned before. Understanding these underlying principles will help you confidently solve more complex equations. So, keep practicing, and don't hesitate to revisit these steps. Remember, the more you practice, the more familiar the steps will become. With the final value of w solved, we can easily determine its value by providing values for x and y.

A Detailed Example: Putting It All Together

Let's put this into practice with a concrete example. Suppose we have the equation x = 3(y + w) - 1, and we know that x = 7 and y = 2. Our goal is to find the value of w. We can use the formula we derived to determine w. We know from the previous section that w = (x + 1) / 3 - y. If we substitute x = 7 and y = 2, the equation becomes w = (7 + 1) / 3 - 2. Now, we can solve for w. First, we calculate the sum in the parentheses, 7 + 1 = 8. Then, we divide this sum by 3, so the result of (7 + 1) / 3 = 8 / 3. Finally, we subtract 2 from 8/3 which is 8/3 - 2 = 8/3 - 6/3 = 2/3. Therefore, w = 2/3. This process provides a clear and straightforward path to the solution. The most important thing is that the process is consistent. By following this method, we can determine the value of w in any situation where x and y are known. If the equation's variables change, the steps will be the same. The steps will give you a clear direction on how to arrive at the solution. With practice and understanding, you can solve similar problems confidently. This method can also be used for other types of equations with the same principles.

The Importance of Order of Operations

Throughout this process, the order of operations (often remembered by the acronym PEMDAS/BODMAS – Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) plays a crucial role. We effectively reverse this order when we solve for a variable, systematically undoing the operations in the reverse order they were applied. The importance lies in ensuring that we isolate w correctly. If we were to apply the operations in the wrong order, we could end up with an incorrect expression for w. Thus, each step is essential and carefully chosen to progressively simplify the equation while maintaining its balance. The best way to practice this is by trying different values to check if the answer is correct.

Common Mistakes to Avoid

While the process of isolating a variable might seem straightforward, some common pitfalls can trip up even experienced math students. Let's look at what mistakes to avoid. Firstly, forgetting to apply the operation to both sides of the equation. This is the cardinal sin of algebra. Remember that an equation is like a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. Secondly, incorrectly distributing or not distributing at all. In the original equation, we had 3(y + w), which represents the multiplication of 3 by the entire term (y + w). Failing to divide both terms on the right-hand side by 3 can lead to an incorrect result. Finally, making arithmetic errors. Simple mistakes in addition, subtraction, multiplication, or division can easily derail your solution. Always double-check your calculations, especially when dealing with negative numbers or fractions. By staying aware of these common mistakes, you can avoid them. Careful attention to detail is key in algebra, and it can help prevent many mistakes.

Conclusion: Mastering the Art of Variable Isolation

And there you have it! We've successfully isolated w and expressed it in terms of x and y. This skill is a fundamental building block in algebra, and it will serve you well as you tackle more advanced problems. Remember to practice regularly, and don't be discouraged if it takes a little time to master. With each equation you solve, you'll gain confidence and understanding. Keep practicing, keep learning, and enjoy the journey of discovery. You're well on your way to becoming a math whiz!

For further practice and more in-depth explanations, check out these resources:

• Khan Academy - This website offers comprehensive video lessons and practice exercises on solving equations.