Solving For X: A Step-by-Step Guide
Hey math enthusiasts! Today, we're diving into a common algebra problem where we need to isolate a variable. We'll be working with the equation (2x + 2) / y = 4w + 2 and our goal is to find the value of x. This kind of problem is super important because it forms the foundation for so many other concepts in algebra and beyond. It's like learning the alphabet before you start reading a book – you gotta know the basics! We'll break it down step-by-step to make sure everyone understands the process, even if algebra isn't your favorite subject. Don't worry, it's easier than it looks! So, grab your pencils and let's get started on this exciting mathematical adventure.
We start with the initial equation: (2x + 2) / y = 4w + 2. Our primary objective is to get 'x' all by itself on one side of the equation. To do this, we need to carefully manipulate the equation, using the rules of algebra to isolate 'x'. It's like peeling an onion; we have to take off each layer one at a time. The main goal here is to remove all the other numbers and variables from the same side of the equation as 'x'. Remember that whatever we do to one side of the equation, we must do to the other side to keep things balanced. This is a fundamental rule in algebra and will be used at every step of the solution process. We will systematically work towards isolating 'x', which is our main target. The end result will be an equation that clearly defines the value of 'x' in terms of the other variables, 'y' and 'w'. Let’s start the journey!
Step 1: Eliminating the Fraction
Alright, guys, our first step is to get rid of that pesky fraction. The equation starts with (2x + 2) / y = 4w + 2. To eliminate the division by 'y', we need to multiply both sides of the equation by 'y'. This will cancel out the 'y' in the denominator on the left side. Remember, doing the same thing to both sides of an equation keeps it balanced and true. It's the golden rule of algebra! So, we multiply both sides by 'y':
y * [(2x + 2) / y] = y * (4w + 2)
This simplifies to:
2x + 2 = y * (4w + 2)
See? The fraction is gone! We've made our equation a little cleaner and easier to handle. This simple step is vital to getting us closer to isolating 'x'. Always remember to perform the same operation on both sides to maintain the equation's balance. This initial step sets the stage for the rest of the solution.
Now, let's keep going! We will carefully address each part of the equation to get to our final goal. We have already addressed the division part, now we can move to the second step of our process. It is important to know that each step builds upon the previous one. We want to remove all the clutter surrounding 'x' to find its value.
Step 2: Expanding the Right Side
In this step, we'll focus on the right side of the equation, which is currently y * (4w + 2). We need to distribute the 'y' across the terms inside the parentheses. This means multiplying 'y' by both 4w and 2. It’s like giving everyone a gift! Let’s do it:
2x + 2 = y * 4w + y * 2
This simplifies to:
2x + 2 = 4wy + 2y
Great job! We've expanded the right side and simplified the equation a bit more. Notice how we're gradually chipping away at the equation, getting closer to isolating 'x'. Each step is designed to simplify and clarify, so don’t worry if you need to take it slow. Just focus on understanding what's happening at each stage. Remember, practice makes perfect! The more you work with these equations, the more comfortable you will become. Let's move onto the next phase.
We have expanded the right side and made it easier to work with. Remember that, in the next step, we'll bring all the terms without 'x' to the right side of the equation. This is the crucial step toward isolating 'x' and solving for its value.
Step 3: Isolating the Term with x
Now we're getting super close, folks! We've got 2x + 2 = 4wy + 2y. The next thing we need to do is get the term with 'x' by itself on one side. This means we want to get rid of the '+ 2' on the left side. To do that, we need to subtract 2 from both sides of the equation. Remember, always keep the balance!
2x + 2 - 2 = 4wy + 2y - 2
This simplifies to:
2x = 4wy + 2y - 2
Awesome! We've successfully isolated the term containing 'x'. Now, the only thing left to do is to solve for 'x'. We’re on the final stretch now! We're almost there. Just one more little adjustment, and we will find the value of 'x'.
It is important to understand each step. Don't be afraid to take your time and review each stage of the process, ensuring you understand every detail. Take note of any questions that arise during your review and research them to deepen your understanding. This method will reinforce the concepts and improve your mathematical skills.
Step 4: Solving for x
Almost done, guys! Our equation now looks like this: 2x = 4wy + 2y - 2. To solve for 'x', we need to get rid of the '2' that's multiplying it. We do this by dividing both sides of the equation by 2. Remember, we always do the same thing to both sides!
(2x) / 2 = (4wy + 2y - 2) / 2
This simplifies to:
x = (4wy) / 2 + (2y) / 2 - 2 / 2
Further simplifying gives us:
x = 2wy + y - 1
And there we have it! We've found the value of x. By following these steps, we've successfully isolated 'x' and expressed it in terms of 'y' and 'w'. This is a fundamental skill in algebra and will help you solve many more complex equations in the future. Congratulations, you did it!
Answer
Therefore, the correct answer is:
B. x = 2yw + y - 1
Great job everyone! You have just solved a challenging algebra problem. The ability to manipulate and solve equations is a cornerstone of mathematics, essential for a wide range of applications from science and engineering to economics and computer science. Keep practicing, and you will become more and more proficient. Understanding these concepts will give you a solid foundation for tackling more complex mathematical challenges. So, keep up the excellent work!