Solving 2y = Y - 5 A Step-by-Step Guide

Hey guys! Let's dive into solving a simple algebraic equation today. We're going to break down how to solve the equation 2y = y - 5 step by step. This is a classic example of a linear equation, and mastering these types of problems is crucial for more advanced math topics. Don't worry, we'll make it super easy and understandable!

Understanding Linear Equations

Before we jump right into the solution, let's quickly recap what a linear equation is. A linear equation is an equation that can be written in the form ax + b = 0, where 'x' is the variable, and 'a' and 'b' are constants. In our case, we have 2y = y - 5, which can be rearranged to fit this form. The key to solving linear equations is to isolate the variable on one side of the equation. This means we want to get 'y' all by itself on either the left or the right side. We achieve this by performing the same operations on both sides of the equation to maintain balance. Think of it like a scale – whatever you do to one side, you have to do to the other to keep it even. This principle is fundamental in algebra, ensuring that the equation remains valid throughout the solving process. Remember, the goal is not just to find an answer, but to understand the process so you can apply it to various problems. So, let's get started and see how we can isolate 'y' in our equation. Understanding this foundational concept makes solving algebraic equations significantly more manageable and less intimidating. We will see how each step we take leads us closer to isolating 'y', making the solution clear and straightforward. This step-by-step approach is designed to build your confidence and understanding in tackling similar algebraic problems.

Step 1: Isolating the Variable

Our equation is 2y = y - 5. The first thing we want to do is get all the 'y' terms on one side of the equation. A great way to do this is by subtracting 'y' from both sides. This keeps the equation balanced and moves us closer to isolating 'y'. So, we perform the operation:

2y - y = y - 5 - y

When we subtract 'y' from both sides, we simplify the equation. On the left side, 2y - y becomes just 'y'. On the right side, y - y cancels each other out, leaving us with '-5'. This simplification is crucial because it reduces the complexity of the equation and brings us one step closer to our solution. It's important to understand that each operation we perform is aimed at making the equation simpler and more manageable. By subtracting 'y' from both sides, we've effectively grouped the 'y' terms on the left side, which is a significant step in isolating the variable. This method highlights the importance of maintaining balance in equations; whatever operation is done on one side must be mirrored on the other. This principle ensures that the equation remains valid and the solution we arrive at is accurate. Now, let's see what our simplified equation looks like and move on to the next step in finding the value of 'y'.

Step 2: Simplifying the Equation

After performing the subtraction, our equation now looks like this:

y = -5

Guess what? We've already solved it! The equation is simplified, and 'y' is isolated on one side. This means we've found the value of 'y' that makes the original equation true. This step is incredibly satisfying because it shows how a series of logical operations can lead to a straightforward solution. It highlights the beauty of algebra in its ability to transform a seemingly complex problem into a simple answer. The fact that 'y' is now equal to '-5' means that if we substitute '-5' for 'y' in the original equation (2y = y - 5), both sides of the equation will be equal. This is the ultimate test of our solution, and it's what gives us confidence that we've solved the equation correctly. This simplicity underscores the power of the initial steps we took to isolate the variable. By methodically reducing the equation's complexity, we've arrived at a clear and concise solution. This reinforces the idea that even the most challenging problems can be broken down into manageable steps, leading to a successful outcome. Now, let's verify our solution to ensure that it holds true.

Step 3: Verifying the Solution

To make sure our solution is correct, we'll substitute y = -5 back into the original equation:

2y = y - 5

Replace 'y' with '-5':

2(-5) = -5 - 5

Now, let's simplify both sides:

-10 = -10

The left side equals the right side, which means our solution y = -5 is correct! This verification step is essential because it confirms the accuracy of our work and provides us with confidence in our answer. It's like the final piece of the puzzle that completes the picture. By substituting our solution back into the original equation, we're essentially double-checking that everything aligns perfectly. This process not only validates our answer but also reinforces our understanding of the equation and the steps we took to solve it. It's a good practice to always verify your solutions, especially in more complex problems, as it helps prevent errors and solidifies your grasp of algebraic principles. This step demonstrates the importance of precision in mathematics and the satisfaction of knowing that you've arrived at the correct answer. So, with the verification complete, we can confidently say that we've successfully solved the equation 2y = y - 5.

Conclusion

So there you have it! We've solved the equation 2y = y - 5 by isolating the variable, simplifying, and verifying our solution. The answer is y = -5. Remember, the key to solving these types of equations is to keep the equation balanced and perform operations that move you closer to isolating the variable. Practice makes perfect, so try solving similar equations to build your skills. You'll become a pro at algebra in no time! This step-by-step approach is not just about getting the right answer; it's about understanding the process and building a solid foundation in algebra. Each step we took, from isolating the variable to verifying the solution, plays a crucial role in solving the equation accurately. This method can be applied to a wide range of algebraic problems, making it a valuable tool in your mathematical toolkit. Keep practicing, and you'll find that solving equations becomes second nature. The more you engage with these types of problems, the more confident and proficient you'll become in your algebraic abilities. Remember, math is a skill that grows with practice, so don't hesitate to tackle new challenges and explore different types of equations. Happy solving, guys! We've successfully navigated this equation together, and you're well on your way to mastering algebra!