Solving The Equation: $-3y(y-8)(y-6) = 0$

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Solving the Equation: $-3y(y-8)(y-6) = 0$

Hey guys! Let's dive into solving the equation −3y(y−8)(y−6)=0-3y(y-8)(y-6) = 0. This is a fun problem in algebra, and it's all about finding the values of y that make this equation true. Don't worry, it's not as scary as it looks. We'll break it down step by step, and I promise, by the end of this, you'll be a pro at solving equations like this. Understanding how to solve such equations is super important because it forms the bedrock for more complex mathematical concepts. So, grab your pencils and let's get started. The key to solving this type of equation is understanding the Zero Product Property. This property states that if the product of several factors is zero, then at least one of the factors must be zero. It's like saying if you multiply a bunch of numbers together and the answer is zero, then at least one of those numbers had to be zero. Pretty straightforward, right? This is the core principle we're going to use to find our solutions for y. We'll be applying this principle to the given equation, setting each factor equal to zero, and then solving for y. This method ensures we don't miss any possible solutions. Understanding this concept not only helps you solve this specific equation but also equips you with a valuable tool for tackling a wide range of algebraic problems. The ability to break down complex equations into simpler parts is a fundamental skill in mathematics, so let's master it together. We're going to walk through this step by step, ensuring you understand not just how to get the answer, but also why we take each step. This approach is not just about getting the right answer; it's about building your math intuition and confidence. Ready to make some math magic?

Step-by-Step Solution

Alright, let's get down to the nitty-gritty and solve this equation step-by-step. Our equation is −3y(y−8)(y−6)=0-3y(y-8)(y-6) = 0. Remember, we're looking for the values of y that make this equation true. First, let's look at the factors. We've got -3, y, (y-8), and (y-6). According to the Zero Product Property, if the product of these factors is zero, then at least one of them must be zero. This is our golden rule for this problem. Let’s start with the easiest part. We have -3 as one of the factors, but since -3 is a constant, it can never equal zero. So, we can safely ignore this factor when we're trying to find solutions for y. Next, we'll consider each of the other factors separately. We'll set each factor equal to zero and solve for y. This process is all about isolating y on one side of the equation. We want to find the individual values of y that satisfy the equation. This involves applying some basic algebraic principles, such as adding or subtracting the same value from both sides of the equation. This ensures that the equality remains true. We'll walk through these steps very carefully, so you won’t miss a thing. The goal here isn't just to get an answer, but to understand the logic behind the solution. This understanding is what will help you in future math problems. Think of each step as a puzzle piece, and by the end, we'll have the complete picture.

Factor 1: y

Let’s start with the first factor: y. We set this equal to zero, which gives us y = 0. This is one of our solutions! This means that if you substitute 0 for y in the original equation, the equation will be true. Easy peasy, right? Finding this first solution is often the simplest step in the process. It's also a great way to double-check your understanding of the Zero Product Property. Remember, this property tells us that we’re looking for any value that makes any part of the equation equal to zero. When we set y = 0, we're directly addressing this requirement. This first part usually serves as a confidence booster, as it's often the most straightforward solution to find. You can think of it as the starting point of your problem-solving journey. It not only provides a solution but also sets the stage for dealing with more complex factors. This initial step is a perfect illustration of how the Zero Product Property works in practice. So, congratulations, you've found your first solution! It’s this factor of the original equation that has led us to the correct answer. We're well on our way to solving the entire problem.

Factor 2: (y-8)

Now, let’s move on to the second factor: (y - 8). We set this equal to zero, so we have y - 8 = 0. To solve for y, we need to isolate it. We can do this by adding 8 to both sides of the equation. This gives us y = 8. Boom! Another solution. This means that if you substitute 8 for y in the original equation, the equation will also be true. Pretty cool, huh? Solving for y in this factor involves a simple application of algebraic principles. Adding 8 to both sides of the equation is a fundamental technique for isolating the variable. This step highlights the importance of keeping the equation balanced. Any operation performed on one side must also be performed on the other side to maintain equality. By taking these steps, you're not just finding an answer but also building a solid foundation in algebra. These basic operations are the building blocks for more advanced concepts, so mastering them is key. Now that we have this solution, we're making excellent progress. We're getting closer to solving the whole problem, and it's all because we're focusing on one factor at a time. Each step we take brings us closer to a complete solution, building your confidence in solving similar problems.

Factor 3: (y-6)

Alright, last but not least, let's consider the third factor: (y - 6). We set this equal to zero, giving us y - 6 = 0. To solve for y, we again need to isolate it. This time, we add 6 to both sides of the equation, which gives us y = 6. And there we have it – our third solution! This means that if you substitute 6 for y in the original equation, the equation will also be true. This step is practically the same as the previous one, reinforcing the methods that we've already learned. Here, we're applying the same principle we used for the factor (y - 8), ensuring a consistent approach to the problem. By adding 6 to both sides, we isolate the y and find its value. This step, just like the others, emphasizes the principles of algebraic manipulation. Having completed this step, we’ve found all the solutions to our original equation. By setting each factor to zero, we’ve found every y value that satisfies the equation. Finding this last solution is like putting the final puzzle piece in place. Congratulations! We've successfully solved the equation and found all possible solutions for y! We did it, and it feels awesome, doesn't it?

Final Solutions

Okay, guys, let’s wrap this up! We've found three solutions for y. They are: y = 0, y = 8, and y = 6. These are the values that make the original equation −3y(y−8)(y−6)=0-3y(y-8)(y-6) = 0 true. You can plug each of these values back into the equation to check if they work. You'll see that each one does indeed satisfy the equation. Pretty neat, right? The beauty of these solutions is that they can be used to describe different real-world situations, depending on what y represents. In some cases, y might represent time, distance, or even the quantity of something. Understanding these solutions is crucial because it allows us to predict and analyze scenarios in a variety of fields. Knowing these values allows us to predict the behavior of the equation under different conditions. This exercise not only provides you with answers but also with a valuable understanding of algebraic principles. This knowledge can be applied to many different areas, making it a very useful skill. We've successfully found all the values of y that make the equation true. Knowing how to solve such equations is an important skill in algebra, which is used in many different fields. These solutions are the culmination of the work we have done, so it's a great feeling to have solved the problem.

Conclusion

And that’s a wrap, everyone! We have successfully solved the equation −3y(y−8)(y−6)=0-3y(y-8)(y-6) = 0. We've learned the importance of the Zero Product Property and how to apply it step by step. We found three solutions: y = 0, y = 8, and y = 6. Not only did we solve the equation, but we also practiced valuable algebraic techniques, like isolating variables and using the properties of zero. This is a big win! You've taken a complex equation and broken it down into manageable parts. This process teaches you how to approach and solve different types of mathematical problems. Remember, the goal here is to understand the process. The more you practice, the easier it becomes. You're building a strong foundation in algebra. Keep up the amazing work! With practice, you'll become more confident in solving similar equations. Keep exploring, keep practicing, and never stop learning. You've got this! Now, go out there and conquer some more equations! Good job, everyone!