Solving 6y - 20 = 2y - 4 A Step-by-Step Guide
Hey guys! Today, we're diving into the world of linear equations and tackling a specific problem: how to solve the equation 6y - 20 = 2y - 4. Don't worry if this looks intimidating – we'll break it down step by step, making it super easy to understand. Whether you're a student grappling with algebra or just brushing up on your math skills, this guide will equip you with the knowledge to conquer similar equations. Linear equations are the building blocks of algebra, and mastering them opens the door to more complex mathematical concepts. So, grab your pencils, and let's get started on this exciting journey of solving for 'y'! We'll explore the fundamental principles behind solving linear equations, including isolating the variable, combining like terms, and applying the properties of equality. By the end of this guide, you'll not only know the solution to this particular equation but also have a solid understanding of how to approach any linear equation with confidence. Remember, math isn't about memorizing formulas; it's about understanding the process and the logic behind it. We'll focus on making the process clear and intuitive so you can apply these skills to various problems. We'll also touch upon common mistakes to avoid, ensuring you're well-prepared for any algebraic challenge that comes your way. So, let’s transform this equation from a puzzle into a piece of cake! Are you ready to unravel the mystery of 'y'? Let’s jump right in and make math fun and accessible for everyone. We'll be using clear, concise language and plenty of explanations to ensure that no step is left unclear. Think of this as your friendly guide to navigating the world of linear equations – let’s solve it together!
Understanding Linear Equations
Before we jump into solving the equation, let's quickly recap what linear equations are. A linear equation is basically a mathematical statement that shows the equality between two expressions. These expressions involve variables (like our 'y') and constants (numbers). The goal is to find the value of the variable that makes the equation true. Think of it like a balancing scale – we need to keep both sides of the equation perfectly balanced while we manipulate it to isolate 'y'. Understanding the core principles of linear equations is crucial for solving more complex algebraic problems. Linear equations pop up everywhere in real life, from calculating simple budgets to understanding more advanced scientific models. The beauty of linear equations lies in their predictability and the clear steps involved in solving them. Each step we take is designed to bring us closer to isolating the variable, and with a bit of practice, you'll become a pro at navigating these equations. Remember, the variable is like the unknown piece of the puzzle, and our job is to find the perfect fit. This involves using operations – addition, subtraction, multiplication, and division – strategically to maintain the balance and reveal the value of 'y'. The structure of a linear equation is key to solving it. Recognizing the different components – the variable, the coefficients (the numbers multiplying the variable), and the constants – helps us plan our approach. We’ll use this understanding to strategically move terms around and simplify the equation until we arrive at the solution. So, let’s keep the balance in mind and start working towards unveiling the mystery of 'y'! With a solid grasp of what linear equations are, we’re now well-equipped to tackle our specific equation and find its solution. Let's delve deeper into the process and see how these principles apply in practice. The next section will guide you through each step, ensuring you understand the logic behind every move we make.
Step-by-Step Solution of 6y - 20 = 2y - 4
Okay, let's get down to business and solve the equation 6y - 20 = 2y - 4. Here’s the breakdown:
Step 1: Group the 'y' terms together.
Our goal is to get all the terms with 'y' on one side of the equation. To do this, we can subtract 2y from both sides. This keeps the equation balanced and moves the 'y' terms closer together. So, we have:
- 6y - 20 - 2y = 2y - 4 - 2y
- This simplifies to 4y - 20 = -4
Grouping the 'y' terms together is a fundamental step in solving linear equations. It's like gathering all the similar puzzle pieces in one place so we can work with them more effectively. Subtracting 2y from both sides ensures that we maintain the equality – what we do to one side, we must do to the other. This principle of equality is the cornerstone of algebraic manipulation. By isolating the 'y' terms, we're setting the stage for isolating 'y' itself. This step is crucial because it consolidates the variable terms, making it easier to manipulate the equation further. Think of it as decluttering one side of the equation so we can clearly see what needs to be done next. The result, 4y - 20 = -4, is a simpler equation that's much easier to work with. We've reduced the complexity by combining like terms, and we're one step closer to finding the value of 'y'. This strategic movement of terms is a key skill in algebra, and mastering it will make solving equations much smoother. So, with the 'y' terms grouped, let's move on to the next step in our journey to solve for 'y'. We’re making great progress, and the solution is just around the corner!
Step 2: Isolate the 'y' term.
Now, we want to get the term with 'y' all by itself on one side. We currently have 4y - 20 = -4. To isolate 4y, we need to get rid of the -20. We can do this by adding 20 to both sides of the equation:
- 4y - 20 + 20 = -4 + 20
- This simplifies to 4y = 16
Isolating the 'y' term is a critical step in solving for 'y'. It’s like clearing the path so we can see the variable clearly and determine its value. By adding 20 to both sides, we're effectively canceling out the -20 on the left side, leaving us with just the term containing 'y'. This step reinforces the principle of equality – whatever we do to one side, we must do to the other to maintain the balance. Think of it as evening the scales; by adding 20 to both sides, we’re keeping the equation in perfect equilibrium. The result, 4y = 16, is a much simpler equation. We’ve successfully isolated the 'y' term, and now we're just one step away from finding the value of 'y'. This process of isolating terms is a fundamental technique in algebra and is used extensively in solving various types of equations. By mastering this step, you're building a strong foundation for more advanced algebraic concepts. So, with the 'y' term isolated, we’re now ready for the final step – solving for 'y' itself. Let’s move on and see how we can reveal the value of our unknown!
Step 3: Solve for 'y'.
We're almost there! We have 4y = 16. To find the value of y, we need to get it completely alone. Since y is being multiplied by 4, we can undo this by dividing both sides of the equation by 4:
- 4y / 4 = 16 / 4
- This gives us y = 4
Solving for 'y' is the final piece of the puzzle! We've successfully navigated through the equation, and now we're ready to unveil the value of our variable. By dividing both sides by 4, we're isolating 'y' completely, revealing its true worth. This step demonstrates the power of inverse operations – using the opposite operation to undo what was done and isolate the variable. It’s like unwrapping a gift; we’re removing the layers until we reveal the treasure inside, which in this case is the value of 'y'. The result, y = 4, is the solution to our equation. We’ve found the value that makes the equation true. This final step is the culmination of all our hard work and strategic manipulation. Knowing how to solve for a variable is a crucial skill in mathematics and has wide-ranging applications in various fields. So, with 'y' now found, we can confidently say that we’ve solved the equation. Congratulations! You’ve successfully navigated through the steps and found the solution. Let’s take a moment to recap and reinforce what we’ve learned. We’ll also discuss how to check our answer to ensure its accuracy. The journey to solving for 'y' has been a rewarding one, and now we have the solution in hand.
Checking Our Solution
It's always a good idea to double-check your answer to make sure it's correct. To do this, we can substitute our solution, y = 4, back into the original equation:
- 6y - 20 = 2y - 4
- 6(4) - 20 = 2(4) - 4
- 24 - 20 = 8 - 4
- 4 = 4
Since both sides of the equation are equal, our solution y = 4 is correct! Checking our solution is like verifying our work – it's a critical step in the problem-solving process. By substituting our answer back into the original equation, we're ensuring that it truly satisfies the equation and that we haven't made any errors along the way. This process not only confirms our solution but also reinforces our understanding of the equation itself. It’s like proofreading a document; we're looking for any inconsistencies or mistakes to ensure our final answer is accurate. The fact that both sides of the equation balance out, 4 = 4, gives us confidence that we've found the correct solution. This confirmation is a rewarding feeling and a testament to our problem-solving skills. Checking our answers is a habit that will serve you well in mathematics and beyond. It teaches us to be meticulous and to trust the process of verification. So, with our solution checked and confirmed, we can move on knowing that we’ve conquered this equation with accuracy and precision. The ability to verify our solutions is a valuable tool in our mathematical toolkit, and it empowers us to tackle even more challenging problems with assurance.
The Answer
So, the correct answer is B. y = 4. Great job, guys! You've successfully navigated the steps to solve the equation 6y - 20 = 2y - 4. The journey from the initial equation to the final solution demonstrates the power of algebraic manipulation and the importance of following a systematic approach. We broke down the problem into manageable steps, ensuring clarity and understanding at each stage. This process not only led us to the correct answer but also reinforced our skills in solving linear equations. Remember, math is a journey, and each equation we solve adds to our knowledge and confidence. The key is to approach each problem with a clear plan, execute the steps with precision, and always check your work. With practice, these techniques will become second nature, and you'll be able to tackle even more complex equations with ease. So, celebrate your success in solving this equation, and carry this confidence forward as you continue your mathematical journey. The world of algebra is vast and exciting, and you're now equipped with the skills to explore it further. Keep practicing, keep learning, and keep pushing the boundaries of your mathematical abilities. The answer, y = 4, is not just a number; it's a testament to your problem-solving skills and your dedication to learning. Well done!
Practice Problems
To solidify your understanding, try solving these similar equations:
- 3x + 5 = x - 1
- 8z - 12 = 4z + 8
- 5a + 7 = 2a - 2
Working through practice problems is the best way to reinforce what you've learned and build your confidence in solving linear equations. These additional equations provide an opportunity to apply the steps and techniques we’ve discussed in this guide. Each problem is a chance to hone your skills and deepen your understanding of algebraic manipulation. Remember, practice makes perfect, and the more you solve, the more comfortable you'll become with the process. Approach each equation with a strategic mindset, break it down into manageable steps, and don't forget to check your answers. Solving these problems will not only solidify your knowledge but also help you identify any areas where you may need further practice. Think of each equation as a puzzle waiting to be solved, and enjoy the challenge of finding the solution. The more you practice, the more fluent you'll become in the language of algebra, and the more confident you'll be in tackling even more complex equations. So, grab a pencil, dive into these practice problems, and continue your journey of mathematical discovery. The solutions to these problems are just a few steps away, and the satisfaction of finding them is well worth the effort. Let's continue to grow our skills and expand our mathematical horizons!