Hey guys! Today, we're diving into a fundamental algebra problem: solving the equation 2x - 2y = 5 for y. This is a common type of problem you'll encounter in algebra, and mastering it will build a strong foundation for more advanced concepts. We'll break down the steps in a clear and easy-to-follow way, ensuring you understand not just the how, but also the why behind each step. Let's get started!
Understanding the Basics: What Does it Mean to Solve for y?
Before we jump into the solution, let's make sure we're all on the same page about what it means to solve for a variable. When we say we want to solve the equation 2x - 2y = 5 for y, we essentially mean we want to isolate y on one side of the equation. Our goal is to rewrite the equation in the form y = [some expression involving x]. This gives us an explicit formula for y in terms of x. This form is super useful because, if you have a value of x, you can just plug it into the expression to find the corresponding value of y. Think of it like a machine: you put in x, and it spits out y. Why is this skill important? Well, imagine you're dealing with a scenario where the relationship between two quantities, say x and y, is described by this equation. Being able to solve for y allows you to easily determine how y changes as x changes. This is essential in many real-world applications, from physics and engineering to economics and finance. Now, the key to isolating y lies in performing operations on both sides of the equation in a way that maintains the balance. Remember, an equation is like a scale, and to keep it balanced, whatever you do to one side, you must do to the other. So, we'll be using addition, subtraction, multiplication, and division strategically to gradually peel away the terms around y until it stands alone. We will also make sure to keep the equation balanced so that we can accurately find the value of y. It's like unwrapping a present, carefully removing each layer until you reveal the gift inside. Speaking of keeping things balanced, let's not forget the golden rule of equation manipulation: whatever you do to one side, you absolutely must do to the other. If you add 3 to the left side, you'd better add 3 to the right side. If you multiply the right side by -1, you've got to multiply the left side by -1 too. This ensures the equation remains true throughout the solving process. Think of it as maintaining equilibrium in a delicate system. Upholding this balance is crucial for arriving at the correct solution. Any misstep in this regard can throw off the entire process, leading to an inaccurate final result. So, let's proceed with caution and precision, keeping this principle at the forefront of our minds as we tackle the equation at hand. This equation is a linear equation, meaning that when you graph it, you get a straight line. Solving for y puts the equation in slope-intercept form (y = mx + b), which makes it very easy to graph. The 'm' represents the slope of the line, and the 'b' represents the y-intercept (the point where the line crosses the y-axis). So, solving for y not only gives you a formula for y but also provides valuable information about the graphical representation of the equation.
Step-by-Step Solution: Isolating y
Okay, let's dive into the nitty-gritty of solving 2x - 2y = 5 for y. We'll walk through each step carefully so you can see exactly how it's done. Remember, our goal is to get y by itself on one side of the equation. Our starting point is the equation 2x - 2y = 5. The first thing we want to do is get rid of the 2x term on the left side. To do this, we'll subtract 2x from both sides of the equation. This keeps the equation balanced and moves us closer to isolating y. Subtracting 2x from both sides gives us: 2x - 2y - 2x = 5 - 2x. On the left side, the 2x and -2x cancel each other out, leaving us with -2y = 5 - 2x. Now, we're one step closer! We have -2y on the left side, but we want just y. The -2 is multiplying y, so to undo this, we'll divide both sides of the equation by -2. Remember, whatever we do to one side, we must do to the other to maintain balance. Dividing both sides by -2 gives us: (-2y) / -2 = (5 - 2x) / -2. On the left side, the -2 in the numerator and the -2 in the denominator cancel each other out, leaving us with just y. On the right side, we have (5 - 2x) / -2. We can simplify this a bit further by dividing each term in the numerator by -2. This gives us: y = (5 / -2) - (2x / -2). Now, let's simplify the fractions. 5 / -2 is simply -5/2. For the second term, -2x / -2, the -2 in the numerator and the -2 in the denominator cancel each other out, leaving us with just x. So, our equation now looks like this: y = -5/2 + x. We can rewrite this in a more standard form by putting the x term first: y = x - 5/2. And there you have it! We've successfully solved the equation 2x - 2y = 5 for y. The solution is y = x - 5/2. This equation tells us exactly how y is related to x. For any value of x, we can simply plug it into this equation to find the corresponding value of y. This is a powerful tool in algebra and is used in many different applications. Let's recap the steps we took: First, we subtracted 2x from both sides to isolate the y term. Then, we divided both sides by -2 to get y by itself. Finally, we simplified the equation to get our final answer, y = x - 5/2. Remember, the key to solving equations is to perform the same operations on both sides to maintain balance. This ensures that the equation remains true throughout the solving process.
Alternative Forms and Interpretations of the Solution
Now that we've found the solution y = x - 5/2, let's explore some alternative ways to express it and what these different forms might tell us. Sometimes, having the solution in a slightly different form can be more useful depending on the context. We arrived at y = x - 5/2 by dividing both terms in the numerator by -2, resulting in a mixed number format. However, we could also express the solution with a common denominator. Take y = (5 - 2x) / -2 from our step-by-step solution above. To get rid of the negative in the denominator, we can multiply both the numerator and denominator by -1, which gives us y = (-5 + 2x) / 2. Reordering the terms in the numerator, we get y = (2x - 5) / 2. This form can be useful in certain situations, such as when you want to see the entire expression divided by a common factor. Another form you might encounter is the slope-intercept form, which is written as y = mx + b, where m is the slope and b is the y-intercept. Our solution y = x - 5/2 is already in slope-intercept form! We can see that the slope m is 1 (since there's an implied 1 in front of the x) and the y-intercept b is -5/2. This form is particularly useful for graphing the equation because you can easily identify the slope and y-intercept. To graph the line, you would start by plotting the y-intercept at (0, -5/2), and then use the slope of 1 to find another point. A slope of 1 means that for every 1 unit you move to the right on the graph, you move 1 unit up. So, from the y-intercept, you could move 1 unit to the right and 1 unit up to find another point on the line. Then, you can draw a line through these two points to graph the equation. Understanding the slope and y-intercept gives you a visual representation of the relationship between x and y. The slope tells you how steep the line is, and the y-intercept tells you where the line crosses the y-axis. These different forms highlight the flexibility of algebraic solutions. Depending on what you want to do with the solution, one form might be more convenient than another. For instance, if you were asked to graph the equation, the slope-intercept form y = x - 5/2 would be the most straightforward to use. If you were asked to find the value of y when x is a certain value, any of the forms would work, but the one that requires the least calculation might be preferred. Let's consider some real-world interpretations of this equation. Imagine that x represents the number of hours you work at a job, and y represents your earnings. The equation y = x - 5/2 could represent a scenario where you have to pay a fixed cost of $2.50 (5/2) before you start earning money, and then you earn $1 for every hour you work. In this case, the y-intercept of -5/2 represents the initial cost, and the slope of 1 represents your hourly wage. This is just one example, but it shows how algebraic equations can be used to model real-world situations. By understanding the different forms of the solution and their interpretations, you can gain a deeper understanding of the relationship between the variables and how they can be applied in various contexts.
Common Mistakes and How to Avoid Them
Alright, let's talk about some common pitfalls people encounter when solving equations like 2x - 2y = 5 for y, and how you can dodge them. Knowing these common mistakes will help you stay sharp and avoid making them yourself. One frequent mistake is forgetting to perform the same operation on both sides of the equation. Remember our balance analogy? If you only subtract 2x from the left side in our original problem, the equation becomes unbalanced, and your solution will be incorrect. You must subtract 2x from both sides to maintain equality. So, always double-check that you're applying the same operation to both sides. Another common error is messing up the signs when dividing or multiplying by a negative number. For example, when we divided (5 - 2x) by -2, it's crucial to remember that the negative sign affects both terms in the parentheses. So, 5 / -2 becomes -5/2, and -2x / -2 becomes +x. Pay close attention to the signs, and maybe even write out the steps explicitly to avoid sign errors. A third mistake is not simplifying the solution completely. For instance, you might stop at y = (5 - 2x) / -2, but it's important to simplify this to y = x - 5/2. Simplifying makes the solution easier to understand and use. Make sure to simplify fractions, combine like terms, and put the equation in a standard form like slope-intercept form if applicable. Another tricky area is when you have to distribute a negative sign. Remember, when you have a negative sign outside parentheses, it's like multiplying each term inside the parentheses by -1. For example, if you had -(3x - 2), you need to distribute the negative sign to both terms, resulting in -3x + 2. Failing to do this correctly can lead to errors in your solution. So, always be careful when you see a negative sign outside parentheses. Finally, one of the most overlooked mistakes is failing to check your answer. Once you've solved for y, plug your solution back into the original equation to see if it holds true. If you substitute your expression for y in the original equation and simplify, you should end up with a true statement (e.g., 5 = 5). If you don't, then you know you've made a mistake somewhere along the way. Checking your answer is a crucial step in the problem-solving process, and it can save you a lot of headaches. In summary, to avoid common mistakes when solving equations, remember to: Always perform the same operation on both sides, watch out for sign errors, simplify your solution completely, distribute negative signs carefully, and always check your answer. By keeping these tips in mind, you'll be well on your way to mastering equation solving! Remember, practice makes perfect, so the more you work through problems like this, the more comfortable and confident you'll become.
Conclusion: Mastering Algebra One Step at a Time
Alright, guys, we've reached the end of our journey of solving the equation 2x - 2y = 5 for y. We've not only found the solution, which is y = x - 5/2, but we've also explored the underlying principles, alternative forms, potential pitfalls, and real-world interpretations. This is what mastering algebra is all about – not just getting the right answer, but understanding the why behind it. We started by understanding the core concept of what it means to solve for a variable, emphasizing the importance of isolating the variable of interest (in this case, y) on one side of the equation. We then walked through the step-by-step process of isolating y, carefully explaining each operation and why it's necessary. We subtracted 2x from both sides, divided both sides by -2, and simplified the resulting expression. Along the way, we highlighted the golden rule of equation manipulation: whatever you do to one side, you must do to the other. We also discussed how the solution can be expressed in different forms, such as slope-intercept form, and how each form can provide different insights into the relationship between the variables. Understanding these different forms allows you to choose the most convenient one for a particular task, whether it's graphing the equation, finding specific values of y for given values of x, or interpreting the equation in a real-world context. We also addressed common mistakes that students often make when solving equations, such as forgetting to perform the same operation on both sides, making sign errors, or failing to simplify the solution completely. By being aware of these pitfalls, you can take steps to avoid them and ensure that you arrive at the correct answer. Remember, one of the best ways to reinforce your understanding and build confidence is to practice. Work through similar problems, and don't be afraid to make mistakes. Mistakes are a natural part of the learning process, and they can actually be valuable learning opportunities. When you make a mistake, take the time to understand why you made it, and what you can do differently next time. Finally, remember that algebra is not just about manipulating symbols and solving equations. It's about developing critical thinking skills, problem-solving abilities, and a deeper understanding of the world around us. The concepts and techniques you learn in algebra can be applied in many different fields, from science and engineering to economics and finance. So, keep practicing, keep exploring, and keep asking questions. The more you engage with algebra, the more you'll appreciate its power and beauty. And most importantly, don't forget to have fun along the way! Algebra can be challenging, but it can also be incredibly rewarding. So, embrace the challenge, enjoy the journey, and celebrate your successes. You've got this! This problem may seem simple, but the principles we've covered here are fundamental to all of algebra. As you move on to more complex problems, you'll find that these basic skills are essential. So, make sure you have a solid grasp of these concepts before moving on. Keep practicing, and you'll be an algebra whiz in no time! Remember, every complex mathematical problem is built upon a foundation of simpler concepts. Mastering these fundamentals is the key to unlocking more advanced topics. So, keep practicing, keep exploring, and never stop learning!