Solving For 'y': A Step-by-Step Guide

Hey guys! Let's dive into a classic algebra problem: solving for y in the equation 7 + 5/(y - 2) = (4y)/(y - 2). Don't worry, it might look a little intimidating at first, but we'll break it down step by step. This is a fundamental concept in mathematics, and understanding how to solve for a variable is super important. We'll go through the process carefully, making sure we understand each move. Remember, the goal is to isolate y on one side of the equation. This means getting y all by itself, so we can figure out its value. Are you ready to get started? Let's do it! Before we get our hands dirty with the equation, it's useful to understand the basics. In algebra, solving for a variable means finding the value of that variable that makes the equation true. An equation is essentially a statement that two expressions are equal. To solve an equation, we use mathematical operations to manipulate the equation until the variable is isolated on one side. This involves performing the same operations on both sides of the equation to maintain balance. Remember the golden rule of equation solving: what you do to one side, you must do to the other! We'll be using addition, subtraction, multiplication, and division to get y by itself. The ultimate aim is to find the value(s) of y that satisfy the original equation. If the solution you find makes the equation true, then you're on the right track. If the solution makes the denominator zero, then that solution is extraneous, and therefore, there is no solution.

Clearing the Fraction

Alright, back to our equation: 7 + 5/(y - 2) = (4y)/(y - 2). The first thing that's bugging me are those fractions, am I right? Let's get rid of them. The best way to do that is to multiply both sides of the equation by the denominator, which is (y - 2). Doing this gets rid of the fractions. So, we'll multiply every single term by (y - 2). This gives us: 7*(y - 2) + [5/(y - 2)]*(y - 2) = [(4y)/(y - 2)]*(y - 2). Let's simplify it step by step. Multiplying 7 by (y - 2) gives us 7y - 14. The second term simplifies beautifully: the (y - 2) in the numerator and the (y - 2) in the denominator cancel out, leaving us with just 5. And on the right side, the (y - 2) in the numerator and denominator cancel out too, leaving us with 4y. Thus our new equation is: 7y - 14 + 5 = 4y. This step is crucial because it transforms the original equation into one that is much easier to manage. Now it involves only whole numbers and variables, which is much simpler. Notice how multiplying by the common denominator cancels the denominators of fractions in the equation. This significantly reduces the complexity of the equation and makes the rest of the process more straightforward.

Simplifying the Equation

Now that we've cleared out the fractions, let's simplify the equation we have: 7y - 14 + 5 = 4y. First, let's combine the constants on the left side of the equation: -14 + 5 = -9. So, our equation becomes 7y - 9 = 4y. This is where the fun really begins! Combining like terms is a basic but essential step in simplifying algebraic equations. It helps streamline the equation and prepares it for the final steps of isolating the variable. This step is about making the equation look cleaner and more manageable, reducing the risk of calculation errors. By combining these terms, we bring the equation one step closer to a solution. You are one step closer to solving the problem. We will combine all the y terms on one side of the equation and the constants on the other side. Let's subtract 4y from both sides: 7y - 4y - 9 = 4y - 4y, which simplifies to 3y - 9 = 0. Then, add 9 to both sides: 3y - 9 + 9 = 0 + 9, giving us 3y = 9. At this point, the equation is significantly simpler. The variable and its coefficient are on one side, while the constant is on the other. It's almost time to isolate y and find its value! It's all starting to come together now, guys!

Isolating the Variable

Okay, we've simplified the equation to 3y = 9. Our ultimate goal is to get y all by itself. Right now, y is being multiplied by 3. So, to undo that, we need to divide both sides of the equation by 3. This gives us: (3y)/3 = 9/3. Now we can simplify this by dividing both sides by the coefficient of the variable. Doing so cancels out the coefficient on the side of the variable, leaving us with just the variable. This is the heart of the solving process, where the variable is finally isolated. Once the variable is isolated, the solution to the equation becomes immediately clear. This is the climax of solving for y. The last step is to find the value of y. Dividing 3y by 3 leaves us with just y, and dividing 9 by 3 gives us 3. Therefore, we get y = 3. Hooray! We've solved for y. This step is about performing the final calculation to get the exact value of the variable. Once the value is obtained, it is a good idea to check if the solution is valid. And finally we have the answer. Let's quickly check our solution to make sure it's valid.

Checking the Solution

Now that we have found that y = 3, it's crucial to verify that this is indeed the correct solution. Let's substitute y = 3 back into the original equation 7 + 5/(y - 2) = (4y)/(y - 2). If y = 3, then the equation becomes: 7 + 5/(3 - 2) = (4*3)/(3 - 2). This simplifies to 7 + 5/1 = 12/1, which is 7 + 5 = 12. This simplifies to 12 = 12. Since the equation holds true, our solution y = 3 is correct. This step is about confirming the validity of the obtained solution. It ensures that the value found for the variable satisfies the initial equation. Always perform this check to avoid errors. This practice is essential because it helps catch any mistakes made during the solution process. It helps ensure confidence in the final answer. Always double-check your solution.

Conclusion

Alright, guys, we did it! We solved for y! We started with the equation 7 + 5/(y - 2) = (4y)/(y - 2), and through a few steps, we found that y = 3. Remember, we cleared the fractions, simplified the equation, isolated the variable, and then checked our answer to make sure it was correct. The key to solving equations like these is to remain systematic, take your time, and always double-check your work. Algebra can be fun if you follow these steps! Keep practicing, and you'll become a pro at solving for variables in no time. I hope this guide has been helpful. Keep up the great work, and good luck with your math adventures! Do you have any other questions? Let me know if you need more help! Keep practicing these types of problems, and soon it will become second nature. It might take a bit of practice, but once you get the hang of it, you'll be able to solve these equations quickly. You've got this, and remember to have fun while learning. The more you practice, the more comfortable you'll become with algebraic equations and the process of solving for a variable.