Solving For Y: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for y. We'll tackle the equation 14y = 6y + 48 step-by-step, making sure you understand every move. Our goal? To find the value of y that makes this equation true. Let's break it down and make it super easy to follow. We'll cover everything from the basic principles to simplifying the final answer, ensuring you not only get the right solution but also grasp the why behind each step. Get ready to flex those math muscles and feel confident in your problem-solving abilities. Ready to jump in? Let's get started!

Understanding the Basics: Equations and Variables

Alright, before we get our hands dirty with the equation, let's quickly recap some fundamental concepts. An equation is simply a mathematical statement that shows two expressions are equal. It's like a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. Our equation, 14y = 6y + 48, is a perfect example. The expression on the left (14y) is equal to the expression on the right (6y + 48). Our main character in this equation is the variable, represented by y. In algebra, a variable is a symbol (usually a letter) that represents an unknown value. Our mission? To find that unknown value of y that makes the equation true. The terms 14y and 6y involve y, and the number 48 stands alone. The entire aim is to isolate y on one side of the equation, revealing its value. Think of it like a treasure hunt; we're trying to find the hidden 'x' that unlocks the secret of the equation. Are you with me so far? Because understanding these basics sets the stage for success. Understanding the equation's structure and the role of the variable is super important before we begin. Are you ready to dive into the problem-solving strategies?

This is where we put our knowledge into action, and we’ll go through the solving process step by step, which will help us solve the equation and find the value of y. So, buckle up!

Step-by-Step Solution: Isolating the Variable

Now, let's get down to business and solve for y in the equation 14y = 6y + 48. The core strategy here is to isolate y on one side of the equation. We want to get y all by itself, so we can reveal its value. This involves a couple of simple steps: Subtracting 6y from both sides, and then dividing to find the value of y.

Step 1: Subtract 6y from both sides.

Remember the balanced scale analogy? We have to treat both sides of the equation equally. To start, we want to eliminate the 6y term from the right side. We can do this by subtracting 6y from both sides of the equation. That gives us:

14y - 6y = 6y + 48 - 6y

When we simplify, this becomes:

8y = 48

Do you see what happened? The 6y on the right side canceled itself out, leaving us with just 48. On the left side, we combined like terms (14y - 6y to get 8y). This is what we wanted! We've taken a big step towards isolating y.

Step 2: Divide both sides by 8.

We're almost there! Now we have 8y = 48. To isolate y, we need to get rid of the 8 that's multiplying it. We can do this by dividing both sides of the equation by 8. This gives us:

(8y) / 8 = 48 / 8

When we simplify, we get:

y = 6

Boom! We've found it! y equals 6. This is the solution to our equation. We've successfully isolated the variable y and revealed its numerical value. The hard work is done, guys! Let's just make sure we got it right.

Checking Your Work: Verification

It's always a smart idea to check our answer to make sure it's correct. We can do this by substituting the value we found for y back into the original equation and seeing if it holds true. If both sides of the equation are equal, then we know our solution is correct. Let's do it:

Original equation: 14y = 6y + 48

Substitute y = 6:

14(6) = 6(6) + 48

Simplify:

84 = 36 + 48

84 = 84

Guess what? The equation is true! Both sides equal 84. This tells us that our solution, y = 6, is correct. Verifying your answers is a great habit to build. It helps catch any errors and boosts your confidence in your math skills. Always take a moment to double-check; it's worth it. Now we know, with absolute certainty, that our solution is accurate. Nice work!

Simplifying Your Answer: Why It Matters

Simplifying your answer is not just about getting the right solution; it's about presenting your work clearly and concisely. When we simplify, we're essentially removing any unnecessary clutter and expressing the answer in its most basic form. In our case, the answer y = 6 is already in its simplest form. There's nothing more we can do to make it look simpler. However, in other problems, you might have to reduce fractions, combine like terms, or perform other simplifications. Always aim to present your answer in the easiest-to-understand way possible. Simplifying shows a deeper understanding of the concepts involved and makes it easier for others (and yourself!) to follow your reasoning. It demonstrates precision and attention to detail, which are super important in math. This also ensures that we present the answer in the clearest and most straightforward manner. So, in our case, we were already set, but keep the simplification principle in mind for other future equations. Remember, the simpler, the better.

Conclusion: You Did It!

Well done, everyone! You successfully solved for y in the equation 14y = 6y + 48. You learned the essential steps of isolating a variable and verifying your solution. You've also learned the importance of simplifying answers and checking your work. Keep practicing, and you'll become a pro at solving equations like these. Remember, math is like a muscle; the more you use it, the stronger you get. Thanks for joining me on this math adventure, guys. Keep up the excellent work, and always remember to double-check those answers!