Algebraic Equation: Solve For Y

Solving for y in Linear Equations: A Step-by-Step Guide

Hey guys! Today we're diving into the exciting world of algebra, specifically tackling how to solve for y in a linear equation. You know, those equations where the highest power of our mystery variable is one? We're going to break down a common type of problem you might see, like the one presented: $3y = 2(4 - 2y) + 6$. This might look a little intimidating at first glance with the parentheses and multiple 'y' terms floating around, but trust me, with a systematic approach, you'll be solving for 'y' like a pro in no time. Our main goal here is to isolate 'y' on one side of the equation, making it the star of the show. We'll achieve this by using the fundamental rules of algebra: whatever we do to one side of the equation, we must do to the other to maintain balance. Think of it like a perfectly balanced scale; you can't just remove a weight from one side without affecting the other, right? We'll start by simplifying both sides of the equation, then gather all the 'y' terms together, and finally, get that lone 'y' all by itself. So, grab your notebooks, get comfy, and let's unravel this algebraic puzzle together. We'll not only solve the specific problem but also equip you with the skills to tackle similar equations, making sure you're well-prepared for any math challenge that comes your way. Remember, practice makes perfect, and understanding the 'why' behind each step is key to true mastery. Let's get started on this algebraic adventure!

Understanding the Equation: The Foundation of Solving for y

Before we even think about isolating 'y', it's crucial to really understand the equation we're working with: $3y = 2(4 - 2y) + 6$. This is what we call a linear equation because the variable 'y' is raised to the power of 1. The goal when we solve for y is to find the specific numerical value that makes this equation true. The equation has a left side, $3y$, and a right side, $2(4 - 2y) + 6$. To maintain the equality, any operation we perform must be applied to both sides. The first hurdle we often encounter with equations like this is the presence of parentheses. The term $2(4 - 2y)$ means we need to multiply the 2 by each term inside the parentheses. This is the distributive property in action, a cornerstone of algebraic manipulation. So, $2$ times $4$ gives us $8$, and $2$ times $-2y$ gives us $-4y$. Once we distribute, the right side of our equation transforms into $8 - 4y + 6$. Now, we can simplify this right side further by combining the constant terms. The constants are numbers that don't have a variable attached to them, in this case, $8$ and $6$. Adding them together, we get $14$. So, the right side simplifies to $-4y + 14$. Our equation now looks much cleaner: $3y = -4y + 14$. See? We're already making progress by simplifying and making the equation more manageable. This initial step of distribution and combining like terms is absolutely vital. It clears up the clutter and allows us to see the core structure of the equation more clearly. Don't skip this part, guys; it sets the stage for all the subsequent steps needed to solve for y effectively. Understanding this foundational step ensures you're building a solid understanding, not just memorizing steps.

Isolating the Variable: Gathering 'y' Terms

Alright, fam, we've successfully simplified our equation to $3y = -4y + 14$. Now comes the exciting part where we start to gather all the terms containing 'y' onto one side of the equation. Remember, our ultimate mission is to get 'y' all by itself. Currently, we have 'y' terms on both sides: $3y$ on the left and $-4y$ on the right. To bring them together, we need to eliminate the $-4y$ from the right side. The opposite operation of subtracting $4y$ is adding $4y$. So, we're going to add $4y$ to both sides of the equation. This is where our balanced scale analogy really shines. On the left side, $3y + 4y$ combines to give us $7y$. On the right side, $-4y + 4y$ cancels each other out, leaving us with just the constant term, $14$. So, after adding $4y$ to both sides, our equation transforms into $7y = 14$. Look at that! All the 'y' terms are now on one side, and we're one step closer to finding our specific value for 'y'. This process of moving variable terms is fundamental. We can move terms across the equals sign by performing the inverse operation. If a term is being added, we subtract it from both sides. If it's being subtracted, we add it. If it's being multiplied, we divide. If it's being divided, we multiply. It's all about using inverse operations to cancel things out and gather like terms. This strategy of isolating the variable is the core principle behind solving most algebraic equations, and mastering it will unlock your ability to tackle a wide array of problems. Keep up the great work, and let's move on to the final step to truly solve for y.

The Final Step: Finding the Value of y

We're in the home stretch, my math enthusiasts! Our equation has been simplified beautifully to $7y = 14$. We've successfully gathered all our 'y' terms on one side, and now we just need to get 'y' completely isolated. Currently, 'y' is being multiplied by 7. To undo multiplication, we use the inverse operation, which is division. So, to get 'y' by itself, we need to divide both sides of the equation by 7. On the left side, $7y$ divided by $7$ leaves us with just $y$. On the right side, $14$ divided by $7$ equals $2$. And voilà! We have successfully isolated 'y', and we find that $y = 2$. This is the solution to our equation. To double-check our work, we can substitute this value of $y=2$ back into the original equation: $3y = 2(4 - 2y) + 6$. Let's see: $3(2) = 2(4 - 2(2)) + 6$. This simplifies to $6 = 2(4 - 4) + 6$, then $6 = 2(0) + 6$, which further simplifies to $6 = 0 + 6$, and finally, $6 = 6$. Since both sides are equal, our solution $y=2$ is correct! This final verification step is super important. It confirms that you haven't made any errors along the way and that you truly understand how to solve for y. So, remember the steps: simplify using the distributive property, combine like terms, gather variable terms using inverse operations, and finally, isolate the variable by dividing. With these skills, you're ready to conquer any linear equation thrown your way. Keep practicing, and you'll become an algebra ace in no time! The answer we found, $y=2$, matches option B, confirming our calculations. Way to go!