Solving 3(y+9)=12y+13 How Many Solutions?

by Felix Dubois 42 views

Hey everyone! Today, we're diving into the world of algebra to figure out how many solutions a particular equation has. We'll be tackling the equation 3(y+9) = 12y + 13. The question asks us to choose one of the following answers:

  • A. No solutions
  • B. Exactly one solution
  • C. Infinitely many solutions

So, let's roll up our sleeves and get started! Our mission? To break down this equation step by step, making sure we understand each move we make. We're not just aiming for the answer; we're after the why behind it. Trust me, understanding the process is way more valuable than just memorizing the solution. Once you grasp the underlying principles, you'll be able to tackle similar problems with confidence. Think of it like learning to ride a bike – once you get the hang of it, you can ride any bike! So, let's jump in and transform this equation from a mystery into a piece of cake. We'll go through each step together, making sure everything is crystal clear. By the end of this guide, you'll not only know the answer but also feel like a total algebra whiz!

Unpacking the Equation: 3(y+9) = 12y + 13

Okay, let's get down to business and really dissect this equation, 3(y+9) = 12y + 13. When we first look at it, it might seem a bit intimidating, but don't worry, we're going to break it down into bite-sized pieces. The first thing that probably jumps out at you is the parentheses, right? Those are like little packages that we need to unwrap before we can move on. To do that, we're going to use the distributive property. This is a fancy term, but all it really means is that we need to multiply the number outside the parentheses (in this case, 3) by each term inside the parentheses (y and +9). Think of it like this: we're making sure everyone inside the package gets a fair share of the 3.

So, let's do it! We multiply 3 by y, which gives us 3y, and then we multiply 3 by +9, which gives us +27. Now, we can rewrite the left side of the equation as 3y + 27. See? We've already made progress! The equation now looks a bit simpler: 3y + 27 = 12y + 13. We've successfully unwrapped our little package. Now, let's take a moment to appreciate what we've done. We've transformed the original expression into something much more manageable, and that's a key skill in solving algebraic equations. Remember, it's all about breaking down complex problems into simpler steps. Now that we've handled the distributive property, we're ready to move on to the next stage of solving this equation. We're building our understanding step by step, and that's the best way to master these concepts. So, let's keep going and see what other algebraic magic we can work!

Simplifying the Equation: Combining Like Terms

Alright, now that we've got our equation looking a little cleaner – 3y + 27 = 12y + 13 – it's time to talk strategy. Our main goal here is to isolate the variable y. What that means is we want to get y all by itself on one side of the equation. Think of it like giving y its own private island! To do that, we're going to use a technique called “combining like terms.” This basically means we want to group all the y terms together and all the constant terms (the plain numbers) together. It's like sorting your socks – you want to keep the pairs together, right? Same idea here.

So, let's start by moving all the y terms to one side of the equation. A common approach is to move them to the side that has the larger coefficient (the number in front of the y) to avoid dealing with negative numbers. In this case, 12y is larger than 3y, so let's aim to get all the y terms on the right side. To do that, we need to get rid of the 3y on the left side. How do we do that? We subtract 3y from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. It's like a seesaw – if you take weight off one side, you need to take the same weight off the other to keep it level. So, subtracting 3y from both sides gives us: 27 = 9y + 13. Awesome! We've made progress in isolating y. Now, let's focus on getting those constant terms together. We're getting closer and closer to giving y its own island!

Isolating the Variable: The Final Countdown

Okay, guys, we're on the home stretch now! We've got our equation looking like this: 27 = 9y + 13. Remember our mission? We're trying to get y all by itself on one side of the equation – giving it its own private island, so to speak. We've already grouped the y terms together, so now it's time to tackle the constant terms. We need to get that +13 away from the 9y. The key here is to do the opposite operation. Since 13 is being added to 9y, we need to subtract 13 from both sides of the equation. Think of it like undoing a puzzle piece by piece.

So, let's subtract 13 from both sides. This gives us: 27 - 13 = 9y + 13 - 13. Simplifying that, we get: 14 = 9y. We're getting so close! Now, we have just one more step to completely isolate y. We have 9y, which means 9 multiplied by y. To undo this multiplication, we need to do the opposite operation: division. We're going to divide both sides of the equation by 9. This will finally get y all by itself.

So, let's do it! Dividing both sides by 9, we get: 14 / 9 = (9y) / 9. Simplifying that, we get: y = 14/9. Boom! We've done it! We've successfully isolated y. Now we know the value of y that satisfies this equation. It's like we've cracked the code and found the secret key. But, before we celebrate too much, let's think about what this result means in terms of our original question. We were asked how many solutions this equation has. We've found one solution, but does that mean there might be more? Let's dive into that question and make sure we fully understand what's going on.

Determining the Number of Solutions

So, we've crunched the numbers and found that y = 14/9. That's awesome! But let's zoom out for a second and think about the big picture. The original question wasn't just asking us to find a solution; it was asking us how many solutions there are. We've landed on one solution, but how do we know if there are more lurking out there? Well, the type of equation we're dealing with gives us a big clue. We started with a linear equation. Linear equations are those friendly equations where the variable (in this case, y) is only raised to the power of 1. There are no sneaky squares, cubes, or other exponents involved. Linear equations are pretty straightforward, and they have a predictable behavior.

Here's the key thing to remember: A linear equation can have one solution, no solutions, or infinitely many solutions. That's it. There are no other possibilities. So, how do we know which one it is? Well, if we've gone through the process of solving the equation and we've arrived at a single value for y, like we did here, that tells us we have exactly one solution. It's like finding the one key that unlocks a specific door. There's no other key that will work. But, what would it look like if there were no solutions? In that case, when we try to solve the equation, we'd end up with a statement that's just plain false. For example, we might end up with something like 5 = 7. That's clearly not true, and it would tell us that there's no value of y that can make the equation work. It's like trying to fit a square peg in a round hole – it just won't happen.

And what about infinitely many solutions? This happens when the equation is actually true for any value of y. When we try to solve it, we'd end up with a statement that's always true, like 0 = 0. This means that no matter what number we plug in for y, the equation will hold. It's like having a magic door that opens for anyone, no matter who they are. So, based on our journey through this equation, what's our final answer? We found one specific value for y, which means...

Final Answer: Exactly One Solution

Alright, guys, we've reached the finish line! We started with the equation 3(y+9) = 12y + 13, and after carefully unpacking it step by step, we discovered that y = 14/9. We conquered the distributive property, combined like terms, and isolated the variable like true algebra pros. But, more importantly, we didn't just stop at finding the solution; we thought about what that solution means in the grand scheme of things. We remembered that linear equations can have one solution, no solutions, or infinitely many solutions, and we used our result to figure out which one we had.

Since we arrived at a single, specific value for y, we know that the equation has exactly one solution. It's like we've solved the mystery and found the one true answer. So, the correct answer to the question “How many solutions does the following equation have?” is B. Exactly one solution. Give yourselves a pat on the back! You've not only solved the equation but also understood the reasoning behind it. That's the real magic of learning mathematics – it's not just about getting the right answer; it's about understanding why the answer is right. And now, you're equipped to tackle similar problems with confidence. You've got the tools, you've got the knowledge, and you've got the skills. So, go out there and conquer those equations!

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