Solving The Equation 3(2x + 2) = 3x - 15 For Water Level Change

Hey guys! Let's dive into a math problem that's got some real-world vibes. We're talking about water levels in a lake and how they change over time. The equation we're tackling today is 3(2x + 2) = 3x - 15, and our mission is to solve for 'x'. This 'x' represents the change in the lake's height, measured in inches. It's super cool how math can help us understand and predict these kinds of natural phenomena, right? So, let's put on our thinking caps and get started!

Breaking Down the Equation: 3(2x + 2) = 3x - 15

When we first look at the equation 3(2x + 2) = 3x - 15, it might seem a bit intimidating, but don't worry, we're going to break it down step by step. The key here is to follow the order of operations and apply some basic algebraic principles. Our main goal is to isolate 'x' on one side of the equation, so we can figure out its value. So, let's start by understanding each part of the equation. On the left side, we have 3 multiplied by the expression (2x + 2). This means we need to distribute the 3 across both terms inside the parentheses. On the right side, we have 3x - 15, which is a simple linear expression. Remember, 'x' is our unknown, the change in the lake's height in inches, and that's what we're trying to find.

Step 1: Distribute on the Left Side

The first thing we need to do is tackle that left side of the equation. We've got 3(2x + 2). To simplify this, we're going to use the distributive property. This means we multiply the 3 by each term inside the parentheses. So, 3 times 2x gives us 6x, and 3 times 2 gives us 6. This transforms our left side into 6x + 6. Distributive property is super important in algebra, and it's a tool we'll use a lot when solving equations. It's like a secret weapon for simplifying things! By distributing, we're essentially getting rid of the parentheses and making the equation easier to work with. So, our equation now looks like this: 6x + 6 = 3x - 15. See? Already, it's starting to look a bit less complicated. This is the power of breaking things down step by step.

Step 2: Gather 'x' Terms on One Side

Now that we've simplified the left side, let's focus on getting all the 'x' terms together. We have 6x + 6 = 3x - 15. To get all the 'x's on one side, we can subtract 3x from both sides of the equation. Why do we do this? Well, it's like a balancing act. Whatever we do to one side of the equation, we have to do to the other to keep things equal. Subtracting 3x from both sides gives us 6x - 3x + 6 = 3x - 3x - 15, which simplifies to 3x + 6 = -15. Now, we've got all our 'x' terms on the left side, which is a big step forward. It's like we're herding all the 'x's into one pen, making them easier to count and manage. This step is crucial because it brings us closer to isolating 'x' and finding its value.

Step 3: Isolate the 'x' Term

We're making great progress! We've got 3x + 6 = -15. Now, we want to isolate the 'x' term, which means getting it all by itself on one side of the equation. To do this, we need to get rid of that +6. We can do that by subtracting 6 from both sides. Remember, it's all about keeping the equation balanced. So, we have 3x + 6 - 6 = -15 - 6, which simplifies to 3x = -21. We're almost there! It's like we're peeling away the layers of an onion, getting closer and closer to the core. Isolating the 'x' term is a crucial step in solving for 'x', and we've just nailed it.

Step 4: Solve for 'x'

Okay, we're in the home stretch! We've got 3x = -21. To finally solve for 'x', we need to get rid of that 3 that's multiplying 'x'. We can do this by dividing both sides of the equation by 3. So, we have 3x / 3 = -21 / 3, which simplifies to x = -7. Boom! We've done it! We've found the value of 'x'. It's like cracking a code and revealing the hidden message. In this case, the hidden message is that 'x' equals -7. This means the change in the lake's height is -7 inches. But wait, what does a negative change mean in the context of our problem? Let's explore that in the next section.

Interpreting the Solution: What Does x = -7 Mean?

So, we've crunched the numbers and arrived at the solution x = -7. But what does this actually mean in the context of our lake level problem? Remember, 'x' represents the change in the lake's height in inches over one month. The fact that we got a negative value for 'x' tells us that the water level in the lake has decreased. Specifically, it has decreased by 7 inches during that month. Isn't it amazing how a simple negative sign can tell us so much about what's happening in the real world? This is why understanding the context of a problem is just as important as being able to solve the equation. We're not just dealing with abstract numbers here; we're talking about a real-world situation where the water level in a lake is changing. So, when you see x = -7, you can picture the lake's water level going down by 7 inches.

Understanding the negative sign is crucial here. In many real-world applications, negative values indicate a decrease or a loss. In this case, the negative sign in -7 tells us that the lake's water level has gone down, not up. This is a key insight that helps us interpret the mathematical solution in a meaningful way. It's like translating from math-speak into plain English. So, the next time you encounter a negative solution in a word problem, remember to think about what that negative sign actually means in the real-world context.

Why Other Options Are Incorrect

It's important to not only know the correct answer but also understand why the other options are incorrect. This helps solidify our understanding of the problem-solving process. In this case, we were given a few potential solutions: x = -17/3, x = 3, and x = 7, in addition to our correct answer, x = -7. Let's briefly discuss why those other options don't work.

• x = -17/3: This is a negative fraction, which might seem plausible since we know the water level decreased. However, if you were to substitute -17/3 back into the original equation, you would find that it does not make the equation true. The left side would not equal the right side, meaning it's not a valid solution.
• x = 3: This is a positive number, which would imply that the lake's water level increased. But based on the equation, we know the water level actually decreased, so this option doesn't fit the context of the problem. Additionally, if you substitute x = 3 into the original equation, you'll find that it doesn't balance the equation.
• x = 7: Similar to x = 3, this is also a positive number and would suggest an increase in water level. This contradicts what the equation tells us about the water level decreasing. Substituting x = 7 into the original equation would also show that it's not a correct solution.

The key takeaway here is that it's always a good idea to check your solution by plugging it back into the original equation. If the equation holds true, then you've likely found the correct answer. If not, then you know you need to go back and review your steps.

Real-World Applications of Solving Equations

Solving equations like 3(2x + 2) = 3x - 15 isn't just an abstract math exercise; it has tons of real-world applications. Think about it: we used it to model the change in a lake's water level, but the same principles can be applied to all sorts of scenarios. For instance, engineers might use equations to calculate stress on a bridge, economists might use them to predict market trends, and scientists might use them to model population growth. The possibilities are endless!

Equations are the language of the universe, and being able to solve them is like learning to speak that language. The skills we've used in this problem, such as distributing, combining like terms, and isolating variables, are fundamental tools that can be applied in many different fields. Whether you're planning a budget, designing a building, or conducting research, the ability to solve equations will be a valuable asset. So, keep practicing and honing your skills. You never know when you'll need them!

Conclusion: The Power of Math in Understanding the World

We've reached the end of our journey through this equation, and hopefully, you've gained a deeper understanding of how math can help us make sense of the world around us. We started with the equation 3(2x + 2) = 3x - 15, and through a series of steps, we solved for 'x', finding that x = -7. We then interpreted this solution in the context of our lake level problem, understanding that the negative sign indicates a decrease in the water level of 7 inches.

But more than just solving a single equation, we've explored the broader power of math as a tool for modeling and understanding real-world phenomena. The same principles we used here can be applied to a wide range of problems, from engineering to economics to environmental science. So, keep practicing, keep exploring, and keep asking questions. The world is full of mathematical mysteries just waiting to be solved!