Solve -1.8 = 0.2 + C/0.2: Step-by-Step Solution

Hey guys! Today, we're diving into a math problem that might seem a bit tricky at first, but I promise, it's totally manageable once we break it down. We're going to tackle the equation -1.8 = 0.2 + c/0.2. This is a classic example of an algebraic equation where we need to isolate a variable, in this case, c. Think of it like solving a puzzle – each step gets us closer to finding that missing piece. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into the solution, let's make sure we really understand the equation. It's not just about crunching numbers; it's about knowing what each part means. We have an equation that states that negative 1.8 is equal to the sum of 0.2 and c divided by 0.2. The goal here is crystal clear: we need to find the value of c that makes this equation true. To do that, we're going to use the magic of algebra – rearranging the equation in a way that c stands alone on one side. This is where the fun begins, like being a math detective on a mission!

Now, let's talk a little bit about why understanding the equation is so crucial. You see, in mathematics, it's not enough to just memorize steps or formulas. If you truly grasp the underlying concepts, you can tackle all sorts of problems, even ones you've never seen before. Understanding the equation is like having a map before you start a journey – it helps you see where you're going and how to get there. It allows you to make informed decisions about which operations to perform and in what order. This understanding is also what separates someone who can simply follow instructions from someone who can solve problems creatively and independently. So, always take a moment to really understand what the equation is saying before you start manipulating it.

And remember, every part of the equation has its role. The -1.8 is our target value, the result of our calculation. The 0.2 is a constant term, a number that doesn't change. And then we have c/0.2, which is the term that contains our variable, the unknown we're trying to find. By understanding how these parts relate to each other, we can strategize our approach to solving for c. Think of it like a balancing act – whatever we do to one side of the equation, we must do to the other to keep the balance. So, with our equation understood and our strategy in mind, let's move on to the next step and start unraveling this mathematical puzzle.

Step 1: Isolate the Term with 'c'

Alright, so the first thing we want to do is isolate the term with 'c'. What does that mean exactly? Well, we want to get the part of the equation that contains 'c' – which is c/0.2 – all by itself on one side of the equation. Think of it like clearing a space on your desk so you can focus on one specific task. To do this, we need to get rid of that pesky 0.2 that's being added on the right side of the equation. Remember, in algebra, we can move terms around as long as we do the same thing to both sides of the equation. It's all about keeping that balance!

So, how do we get rid of the 0.2? We use the inverse operation. Since 0.2 is being added, we're going to subtract 0.2 from both sides of the equation. This is a fundamental principle in algebra – using inverse operations to undo what's been done. It's like having a mathematical undo button! This step is crucial because it simplifies the equation and brings us closer to isolating 'c'. Without this step, we'd be trying to solve for 'c' with extra baggage, making the process much more complicated.

When we subtract 0.2 from both sides, the equation transforms. On the left side, we have -1.8 - 0.2, and on the right side, the 0.2s cancel each other out, leaving us with just c/0.2. This is exactly what we wanted! We've successfully isolated the term with 'c'. It's like separating the wheat from the chaff, focusing our attention on what's truly important. Now, we're in a much better position to find the value of 'c'. We've taken a complex equation and made it simpler, more manageable. This is the power of algebraic manipulation – breaking down problems into smaller, more digestible steps. So, with the term containing 'c' now isolated, we're ready to move on to the next step and finally solve for 'c'.

Step 2: Simplify the Equation

Okay, now that we've isolated the term with 'c', it's time to simplify the equation. This means performing any necessary calculations to make our equation as clean and straightforward as possible. Remember, after subtracting 0.2 from both sides, we have -1.8 - 0.2 on the left side. So, let's tackle that first. What is -1.8 minus 0.2? If you think of it on a number line, you're moving further into the negative numbers. So, -1.8 - 0.2 equals -2.0. Simplifying numerical expressions like this is a key skill in algebra, and it's something you'll use over and over again.

Now our equation looks much neater: -2.0 = c/0.2. See how much cleaner that is? It's like decluttering your workspace – with fewer distractions, we can focus better on the task at hand. Simplifying the equation not only makes it easier to look at but also easier to work with. It reduces the chances of making mistakes and helps us see the next steps more clearly. Think of it as taking a deep breath before diving into the next part of the problem. By simplifying, we're setting ourselves up for success.

This step is also a good opportunity to check our work. Did we make any mistakes in the previous step? Did we subtract correctly? It's always a good idea to double-check your calculations, especially when dealing with negative numbers, as they can sometimes be a bit tricky. Catching errors early can save you a lot of time and frustration later on. So, with our equation simplified to -2.0 = c/0.2, we can confidently move on to the next step, knowing that we've done our due diligence. We're getting closer and closer to finding the value of 'c', and with each step, the solution becomes clearer. Simplifying is like sharpening our tools before the final cut – it ensures that we're prepared to tackle the last part of the problem with precision and accuracy.

Step 3: Solve for 'c'

Alright, we've reached the final stretch! We're ready to solve for 'c'. Remember, our equation is now -2.0 = c/0.2. We need to get 'c' all by itself on one side of the equation. Right now, 'c' is being divided by 0.2. So, how do we undo division? You guessed it – we multiply! This is another application of inverse operations, a fundamental concept in algebra. Just like we used subtraction to undo addition, we're now going to use multiplication to undo division.

To get 'c' by itself, we need to multiply both sides of the equation by 0.2. This is crucial – whatever we do to one side, we must do to the other to maintain the balance of the equation. Think of it like a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. Multiplying both sides by 0.2 is the key to unlocking the value of 'c'.

When we multiply the right side, c/0.2, by 0.2, the 0.2 in the numerator and the 0.2 in the denominator cancel each other out, leaving us with just 'c'. This is exactly what we wanted! We've isolated 'c'. On the left side, we have -2.0 multiplied by 0.2. So, what is -2.0 times 0.2? Well, 2 times 0.2 is 0.4, and since we have a negative sign, the result is -0.4. Therefore, the solution to our equation is c = -0.4. We've done it! We've successfully solved for 'c'. It's like reaching the top of a mountain after a challenging climb – the view is definitely worth the effort!

Step 4: Verify the Solution

But wait, we're not quite done yet! It's always a good idea to verify the solution. What does that mean? It means plugging our answer, c = -0.4, back into the original equation to make sure it works. This is like checking your work in any other subject – it ensures that you haven't made any mistakes along the way. Verifying the solution is a crucial step in problem-solving, and it's something that good mathematicians always do.

So, let's go back to our original equation: -1.8 = 0.2 + c/0.2. We're going to substitute -0.4 for 'c' and see if the equation holds true. That gives us: -1.8 = 0.2 + (-0.4)/0.2. Now, let's simplify the right side of the equation. -0.4 divided by 0.2 is -2. So, the equation becomes: -1.8 = 0.2 + (-2). And what is 0.2 plus -2? It's -1.8! So, we have: -1.8 = -1.8. The equation holds true! This confirms that our solution, c = -0.4, is correct. It's like getting the green light after a thorough inspection – we can be confident that our answer is accurate.

Verifying the solution not only gives us peace of mind but also helps us catch any potential errors. If the equation didn't hold true, we would know that we need to go back and check our steps. This is why it's such an important part of the problem-solving process. It's like having a safety net – it protects us from making mistakes and ensures that we're on the right track. So, with our solution verified and our confidence high, we can proudly say that we've mastered this equation. We've not only found the answer but also confirmed its accuracy. That's the mark of a true math whiz!

Conclusion

And there you have it, guys! We've successfully solved the equation -1.8 = 0.2 + c/0.2, and found that c = -0.4. We walked through each step carefully, from understanding the equation to verifying our solution. Remember, the key to solving algebraic equations is to take it one step at a time, use inverse operations to isolate the variable, and always double-check your work. Math can be like a fun adventure when you approach it with confidence and a clear strategy. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!