Solving Equations: Double And Opposite Numbers

Hey guys! Let's dive into a cool math problem where we'll figure out how to find two special numbers. These numbers have a neat relationship: one is double the other. Then, we throw in a twist – if we add 4.8 to both of them, they magically become opposites! Sounds like a fun challenge, right? We're going to break down this problem step by step, making sure everything is super clear and easy to understand. We'll use our knowledge of algebra to solve this puzzle. So, grab your pencils and let's get started. By the end, you'll be a pro at finding these kinds of numbers, and it's super useful for all sorts of other math problems too.

Understanding the Problem: The Core Concepts

Alright, before we jump into the math, let's make sure we're all on the same page. The heart of this problem lies in understanding a few key concepts. First, we need to grasp what it means for one number to be double another. That means one number is exactly twice the size of the other. For example, if one number is 5, the other is 10. Simple, right? Next up, we have opposites. Opposites are numbers that are the same distance from zero on the number line but on opposite sides. For instance, 3 and -3 are opposites. They cancel each other out when added together (3 + (-3) = 0). Got it? Finally, we need to know how to set up an equation. An equation is like a mathematical sentence that says two things are equal. We'll use variables (like x) to represent our unknown numbers, and then use mathematical operations (addition, subtraction, multiplication, division) to build our equation. The goal is to isolate the variable and find its value. This is the art of algebra, where we use symbols to represent numbers and solve problems. Don't worry if it sounds complicated – we'll go slowly and make sure everything is crystal clear. Remember, practice makes perfect, so don't be afraid to try some examples on your own.

Now, let's go back to our problem. We are looking for two numbers that have a special relationship. One number is twice as big as the other. If we take both numbers and add 4.8, we get two opposite numbers. To solve this, we can set up an algebraic equation. We'll represent the smaller number as 'x', and the larger number (which is double the smaller one) as '2x'. Then, we can create an equation that reflects the second part of the problem. Adding 4.8 to each of our numbers turns them into opposites. So we get x + 4.8 = - (2x + 4.8). This is our equation. The first step in solving this equation is to get all the terms containing 'x' to one side of the equation and the constants to the other. By doing so, we're essentially grouping similar elements together to simplify and solve for 'x'. Once we solve for 'x', we find one of the required numbers. To find the second number, we apply the initial condition: double the value of x. Finally, we'll want to check our answer to make sure we did it right. This is done by substituting the numbers in the equation to confirm that adding 4.8 produces opposite numbers.

Setting Up the Equation: Translating Words to Math

Okay, guys, let's translate the problem into mathematical language. This is where we use variables and symbols to represent the words. We know we have two numbers, and one is double the other. Let's call the smaller number 'x'. Since the other number is double, it will be '2x'. Easy peasy! Now, the problem says that if we add 4.8 to each of these numbers, we get opposites. So, let's add 4.8 to both of our expressions. The first number becomes 'x + 4.8', and the second becomes '2x + 4.8'. Remember, opposites add up to zero. This means that the sum of these two new expressions should be zero. Or, one of them is the negative of the other. Here's our equation: x + 4.8 = -(2x + 4.8). Another way to write it is x + 4.8 + 2x + 4.8 = 0. Notice how the problem's conditions translate directly into this equation. The key here is to take each piece of information and represent it mathematically. The word "double" becomes "2x", adding 4.8 is "+ 4.8", and opposites mean that the two expressions are equal in magnitude but opposite in sign. So, taking the time to translate each piece of the sentence into an equation is very important to solve the problem. Practice this a bit, and you'll find that it becomes second nature. And hey, even if you make a mistake, it's a great learning opportunity. Now we know what our equation is, it's time to solve for x!

To solve this equation, let's start by removing the parenthesis in the expression -(2x + 4.8). That becomes -2x - 4.8. Now our equation looks like this: x + 4.8 = -2x - 4.8. Our next goal is to get all the 'x' terms on one side of the equation and all the constants on the other side. Let's add 2x to both sides. Doing this gives us: x + 2x + 4.8 = -2x + 2x - 4.8, which simplifies to 3x + 4.8 = -4.8. Now, let's subtract 4.8 from both sides. This gives us: 3x + 4.8 - 4.8 = -4.8 - 4.8, which further simplifies to 3x = -9.6. To find the value of x, we need to isolate it by dividing both sides by 3. Doing this gives us x = -9.6 / 3. Finally, x = -3.2. Therefore, our smaller number is -3.2. Now, using the first conditions, since the second number is double the first one, it will be -3.2 * 2 = -6.4. Those are our two numbers. But are they correct? Let's check!

Solving for the Unknown: Finding the Value of 'x'

Now, let's roll up our sleeves and solve the equation we created. Remember, our equation is x + 4.8 = -(2x + 4.8). This is where the real fun begins! Our goal is to find the value of 'x', which represents the smaller of our two numbers. To do this, we need to isolate 'x' on one side of the equation. This involves using inverse operations to undo the operations that are applied to 'x'. First, let's simplify the right side of the equation by distributing the negative sign. That is, we multiply the term inside the parenthesis by -1. So, -(2x + 4.8) becomes -2x - 4.8. Now our equation is x + 4.8 = -2x - 4.8. Next, let's get all the 'x' terms on one side and the constant terms on the other side. We can start by adding 2x to both sides of the equation. This gives us x + 2x + 4.8 = -2x + 2x - 4.8. This simplifies to 3x + 4.8 = -4.8. Now, we want to get the constant terms to the right side of the equation. To do that, we can subtract 4.8 from both sides. We then have: 3x + 4.8 - 4.8 = -4.8 - 4.8. This simplifies to 3x = -9.6. Finally, we isolate 'x' by dividing both sides of the equation by 3. This gives us x = -9.6 / 3, which equals x = -3.2. Great job! We've found the value of x. Now we have one of the numbers.

To find the second number, we need to use the relationship we know: one number is double the other. Therefore, if x = -3.2, then the other number is 2 * (-3.2) = -6.4. So, our two numbers are -3.2 and -6.4. See how organized steps and attention to detail make a problem easier to solve. Don't worry if it takes a bit of practice. The most important thing is to understand each step. Take your time, break down the problem, and you'll find that solving equations is a lot more manageable than it might seem at first. Now that we have our answer, let's make sure it's correct.

Verifying the Solution: Checking Our Work

Alright, folks, we've done the hard work, and we've got our two numbers: -3.2 and -6.4. But, hold on a second! Math is all about accuracy, so we need to make sure our solution is correct. This is where we verify our solution by plugging the numbers back into the original problem and seeing if everything checks out. Let's recap the problem: we're looking for two numbers where one is double the other, and when we add 4.8 to both, they become opposites. First, let's check the double relationship. Is -6.4 double -3.2? Yes, it is! (-3.2 * 2 = -6.4). Great. Now, let's add 4.8 to each number and see if they become opposites. Adding 4.8 to -3.2 gives us -3.2 + 4.8 = 1.6. Adding 4.8 to -6.4 gives us -6.4 + 4.8 = -1.6. Are 1.6 and -1.6 opposites? Yes, they are! They are the same distance from zero, but on opposite sides of the number line. Bingo! Our solution is correct. We successfully found two numbers that satisfy all the conditions of the problem.

This verification step is crucial. It gives us confidence in our answer and helps us catch any mistakes we might have made along the way. Always remember to check your work. It's like proofreading your essay – it helps you catch errors and make sure your ideas are clear and accurate. So, in summary, we found that -3.2 and -6.4 are the two numbers that solve the problem. Remember the process? First, we translated the problem into an equation, then we solved for our variable (x), and finally, we checked our answer to make sure it was right. That's the formula for success in many math problems! Now, give it a try with different numbers or conditions. With practice, you'll become a master of solving these kinds of problems.

Conclusion: Wrapping It Up and Next Steps

Awesome work, everyone! We've successfully navigated the world of equations, doubles, and opposites. We started with a word problem, broke it down into smaller, manageable parts, translated it into mathematical language, and finally, found the solution. We learned how to identify key information, set up an equation, solve for an unknown variable, and verify our answer. Remember, the process is key! Don't get discouraged if it takes a bit of time to grasp the concepts. Practice makes perfect, and with each problem you solve, you'll become more confident in your abilities. Now, you can use these skills to tackle similar problems where you need to find numbers that satisfy specific conditions. Try creating your own variations of this problem. What happens if you change the relationship between the numbers? Or the amount you add? Experimenting is a fantastic way to solidify your understanding and discover new mathematical insights. Consider what if you change the relationship to the triple or quadruple. Try using decimals or fractions. Use these as a starting point. Keep practicing, keep exploring, and keep the curiosity alive. Math is like a puzzle, and it's incredibly rewarding to find the missing pieces. So, keep up the great work and have fun exploring the wonders of mathematics!