Solve The Equation: -6.8 + 6.4 + ___ = 0
Hey guys! Today, we're diving into a fun math problem. We've got an incomplete equation, and our mission, should we choose to accept it, is to figure out which expression will make it true. It's like a puzzle, and who doesn't love puzzles? So, let's break it down step by step and get to the solution. This is a foundational concept in mathematics, and mastering it will help you tackle more complex problems later on. We'll go through the basics of adding and subtracting decimals, how to balance equations, and the importance of understanding number properties. Ready to get started? Let's do this!
Understanding the Problem
The equation we're tackling is:
-6.8 + 6.4 + ___ = 0
Our goal is crystal clear: we need to find the expression that, when added to -6.8 and 6.4, results in zero. Think of it like balancing a scale. The left side of the equation needs to perfectly balance the right side, which in this case, is zero. To make this happen, we first need to simplify the known part of the equation. This involves adding -6.8 and 6.4. Remember, when adding numbers with different signs, we're essentially finding the difference between their absolute values and then using the sign of the number with the larger absolute value. So, let's get into the nitty-gritty of the math and see what we get!
Step-by-Step Breakdown
Let's take it slow and easy, guys. First, focus on the initial part of the equation:
-6.8 + 6.4
To solve this, we subtract the absolute values: |6.8| - |6.4| = 6.8 - 6.4 = 0.4. Now, because -6.8 has a larger absolute value than 6.4, the result will be negative. Therefore:
-6.8 + 6.4 = -0.4
Now our equation looks simpler:
-0.4 + ___ = 0
What number do we need to add to -0.4 to get zero? This is where our understanding of additive inverses comes into play. An additive inverse is a number that, when added to another number, results in zero. In simple terms, it's the opposite of the number. So, the additive inverse of -0.4 is 0.4. This means we need an expression that equals 0.4. Now, let's look at the answer choices and see which one fits the bill!
Evaluating the Answer Choices
We've simplified the equation and know we need an expression that equals 0.4. Now, let's put on our detective hats and evaluate each answer choice. We'll add the numbers in each option and see if we get 0.4. This is where careful calculation is key, guys. A small mistake can throw us off, so let's double-check our work as we go. We'll take each option one by one, break it down, and see if it leads us to the correct answer.
Option A: -4.3 + (-4.7)
Let's start with the first option: A. -4.3 + (-4.7). When we add two negative numbers, we simply add their absolute values and keep the negative sign. So,
-4.3 + (-4.7) = -(4.3 + 4.7) = -9.0
This result, -9.0, is definitely not equal to 0.4. So, Option A is not the correct answer. We can cross this one off our list and move on to the next option. Remember, it's all about systematically checking each possibility until we find the one that fits perfectly.
Option B: -6.5 + 6.7
Next up is Option B: -6.5 + 6.7. Here, we're adding numbers with different signs. As we discussed earlier, we find the difference between their absolute values and use the sign of the number with the larger absolute value. So,
|-6.5| = 6.5
|6.7| = 6.7
6.7 - 6.5 = 0.2
Since 6.7 has a larger absolute value and is positive, the result is positive 0.2. While 0.2 is closer to our target of 0.4 than -9.0, it's still not the right answer. So, we'll keep looking!
Option C: -4.3 + 4.7
Let's take a look at Option C: -4.3 + 4.7. This one looks promising! Again, we're adding numbers with different signs, so we find the difference between their absolute values:
|4.7| = 4.7
|-4.3| = 4.3
4.7 - 4.3 = 0.4
And there we have it! The result is 0.4, which is exactly what we needed. So, Option C is the correct answer. But, just to be thorough, let's check Option D as well.
Option D: -6.5 + (-6.7)
Finally, let's check Option D: -6.5 + (-6.7). Similar to Option A, we're adding two negative numbers. We add their absolute values and keep the negative sign:
-6.5 + (-6.7) = -(6.5 + 6.7) = -13.2
This result, -13.2, is nowhere near 0.4. So, Option D is definitely not the correct answer. We've confirmed that Option C is indeed the solution.
The Solution
We've cracked the code, guys! After carefully evaluating each option, we found that the expression that makes the equation true is:
C. -4.3 + 4.7
When we place this expression in the blank, the equation becomes:
-6.8 + 6.4 + (-4.3 + 4.7) = 0
-6.8 + 6.4 + 0.4 = 0
-0.4 + 0.4 = 0
0 = 0
The equation is balanced! We did it! Understanding how to solve equations like this is a crucial skill in mathematics. It involves not just arithmetic but also a solid grasp of number properties and how to manipulate equations to find the unknown. So, give yourselves a pat on the back for sticking with it and solving this problem with me!
Key Takeaways
Before we wrap up, let's highlight some key takeaways from this problem. These are the golden nuggets of knowledge that you can carry forward and apply to other math challenges. Understanding these concepts will not only help you solve similar problems but will also build a stronger foundation for more advanced math topics.
1. Adding Numbers with Different Signs
When you're adding numbers with different signs (one positive and one negative), remember to find the difference between their absolute values. The result will have the same sign as the number with the larger absolute value. This is a fundamental rule in arithmetic and will come in handy time and time again.
2. Additive Inverses
The concept of additive inverses is super important. An additive inverse is a number that, when added to another number, results in zero. For example, the additive inverse of 5 is -5, and the additive inverse of -3.2 is 3.2. Recognizing additive inverses can simplify equations and make problem-solving much easier.
3. Systematic Evaluation
When faced with multiple answer choices, like in this problem, it's crucial to evaluate each option systematically. Don't jump to conclusions! Take the time to work through each choice, eliminating the ones that don't fit until you find the correct answer. This methodical approach will help you avoid mistakes and build confidence in your solutions.
4. Double-Check Your Work
Last but not least, always double-check your work! Math problems can be tricky, and a small error in calculation can lead to the wrong answer. Take a moment to review your steps, make sure your arithmetic is correct, and that you haven't missed any details. This simple habit can significantly improve your accuracy and your overall problem-solving skills.
Practice Makes Perfect
So there you have it, guys! We've successfully solved the equation and learned some valuable math lessons along the way. Remember, practice makes perfect. The more you work on problems like this, the more confident and skilled you'll become. So, keep practicing, keep asking questions, and keep challenging yourselves. Math can be fun, and with a little effort, you can master it. Keep up the great work, and I'll see you in the next problem!