Addition Expression With Sum 8-3i Step-by-Step Solution

Hey guys! Today, we're diving into a fun little math problem where we need to figure out which addition expression results in the complex number 8 - 3i. Complex numbers might sound intimidating, but don't worry, they're actually pretty straightforward once you get the hang of them. We'll go through each option step-by-step, making sure you understand exactly how to add complex numbers and find the right answer. So, let's put on our math hats and get started!

Understanding Complex Numbers

Before we jump into the expressions, let's quickly recap what complex numbers are all about. A complex number has two parts: a real part and an imaginary part. The real part is just a regular number, like 8 or -1, while the imaginary part is a number multiplied by 'i', where 'i' is the imaginary unit defined as the square root of -1. So, things like 2i, -3i, and 4i are imaginary numbers. A complex number is usually written in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. For example, in the complex number 8 - 3i, 8 is the real part, and -3 is the coefficient of the imaginary part.

When we add complex numbers, we treat the real and imaginary parts separately. It's like combining like terms in algebra. We add the real parts together and the imaginary parts together. For instance, if we have (2 + 3i) + (4 - i), we would add the real parts (2 + 4) to get 6 and the imaginary parts (3i - i) to get 2i. So, the sum would be 6 + 2i. Remember, we can only add real parts with real parts and imaginary parts with imaginary parts. Mixing them up would be like trying to add apples and oranges – they just don't go together! Keeping this in mind will help us solve our problem efficiently and accurately.

The Target: 8 - 3i

Our goal is to find the addition expression that equals 8 - 3i. This means we need to look for the expression where the real parts add up to 8, and the imaginary parts add up to -3i. Think of it as a puzzle where we need to match the pieces correctly. We'll examine each option to see if it fits the target. We'll add the real components and imaginary components of each option and check if the sums equal 8 and -3i, respectively. This will involve basic addition and subtraction, but it’s crucial to be meticulous to avoid any simple arithmetic errors. Accuracy is key in these calculations, as a small mistake can lead to the wrong answer. Always double-check your work to ensure you've added the numbers correctly and haven't missed any negative signs. By systematically evaluating each option, we can confidently identify the expression that matches our desired sum of 8 - 3i.

Evaluating the Expressions

Okay, let's get our hands dirty and evaluate each expression. We'll take them one by one, add the real and imaginary parts, and see if we get our target sum of 8 - 3i. Ready? Let's go!

Option 1: (9 + 2i) + (1 - i)

First up, we have (9 + 2i) + (1 - i). To add these complex numbers, we'll add the real parts together and the imaginary parts together. The real parts are 9 and 1, so 9 + 1 equals 10. The imaginary parts are 2i and -i, so 2i + (-i) equals i (since 2 - 1 = 1). Combining these results, we get 10 + i. Now, let's compare this to our target sum, which is 8 - 3i. Clearly, 10 + i is not equal to 8 - 3i. The real parts don't match (10 ≠ 8), and the imaginary parts don't match either (1 ≠ -3). So, this option is not the correct one. We need to keep searching! It’s important to note how systematically we are approaching this problem. By breaking down the complex numbers into their real and imaginary parts, we make the addition process more manageable and less prone to error. This method allows us to clearly see if the resulting sum matches our target complex number.

Option 2: (9 + 4i) + (-1 - 7i)

Next, we're tackling (9 + 4i) + (-1 - 7i). Again, we'll add the real parts and the imaginary parts separately. The real parts are 9 and -1, so 9 + (-1) equals 8. The imaginary parts are 4i and -7i, so 4i + (-7i) equals -3i (since 4 - 7 = -3). Combining these, we get 8 - 3i. Bingo! This is exactly what we're looking for. The real part is 8, and the imaginary part is -3i, which matches our target sum perfectly. So, it looks like we've found our answer! But just to be sure, let's quickly check the remaining options. It's always a good idea to be thorough and confirm our solution, especially in math problems where a small mistake can lead to the wrong answer. This approach ensures that we not only find an answer but also verify its correctness.

Option 3: (7 + 2i) + (1 - i)

Let's move on to the third option: (7 + 2i) + (1 - i). Adding the real parts, we have 7 + 1, which equals 8. Adding the imaginary parts, we have 2i + (-i), which equals i. So, the sum is 8 + i. Comparing this to our target of 8 - 3i, we see that while the real parts match (8 = 8), the imaginary parts do not (i ≠ -3i). Therefore, this option is not the correct one. Although the real part of the sum matches our target, the mismatch in the imaginary part disqualifies this option. This highlights the importance of checking both the real and imaginary parts when dealing with complex numbers. A correct sum requires both components to match the target.

Option 4: (7 + 4i) + (-1 - 7i)

Finally, let's check the last option: (7 + 4i) + (-1 - 7i). Adding the real parts, we get 7 + (-1), which equals 6. Adding the imaginary parts, we get 4i + (-7i), which equals -3i. So, the sum is 6 - 3i. Comparing this to our target of 8 - 3i, we see that the real parts do not match (6 ≠ 8), even though the imaginary parts do (-3i = -3i). This means this option is also incorrect. The real part of the sum, 6, is different from the real part of our target, which is 8. Thus, even though the imaginary parts match, the expression as a whole does not sum up to the target complex number. This underscores the importance of ensuring that both the real and imaginary parts align with the target when adding complex numbers.

The Winner! (9 + 4i) + (-1 - 7i)

Alright guys, we've gone through all the options, and it's clear that the expression (9 + 4i) + (-1 - 7i) is the winner! When we add these complex numbers together, we get exactly 8 - 3i, which is the target sum we were looking for. We added the real parts (9 and -1) to get 8, and we added the imaginary parts (4i and -7i) to get -3i. This matches our target perfectly. So, we can confidently say that this is the correct answer. We systematically evaluated each option, ensuring we didn't miss any details, and accurately identified the expression that meets our criteria. This meticulous approach is crucial in solving mathematical problems, as it minimizes the chances of errors and confirms the validity of our solution.

Final Answer

So, to wrap it all up, the addition expression that has the sum 8 - 3i is indeed (9 + 4i) + (-1 - 7i). Great job, everyone! You tackled complex numbers like pros! Keep practicing, and these problems will become second nature. Remember, the key is to break down the problem into smaller parts, add the real and imaginary parts separately, and compare the result to the target sum. You've got this! Keep up the awesome work, and let's keep exploring the fascinating world of mathematics together.