Solving 821 = 11x: A Step-by-Step Guide

Hey guys! Let's break down how to solve the equation 821 = 11x. This is a classic algebra problem, and we'll walk through it together step by step. We need to isolate 'x' to find its value. So, let's dive right in!

Understanding the Equation

First off, let's make sure we all understand what the equation 821 = 11x means. In this equation, 'x' is our unknown variable – the number we're trying to find. The number 11 is multiplying 'x,' and the result of this multiplication should equal 821. Our main goal here is to get 'x' by itself on one side of the equation. To do that, we'll use a little algebraic magic, specifically, the concept of inverse operations. Remember those? What we do to one side of the equation, we must do to the other to keep everything balanced. This is super important because equations are all about balance! Think of it like a scale – if you add or subtract weight from one side, you need to do the same on the other to keep it level. Got it? Cool! Now, let’s move on to the next part where we'll actually start solving this thing. We'll use the inverse operation of multiplication to isolate 'x'. Keep reading, you're doing great!

Isolating the Variable

The core of solving any algebraic equation, especially one like 821 = 11x, is isolating the variable. In our case, the variable is 'x,' and it's currently being multiplied by 11. To get 'x' all by its lonesome on one side of the equation, we need to undo that multiplication. And how do we do that? With division! Division is the inverse operation of multiplication. This means that if we divide both sides of the equation by 11, we can effectively "cancel out" the 11 that's attached to 'x.' Remember that golden rule of equations: whatever you do to one side, you must do to the other. So, we're dividing both 821 and 11x by 11. This keeps the equation balanced and fair. Now, let's write that out mathematically: 821 / 11 = (11x) / 11. See what we did there? On the right side, the 11 in the numerator and the 11 in the denominator cancel each other out, leaving us with just 'x.' Awesome! But we're not done yet. We still need to perform the division on the left side to find the actual value of 'x.' So, let's grab our calculators (or our long division skills!) and figure out what 821 divided by 11 is. Keep going; we're almost there!

Performing the Division

Alright, let's tackle the division: 821 ÷ 11. This is where our basic arithmetic skills come into play. If you're reaching for a calculator, that's totally fine, but it's also great practice to do this by hand! When we divide 821 by 11, we're essentially asking, "How many times does 11 fit into 821?" You might remember the process of long division from school – starting from the left, we see how many times 11 goes into 82. It goes in 7 times (7 x 11 = 77). Then, we subtract 77 from 82, which gives us 5. We bring down the next digit, which is 1, making our new number 51. Now, how many times does 11 go into 51? It goes in 4 times (4 x 11 = 44). Subtracting 44 from 51 gives us 7. So, we have a quotient of 74 and a remainder of 7. But wait! We're looking for a precise answer here, and it seems we don't have a whole number. This is a good reminder that sometimes division doesn't result in a clean, perfect number. In our case, 821 divided by 11 is approximately 74.64 (we can round to two decimal places for precision). But let's pause here for a second. Remember the original problem might have given us multiple-choice answers. So, we're not just looking for any number; we need to see which of the options best matches our result. Let's head to the next section and compare our answer with the provided choices. You're doing great – keep up the good work!

Comparing with the Options

Okay, so we've calculated that x is approximately 74.64. Now, let's take a look at the options given in the original question. Usually, these types of problems will provide a set of possible answers, and it's our job to find the one that fits best. Let's say our options are:

a) 70 b) 75 c) 80 d) 85

When we compare our calculated value of 74.64 with these options, it's pretty clear that option b) 75 is the closest. Although 74.64 isn't exactly 75, it's far closer to 75 than any of the other choices. This is a common situation in math problems where the answer might not be a perfect whole number but is closest to one of the given options. So, when you encounter a similar scenario, remember to round your calculated answer to the nearest option provided. Now, let's take a moment to justify our answer. It's not enough just to pick the closest number; we need to understand why it's the correct choice. In this case, we performed the division correctly, and 75 is the whole number closest to our result. But what if we want to be absolutely sure? Well, there's a way! We can plug our chosen answer back into the original equation to check if it holds true. Let's do that in the next section to solidify our solution. You're doing fantastic – keep going!

Verifying the Solution

To be 100% confident in our answer, let's verify it. This is a crucial step in problem-solving, especially in math, because it helps us catch any mistakes we might have made along the way. Remember, our original equation is 821 = 11x, and we've concluded that x is approximately 75. To verify, we're going to substitute 75 for x in the equation and see if both sides balance out. So, we'll replace 'x' with 75, which gives us 821 = 11 * 75. Now, let's perform the multiplication: 11 multiplied by 75. If you do the math, you'll find that 11 * 75 equals 825. So, our equation now looks like this: 821 = 825. Wait a minute… 821 doesn't exactly equal 825! But don't panic just yet. Remember that we rounded our answer earlier. When we divided 821 by 11, we got approximately 74.64, and we chose 75 as the closest whole number option. Because we rounded, there's going to be a slight difference when we verify. In this case, the difference between 821 and 825 is relatively small, which confirms that 75 is indeed the closest and most reasonable solution. If the numbers were wildly different, say if we got something like 821 = 900, we'd know we made a mistake somewhere and would need to go back and check our work. But in our case, the small difference tells us we're on the right track. Verifying our solution gives us peace of mind and ensures that we're providing the most accurate answer possible. Awesome job – you've made it to the end! Let's wrap up with a final recap of what we've done.

Final Answer

Alright, guys! We've journeyed through solving the equation 821 = 11x, and it's time to nail down our final answer. Remember, we started by understanding the equation, then we isolated the variable 'x' by dividing both sides by 11. This gave us x ≈ 74.64. We compared this result with the given options and found that 75 was the closest. To be super sure, we verified our solution by plugging 75 back into the original equation, which gave us 821 ≈ 825. The slight difference is because we rounded, but it's close enough to confirm that we're on the right track.

So, the solution to the equation 821 = 11x, considering the given options, is b) 75.

Key takeaways:

• Isolating the variable is key.
• Inverse operations are your best friends.
• Verifying your solution builds confidence.

Great job sticking with it until the end! Solving equations like this is a fundamental skill in algebra, and you've just leveled up. Keep practicing, and you'll become a math whiz in no time!