Solve For X: 8x + 14x = 440

Hey guys, let's dive into a super fun math puzzle today! We've got this intriguing problem where a number is the key to unlocking a specific equation. The challenge is to find that mystery number when we know that the sum of eight times this number and fourteen times the same number adds up to a grand total of 440. It sounds a bit like a detective story, right? We're on a mission to uncover the value of this unknown quantity. To tackle this, we'll break it down step-by-step, using some basic algebra that’ll make finding our number a piece of cake. Think of it as a treasure hunt, and the treasure is the correct value for our number!

Understanding the Problem: Decoding the Math

Alright, let's get real with this problem. We're told there's a number, and we need to figure out what it is. The clue we have is about its relationship with other numbers when multiplied and then added together. Specifically, if you take this unknown number, multiply it by 8, and then take the same unknown number and multiply it by 14, and then add those two results together, you get 440. Pretty straightforward, once you translate the words into mathematical symbols. This is where algebra shines, guys! We can represent our unknown number with a variable, typically 'x'. So, 'eight times the number' becomes 8x, and 'fourteen times the number' becomes 14x. The phrase 'the sum of... is 440' means we add these two expressions together and set them equal to 440. So, the equation we need to solve is 8x + 14x = 440. See? We've turned a word problem into a solvable equation. It’s all about breaking down the information and representing it clearly. This initial step is crucial because it sets the foundation for the entire solution. Without a clear understanding of what the problem is asking and how to represent it mathematically, we’d be lost at sea!

Setting Up the Equation: Your Algebraic Blueprint

Now that we've got the gist of the problem, let's build our algebraic blueprint. As I mentioned, we'll use 'x' to represent our mystery number. The problem states, "a number is such that the sum of 8 times the number and 14 times the number is 440." Translating this into mathematical terms is super important. We have '8 times the number', which we write as 8x. Then we have '14 times the number', which is 14x. The word 'sum' tells us to add these two parts together. So, we get 8x + 14x. Finally, the problem says this sum 'is 440', which means it's equal to 440. Therefore, our complete equation is 8x + 14x = 440. This equation is our roadmap, guys. It visually represents all the information given in the problem. It's concise, it's clear, and it's ready for us to solve. Setting up the equation correctly is arguably the most critical step in solving any algebra problem. If your equation isn't accurate, your answer will be too, no matter how well you do the calculations. So, take a moment, double-check your translation from words to symbols. Does 8x really represent '8 times the number'? Yes. Does 14x represent '14 times the number'? Absolutely. Does + represent 'sum'? You bet. And does = represent 'is'? You got it. This equation is solid gold!

Simplifying the Equation: Combining Like Terms

Alright, math wizards, we've got our equation: 8x + 14x = 440. Now, before we go trying to isolate 'x', let's make things a little tidier. See how we have 'x' terms on the left side of the equation? We have 8x and 14x. These are what we call 'like terms' because they both involve the variable 'x' raised to the same power (in this case, to the power of 1). When you have like terms, you can combine them, just like you can combine apples with apples and oranges with oranges. So, 8x + 14x can be simplified by adding the coefficients (the numbers in front of the 'x'). Think of it as having 8 apples and then getting 14 more apples; now you have a total of 22 apples! So, 8x + 14x simplifies to 22x. Our equation now looks much cleaner: 22x = 440. This step is super important for making the rest of the solving process easier. It reduces the complexity of the equation, bringing us closer to finding that mystery number. Simplifying like terms is a fundamental skill in algebra that saves a lot of headache down the line. It’s like prepping your ingredients before you start cooking; you make the process smoother and the final result better. So, always look for opportunities to combine like terms whenever you see them!

Solving for x: The Grand Reveal

We're in the home stretch, guys! Our simplified equation is 22x = 440. Our goal is to get 'x' all by itself on one side of the equation. Right now, 'x' is being multiplied by 22. To undo multiplication, we use its opposite operation, which is division. So, to isolate 'x', we need to divide both sides of the equation by 22. This is a fundamental rule in algebra: whatever you do to one side of the equation, you must do to the other side to keep it balanced. So, we'll divide the left side by 22: (22x) / 22. This simplifies to just 'x' because 22 divided by 22 is 1, and 1 times 'x' is 'x'. Now, we do the same to the right side: 440 / 22. Let's do that division. If you calculate 440 divided by 22, you get 20. So, our equation becomes x = 20. Ta-da! We've found our mystery number! It was hiding all along, and algebra helped us uncover it. This is the power of solving equations – turning unknowns into knowns. The 'grand reveal' is always the most satisfying part, seeing that 'x' finally stand alone with its value revealed.

Verification: Checking Our Work

We found our number, x = 20. But in math, especially when you're learning, it's always a super good idea to check your answer. Did we actually find the right number? Let's plug '20' back into our original problem statement or, even better, our original equation: 8x + 14x = 440. We substitute 20 for 'x': 8(20) + 14(20). First, let's calculate 8 * 20. That's 160. Next, let's calculate 14 * 20. That's 280. Now, we add those two results together: 160 + 280. And guess what? 160 + 280 equals 440! It matches the number given in the problem! This means our solution, x = 20, is absolutely correct. Verification is like getting a gold star for your homework; it confirms you did a stellar job. It builds confidence in your mathematical abilities and helps catch any silly mistakes. So, never skip this step, guys! It’s a small effort for a big reward of certainty.

Conclusion: The Number Revealed!

So, there you have it, folks! We took a word problem, translated it into a clear algebraic equation (8x + 14x = 440), simplified it by combining like terms (22x = 440), and then solved for our unknown variable by isolating 'x' (x = 20). Finally, we verified our answer by plugging it back into the original equation, confirming that our solution is indeed correct. The number we were looking for is 20. It's amazing how a little bit of algebra can solve these puzzles, isn't it? Keep practicing, and you'll become a math whiz in no time! Remember, every problem is an opportunity to learn and grow your skills. So, next time you see a math problem, tackle it with confidence and a smile!