Solving Equations: Finding Unknowns In Math Problems

Hey guys! Let's dive into the exciting world of algebra and tackle some equations together. We're going to break down how to find those sneaky unknown numbers hiding in plain sight. Think of it like being a math detective – super fun, right? We'll look at each equation step-by-step, so even if you're just starting out, you'll feel like a pro in no time. Get ready to sharpen those pencils and put on your thinking caps! We've got three awesome equations to solve today. It might sound intimidating, but trust me, with a little bit of know-how, we can crack these problems wide open. So, buckle up and let’s get started on our mathematical adventure!

1. Solving for X: (X - 256) : 6 = 240

Let's start with the first equation: (X - 256) : 6 = 240. The key here is to isolate 'X' on one side of the equation. We need to undo the operations that are affecting 'X'. Remember PEMDAS or BODMAS? We'll be working in reverse! First off, we see that the entire expression (X - 256) is being divided by 6. To undo this division, we'll multiply both sides of the equation by 6. This keeps the equation balanced, which is super important in algebra! So, we have (X - 256) : 6 * 6 = 240 * 6. On the left side, the division by 6 and multiplication by 6 cancel each other out, leaving us with just (X - 256). On the right side, 240 multiplied by 6 gives us 1440. Now our equation looks a lot simpler: X - 256 = 1440. We're almost there! The next step is to get rid of that -256 that's hanging out with our 'X'. To do this, we'll add 256 to both sides of the equation. Again, balance is key! So, we have X - 256 + 256 = 1440 + 256. On the left side, -256 and +256 cancel each other out, leaving us with just 'X'. On the right side, 1440 plus 256 equals 1696. And there you have it! We've solved for 'X'. X = 1696. To make sure we're right, we can plug this value back into the original equation and see if it holds true. Let's try it: (1696 - 256) : 6 = 240. 1696 minus 256 is 1440. 1440 divided by 6 is indeed 240! So, our answer checks out. We've successfully found the value of 'X'. See? Not so scary after all! This method of working backwards, using inverse operations, is a fundamental skill in algebra. You'll be using it all the time, so it's great to get comfortable with it now. We essentially unwrapped the equation like a present, one layer at a time, until we revealed the mystery number inside. Keep practicing, and you'll be a pro at solving for 'X' in no time. Remember the key steps: undo division by multiplying, undo subtraction by adding, and always keep the equation balanced by doing the same thing to both sides. Now, let's move on to our next equation and tackle another unknown!

2. Cracking the Code for Y: (148 + y) × 3 = 1056

Alright, let's move onto our next challenge: (148 + y) × 3 = 1056. We're on the hunt for 'y' this time! Just like before, our main goal is to isolate 'y' on one side of the equation. Think of it like a treasure hunt, and 'y' is the treasure we're after. We need to carefully follow the clues and unravel the equation to find it. Looking at the equation, we see that the expression (148 + y) is being multiplied by 3. So, our first move should be to undo this multiplication. How do we do that? You guessed it – we divide! We'll divide both sides of the equation by 3. Remember, we always have to keep the equation balanced. Whatever we do to one side, we must do to the other. So, we have (148 + y) × 3 / 3 = 1056 / 3. On the left side, the multiplication by 3 and division by 3 cancel each other out, leaving us with just (148 + y). On the right side, 1056 divided by 3 gives us 352. Now our equation is looking much simpler: 148 + y = 352. We're getting closer to finding 'y'! We just need to get rid of that pesky 148 that's hanging out with it. Since 148 is being added to 'y', we need to do the opposite – we need to subtract 148. Again, we subtract from both sides of the equation to maintain that crucial balance. So, we have 148 + y - 148 = 352 - 148. On the left side, +148 and -148 cancel each other out, leaving us with 'y' all alone! On the right side, 352 minus 148 equals 204. Bingo! We've found our treasure. y = 204. To be super sure we're right (because it's always good to double-check!), let's plug this value back into the original equation: (148 + 204) × 3 = 1056. 148 plus 204 is 352. 352 multiplied by 3 is indeed 1056! Our answer is correct. We successfully solved for 'y'. High five! This problem reinforces the importance of using inverse operations to unravel equations. We undid the multiplication by dividing, and we undid the addition by subtracting. It's like peeling an onion, one layer at a time, until we get to the core – which in this case, is the value of 'y'. Keep practicing these techniques, and you'll become a master equation solver. Now, let's tackle our final equation and find that last unknown!

3. Unlocking Z's Value: 697 - (5 : z) = 342

Okay, guys, let’s get to our final equation: 697 - (5 : z) = 342. This one looks a little trickier, but don't worry, we can handle it! This time, we're searching for 'z'. Just like the previous problems, we need to isolate 'z' on one side of the equation. It's like a puzzle, and we need to figure out the right moves to reveal the hidden number. Looking at the equation, we see that the term (5 : z) is being subtracted from 697. So, the first thing we want to do is get that (5 : z) term by itself. To do this, we'll subtract 697 from both sides of the equation. Remember, balance is key! So, we have 697 - (5 : z) - 697 = 342 - 697. On the left side, 697 and -697 cancel each other out, leaving us with -(5 : z). Be careful with that negative sign! On the right side, 342 minus 697 equals -355. Now our equation looks like this: -(5 : z) = -355. Now, we have a negative sign on both sides of the equation. We can get rid of these by multiplying both sides by -1. This is because a negative times a negative equals a positive. So, we have -(5 : z) * -1 = -355 * -1. This simplifies to 5 : z = 355. We're getting closer! Now we have 5 divided by 'z' equals 355. To isolate 'z', we need to get it out of the denominator. We can do this by multiplying both sides of the equation by 'z'. So, we have (5 : z) * z = 355 * z. On the left side, the division by 'z' and multiplication by 'z' cancel each other out, leaving us with just 5. On the right side, we have 355z. So our equation is now 5 = 355z. Almost there! Now we just need to get 'z' by itself. Since 'z' is being multiplied by 355, we need to do the opposite – we need to divide both sides by 355. So, we have 5 / 355 = 355z / 355. On the right side, the multiplication by 355 and division by 355 cancel each other out, leaving us with 'z'. On the left side, 5 divided by 355 simplifies to 1/71. And there we have it! z = 1/71. Let's check our answer by plugging it back into the original equation: 697 - (5 : (1/71)) = 342. Dividing by a fraction is the same as multiplying by its reciprocal, so 5 : (1/71) is the same as 5 * 71, which equals 355. So we have 697 - 355 = 342. And 697 minus 355 does indeed equal 342! Our answer is correct. We've successfully found the value of 'z'. This equation had a few more steps than the previous ones, but we broke it down and conquered it! The key was to work step-by-step, using inverse operations and keeping that equation balanced. You guys are awesome! We've now solved for X, Y, and Z. Give yourselves a pat on the back! Remember, practice makes perfect, so keep working on these types of problems, and you'll become a master of algebra in no time.

Final Thoughts: Keep Practicing!

So, guys, we've tackled three different equations today and successfully found the unknown numbers. Remember, the most important thing is to practice! The more you work with equations, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a crucial part of learning. And always remember to double-check your answers to make sure they're correct. Keep up the great work, and you'll be solving even the trickiest equations in no time!