Solving For The Unknown Finding A Number With A Difference Of 13 Equal To 5 Cubed

Unraveling the Mystery Number: When the Difference Between a Number and 13 Equals 5 Cubed

Hey guys! Ever stumbled upon a math problem that sounds like a riddle? This is one of those! We're diving into a fascinating mathematical exploration where we need to find the number that, when its difference with 13 is calculated, gives us 5 cubed. Sounds intriguing, right? Let's break it down step by step. To kick things off, we need to understand the basic components of the problem. What does it mean when we say "the difference between a number and 13"? In mathematical terms, it means we're subtracting 13 from a certain number or vice versa. The key here is to figure out which way the subtraction goes. Is it the number minus 13, or 13 minus the number? This will significantly impact our final answer. Then we encounter the phrase "5 cubed." What does that mean? Cubing a number means multiplying it by itself three times. So, 5 cubed is 5 * 5 * 5. Calculating this value is our next crucial step. Once we know what 5 cubed equals, we can set up our equation and start solving for the unknown number. Now, here’s where it gets interesting. We actually have two possible scenarios to consider. The difference between a number and 13 can be expressed in two ways: as the number minus 13, or as 13 minus the number. This gives us two separate equations to solve, each potentially leading to a different solution. It's like a mathematical fork in the road! We'll explore both paths to make sure we find all the possible answers. As we work through the problem, we'll be using some fundamental algebraic principles. Remember, algebra is like a puzzle, and each step we take gets us closer to the solution. We'll isolate the unknown number on one side of the equation, using inverse operations to undo the subtraction and ultimately reveal the value of our mystery number. So, grab your thinking caps, guys, and let's embark on this mathematical adventure together! We'll decode the language of the problem, translate it into equations, and then solve those equations to find the number we're looking for. It's going to be an exciting journey of mathematical discovery!

Setting Up the Equation: Translating Words into Math

Alright, let's get down to brass tacks and translate this word problem into a real mathematical equation. This is like turning a secret code into plain English, and it's a crucial step in solving any math puzzle. Remember, our mission is to find the number, so let's call that number "x." This is a classic algebraic move – using a letter to represent the unknown. Now, let's dissect the sentence: "The difference between a number and 13 equals 5 cubed." We've already established that "the difference between a number and 13" can be interpreted in two ways: x - 13 or 13 - x. And we know that "5 cubed" means 5 * 5 * 5, which equals 125. So, we've got two potential equations brewing: x - 13 = 125 and 13 - x = 125. These are our two paths forward, our two mathematical avenues to explore. It's like having two different keys that might unlock the same treasure chest! We need to consider both equations because the phrase "the difference between" doesn't explicitly tell us which number is being subtracted from which. It's this ambiguity that creates the two possibilities, and we have to be thorough mathematicians and investigate them both. Now, why is this step of setting up the equation so important? Well, it's the foundation of our entire solution. If we misinterpret the words and create the wrong equation, we're going to end up with the wrong answer. Think of it like building a house – if the foundation is shaky, the whole structure is at risk. So, we need to be meticulous and ensure our equations accurately reflect the problem's statement. Once we have our equations, we can unleash the power of algebra to solve for x. We'll use inverse operations, those mathematical opposites that undo each other, to isolate x on one side of the equation. It's like peeling away layers of an onion, gradually revealing the core, which in this case is the value of our unknown number. But before we jump into solving, let's pause and appreciate the beauty of this process. We've taken a verbal description, a story if you will, and transformed it into a precise mathematical statement. This is the power of algebra – to abstract and represent real-world situations with symbols and equations. So, with our equations firmly in place, we're ready to embark on the next stage of our journey: solving for x and uncovering the mystery number. Let's do it, guys!

Solving the Equations: Unlocking the Value of 'x'

Okay, guys, we've arrived at the heart of the matter: solving the equations! We've got two potential equations on our hands, each representing a possible solution to our mystery number. Remember, our equations are: 1. x - 13 = 125 2. 13 - x = 125 Let's tackle the first equation: x - 13 = 125. Our goal here is to isolate x, to get it all by itself on one side of the equation. To do this, we need to undo the subtraction of 13. And what's the opposite of subtraction? Addition! So, we're going to add 13 to both sides of the equation. Why both sides? Because in algebra, we need to maintain balance. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, adding 13 to both sides, we get: x - 13 + 13 = 125 + 13. The -13 and +13 on the left side cancel each other out, leaving us with x = 138. Boom! We've found our first potential solution: x = 138. Now, let's move on to the second equation: 13 - x = 125. This one's a little trickier because we're subtracting x. There are a couple of ways we can approach this. One way is to add x to both sides, which gives us 13 = 125 + x. Then, we can subtract 125 from both sides to isolate x. This gives us 13 - 125 = x, which simplifies to -112 = x. So, our second potential solution is x = -112. Another way to solve this equation is to subtract 13 from both sides first, resulting in -x = 112. Then, multiply both sides by -1 to get x = -112. Same answer, different path! It's like taking two different routes to the same destination. Now, we have two possible values for x: 138 and -112. But are both of these solutions valid? This is where we need to go back to the original problem and make sure our answers make sense in the context of the question. We'll do this in the next section when we check our answers. But for now, let's celebrate our algebraic victory! We've successfully navigated the equations and unearthed two potential solutions. We're like mathematical detectives, uncovering clues and piecing together the puzzle. So, pat yourselves on the back, guys! We're one step closer to cracking this case.

Checking the Answers: Ensuring Our Solutions Make Sense

Alright, we've done the hard work of solving the equations, but our job isn't quite done yet. We need to be meticulous mathematicians and check our answers to make sure they actually fit the original problem. This is like proofreading your work before submitting it – you want to catch any errors and ensure everything is perfect. Remember, we found two potential solutions: x = 138 and x = -112. Let's start with x = 138. The original problem stated: "The difference between a number and 13 equals 5 cubed." So, let's plug 138 into that statement. The difference between 138 and 13 is either 138 - 13 or 13 - 138. Let's calculate both: 138 - 13 = 125 And 13 - 138 = -125 5 cubed is 5 * 5 * 5 = 125. Aha! 138 - 13 does indeed equal 125. So, x = 138 is a valid solution. It fits the puzzle perfectly! Now, let's check x = -112. The difference between -112 and 13 is either -112 - 13 or 13 - (-112). Let's calculate: -112 - 13 = -125 And 13 - (-112) = 13 + 112 = 125 Bingo! 13 - (-112) equals 125. So, x = -112 is also a valid solution. It's like finding a second key that unlocks the same door! Why is checking our answers so crucial? Well, it's easy to make a small mistake in the solving process – a dropped negative sign, a miscalculation, etc. Checking our answers helps us catch these errors and ensures we're submitting the correct solutions. It's like a safety net, preventing us from falling into the trap of an incorrect answer. Furthermore, checking our answers helps us develop a deeper understanding of the problem. We're not just blindly applying formulas; we're thinking critically about the relationship between the numbers and the words in the problem. This kind of critical thinking is what makes you a true mathematical master! So, we've checked both our solutions, and they both work. We've successfully navigated the problem from start to finish, from translating the words into equations to solving those equations and verifying our answers. Give yourselves a big round of applause, guys! You've conquered this mathematical challenge!

Final Answer: The Numbers Are 138 and -112

Drumroll, please! We've reached the grand finale, the moment of truth where we reveal the answer to our mathematical mystery. After all our careful calculations, equation solving, and answer checking, we've arrived at the solution. The numbers that fit the description "the difference between a number and 13 equals 5 cubed" are 138 and -112. There you have it, guys! We've not just found one solution, but two! This highlights the importance of considering all possibilities and not settling for the first answer you find. In this case, the phrase "the difference between" was the key to unlocking the two solutions. It created an ambiguity that led us down two different paths, both of which ultimately led to valid answers. Now, let's take a moment to appreciate the journey we've been on. We started with a word problem, a seemingly simple statement, and transformed it into a mathematical exploration. We translated words into equations, wielded the power of algebra to solve those equations, and then rigorously checked our answers to ensure their validity. It's like we've been on a mathematical quest, and we've emerged victorious! This problem, while seemingly straightforward, touches on some fundamental mathematical concepts: * Variables: We used the letter 'x' to represent the unknown number, a cornerstone of algebra. * Equations: We created equations to represent the relationships described in the problem, the language of mathematics. * Inverse Operations: We used addition and subtraction to isolate the variable, the tools of our trade. * Critical Thinking: We analyzed the problem statement, considered multiple interpretations, and checked our answers, the hallmarks of a skilled problem-solver. So, what's the takeaway from all this? Math isn't just about memorizing formulas and crunching numbers. It's about understanding concepts, thinking critically, and solving problems creatively. It's about taking a complex situation and breaking it down into manageable steps. And most importantly, it's about the thrill of discovery, the satisfaction of finding the solution. So, congratulations, guys! You've not only solved this problem, but you've also honed your mathematical skills and deepened your understanding of the subject. Keep exploring, keep questioning, and keep solving! The world of mathematics is vast and fascinating, and there's always something new to discover. Until next time, happy problem-solving!