Solving Math Equations The Sum Of Twice A Number And 13 Is 75

Hey guys! Ever stumbled upon a math problem that looks like it's straight out of a detective novel? Well, today we're diving into one of those mysteries: "The sum of twice a number and 13 is 75." Sounds intriguing, right? Don't worry, we're going to break it down step by step, so even if you're not a math whiz, you'll be able to solve it like a pro. So, grab your thinking caps, and let's get started!

Unraveling the Enigma: Translating Words into Math

Okay, so the first step in tackling any word problem is to translate those words into the language of math. Think of it like learning a new language – we need to understand the vocabulary and grammar. In this case, our vocabulary consists of mathematical operations and variables. Let's dissect the sentence: "The sum of twice a number and 13 is 75."

• "A number" – This is our mystery element, the thing we're trying to find. We'll call it 'x' because that's a classic math variable. You can think of 'x' as a placeholder, like a blank space waiting to be filled with the correct number.
• "Twice a number" – This means we're multiplying our mystery number 'x' by 2. So, "twice a number" translates to 2 * x, or simply 2x. Easy peasy, right?
• "The sum of" – This is a big clue! It tells us we're adding things together. So, we're adding "twice a number" (2x) to something else.
• "and 13" – We're adding 13 to our 2x. Now we're building our equation! We have 2x + 13.
• "is 75" – This is the final piece of the puzzle. "Is" in math often means "equals." So, our entire expression, 2x + 13, equals 75. We can write this as 2x + 13 = 75.

There you have it! We've successfully translated the words into a mathematical equation. Our equation, 2x + 13 = 75, is the key to unlocking the mystery. It's like having a secret code that we can now decipher. Writing the equation correctly is super crucial, guys, because it sets the foundation for everything else. If we mess up the equation, we're going to end up with the wrong answer, and nobody wants that! So, always take your time and double-check that you've translated everything accurately. Think of each part of the sentence as a separate piece of a jigsaw puzzle, and the equation is the complete picture. Once you've got that picture, you're well on your way to solving the problem.

The Art of Isolation: Solving for 'x'

Alright, detectives, now that we've got our equation (2x + 13 = 75), it's time to put our solving skills to the test! The goal here is to isolate 'x,' which basically means getting 'x' all by itself on one side of the equation. Think of it like giving 'x' some personal space so we can finally see its true value. To do this, we're going to use the magic of inverse operations.

• Inverse Operations – Our Secret Weapon: In math, every operation has an inverse, which is like its opposite. Addition's inverse is subtraction, and multiplication's inverse is division. We're going to use these inverses to undo the operations that are messing with 'x'.
• Step 1: Subtract 13 from both sides: Remember that golden rule of equations? What you do to one side, you gotta do to the other! We have 2x + 13 = 75. To get rid of that +13, we subtract 13 from both sides. This gives us: 2x + 13 - 13 = 75 - 13. Simplifying, we get 2x = 62. We're one step closer to isolating 'x'!
• Step 2: Divide both sides by 2: Now we have 2x = 62. 'x' is being multiplied by 2, so to undo that, we divide both sides by 2. This gives us: 2x / 2 = 62 / 2. Simplifying, we finally get x = 31. Boom! We've done it!

We've successfully isolated 'x' and discovered its value: 31. It's like cracking a code, guys, and the feeling of accomplishment is awesome! The key takeaway here is the power of inverse operations. They're your best friends when it comes to solving equations. Remember, we subtracted to undo addition and divided to undo multiplication. By strategically applying these inverse operations, we can systematically peel away the layers surrounding 'x' until it stands alone and reveals its true identity. And always, always remember to do the same thing to both sides of the equation. This keeps the equation balanced, like a perfectly balanced scale. If you only change one side, the whole thing goes haywire, and you won't get the right answer.

The Grand Finale: Verifying Our Solution

Okay, detectives, we've found our suspect – x = 31! But before we close the case, we need to make absolutely sure we've got the right guy. This is where verification comes in. It's like our final check to ensure our solution is rock solid.

• Plugging it Back In: To verify our solution, we take the value we found for 'x' (which is 31) and plug it back into our original equation: 2x + 13 = 75. So, we replace 'x' with 31, giving us: 2 * 31 + 13 = 75. Now, we just need to do the math and see if both sides of the equation are equal.
• The Moment of Truth: Let's simplify the left side of the equation. 2 * 31 = 62, and 62 + 13 = 75. So, the left side equals 75. The right side of the equation is already 75. Drumroll, please… 75 = 75! It checks out!

Our solution, x = 31, makes the equation true. We've cracked the case! This step, verification, is absolutely crucial, guys. It's like having a safety net that catches you if you've made a mistake somewhere along the way. It's so easy to make a small error in the calculations, especially when you're dealing with multiple steps. Plugging the solution back into the original equation is the best way to catch those errors and ensure that your answer is correct. Think of it like proofreading an essay before you submit it – you want to make sure everything is perfect before you call it done. Plus, verification gives you confidence in your answer. You know you've solved the problem correctly, and that's a great feeling!

Real-World Math: Why This Matters

So, we've solved a math problem, which is awesome! But you might be thinking, "Okay, that's cool, but when am I ever going to use this in real life?" That's a fair question! The truth is, math is all around us, even when we don't realize it. Understanding how to solve equations like this one helps us develop critical thinking and problem-solving skills, which are super valuable in all sorts of situations.

• Everyday Scenarios: Imagine you're trying to figure out how much you can spend on a new gadget. You have a certain amount of money saved, and you know you need to set aside some for other expenses. You can use an equation to figure out exactly how much you can spend. Or, let's say you're planning a road trip and need to calculate how much gas you'll need. Equations can help with that too!
• Beyond the Classroom: These skills aren't just useful for math class. They're essential in fields like science, engineering, finance, and even cooking! Scientists use equations to model the world around us, engineers use them to design structures, and financial analysts use them to make investment decisions. Even when you're baking a cake, you're using math to measure ingredients and adjust cooking times.

Solving equations teaches us to break down complex problems into smaller, more manageable steps. It teaches us to think logically and systematically, and to persevere even when things get tough. These are skills that will serve you well in all aspects of your life, guys, whether you're solving a math problem, planning a project at work, or making important decisions. So, the next time you encounter an equation, don't shy away from it. Embrace the challenge, and remember that you have the tools to solve it!

Final Thoughts: You've Got This!

We've journeyed through the mystery of the equation "The sum of twice a number and 13 is 75," and we've emerged victorious! We learned how to translate words into math, how to isolate 'x' using inverse operations, and how to verify our solution to make sure we got it right. We even talked about how these skills apply to the real world.

Remember, math can be challenging, but it's also incredibly rewarding. Every time you solve a problem, you're building your problem-solving muscles and becoming a more confident thinker. So, keep practicing, keep exploring, and never be afraid to ask questions. You've got this!