Solving For N In 24 = 3(n - 5) A Step-by-Step Guide

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Hey guys! Today, we're diving deep into a common algebraic problem: solving for a variable. In this case, we're tackling the equation 24 = 3(n - 5). Don't worry if algebra feels like a puzzle sometimes; we're going to break it down step by step, making sure everyone understands the process. Whether you're a student brushing up on your skills or just someone who enjoys a good mathematical challenge, this guide will equip you with the knowledge to solve this type of equation with confidence. Let's get started and unravel the mystery of 'n'!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what the equation 24 = 3(n - 5) is telling us. In algebraic terms, this equation states that the quantity 24 is equal to 3 times the difference between a number 'n' and 5. Our goal is to isolate 'n' on one side of the equation to find its value. Think of it like a balancing act; whatever we do to one side of the equation, we must also do to the other side to maintain the balance.

To truly grasp the equation, let’s break it down further. The left side, 24, is a constant – a fixed value. The right side, 3(n - 5), is where the magic happens. Here, 'n' is our variable, the unknown we're trying to find. The expression (n - 5) means we're subtracting 5 from 'n', and the 3 outside the parentheses indicates that the entire result of (n - 5) is being multiplied by 3. This understanding is crucial because it dictates the order in which we'll perform the operations to solve for 'n'. We need to carefully undo these operations, step by step, to isolate 'n' and reveal its value. Remember, each step we take is designed to simplify the equation while maintaining its balance, bringing us closer to our solution. With this foundational understanding, we're ready to roll up our sleeves and start solving!

Step-by-Step Solution

Okay, let's get down to the nitty-gritty and solve for 'n' in the equation 24 = 3(n - 5). We'll take it one step at a time, making sure each move is clear and logical. Remember, the key is to isolate 'n' on one side of the equation, and we'll do that by carefully unwrapping the operations that are currently affecting it.

Step 1: Distribute the 3

The first thing we need to do is get rid of those parentheses. To do that, we'll use the distributive property. This means we multiply the 3 outside the parentheses by each term inside the parentheses. So, 3 multiplied by 'n' is 3n, and 3 multiplied by -5 is -15. This gives us a new equation: 24 = 3n - 15. Now, the equation looks a bit simpler, and we're one step closer to isolating 'n'. Distributing the 3 was crucial because it allows us to separate the terms and start moving things around to get 'n' by itself.

Step 2: Isolate the Term with 'n'

Next up, we want to isolate the term with 'n', which is 3n in our equation 24 = 3n - 15. To do this, we need to get rid of the -15 on the right side. Remember, we maintain balance by doing the same thing to both sides. So, we'll add 15 to both sides of the equation. This gives us: 24 + 15 = 3n - 15 + 15. Simplifying, we get 39 = 3n. See how the -15 and +15 canceled each other out on the right side, leaving us with just 3n? We're making great progress! Isolating the term with 'n' is a pivotal step because it sets us up for the final move: getting 'n' all by itself.

Step 3: Solve for 'n'

Alright, we're in the home stretch! We now have the equation 39 = 3n. To finally solve for 'n', we need to get rid of the 3 that's multiplying it. To do this, we'll divide both sides of the equation by 3. This gives us: 39 / 3 = 3n / 3. Simplifying, we find that 13 = n. And there you have it! We've successfully solved for 'n'. This final step is the culmination of all our efforts, neatly isolating 'n' and revealing its value. We've unwrapped all the operations surrounding 'n', leaving us with the clear and concise solution.

The Solution: n = 13

So, after all that careful stepping through, we've arrived at our solution: n = 13. That's it! We've successfully solved the equation 24 = 3(n - 5). But before we celebrate, let's take a moment to verify our answer. It's always a good practice to plug our solution back into the original equation to make sure it holds true. This gives us confidence that we haven't made any mistakes along the way. Think of it as a final check, a seal of approval on our hard work.

To verify, we'll substitute 13 for 'n' in the original equation: 24 = 3(13 - 5). Now, let's simplify. Inside the parentheses, 13 - 5 equals 8, so we have 24 = 3(8). Next, 3 multiplied by 8 is 24, giving us 24 = 24. This is a true statement! Both sides of the equation are equal, which confirms that our solution, n = 13, is indeed correct. Verifying our solution is not just a formality; it's a crucial step in the problem-solving process. It ensures accuracy and reinforces our understanding of the equation and the steps we took to solve it. With this verification, we can confidently say that we've mastered this equation!

Common Mistakes to Avoid

Alright, guys, let's talk about some common pitfalls that students often encounter when solving equations like 24 = 3(n - 5). Being aware of these mistakes can help you avoid them and ensure you get the correct answer every time. Think of it as learning from others' experiences, so you don't have to make the same errors yourself. It's all about sharpening your algebraic skills and becoming a more confident problem solver.

Forgetting the Distributive Property

One frequent error is forgetting to distribute the number outside the parentheses to both terms inside. In our equation, it's crucial to multiply the 3 by both 'n' and -5. Some students might only multiply by 'n', resulting in an incorrect equation. Remember, the distributive property is key to unlocking the equation and simplifying it correctly. It ensures that every term inside the parentheses is properly accounted for. So, always double-check that you've distributed the number to all terms, no exceptions!

Incorrect Order of Operations

Another common mistake is messing up the order of operations. You know, PEMDAS or BODMAS – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our equation, we need to distribute first, then deal with addition and subtraction. Jumping the gun and adding 5 to both sides before distributing the 3 would lead to a wrong path. Sticking to the correct order ensures that you're unwrapping the equation in the right sequence, step by step. So, always keep PEMDAS/BODMAS in mind, and you'll be on the right track!

Arithmetic Errors

Simple arithmetic errors can also derail your solution. A small mistake in addition, subtraction, multiplication, or division can lead to a completely wrong answer. This is why it's so important to be careful with your calculations and double-check your work. Even the most seasoned mathematicians can make a slip-up, so don't feel bad if it happens. The key is to be vigilant and catch those errors before they throw off your entire solution. Consider using a calculator for more complex calculations, and always take your time to review each step.

By being mindful of these common mistakes – forgetting the distributive property, messing up the order of operations, and making arithmetic errors – you can significantly improve your accuracy and confidence in solving algebraic equations. Remember, practice makes perfect, and each problem you solve is a step towards mastering algebra!

Practice Problems

Alright, guys, now that we've conquered the equation 24 = 3(n - 5) and discussed common pitfalls, it's time to put your skills to the test! Practice is the name of the game when it comes to mastering algebra. The more you practice, the more comfortable you'll become with the steps involved, and the faster you'll be able to solve equations. Think of it like learning a new language or a musical instrument; the more you use it, the better you get. So, let's dive into some practice problems to solidify your understanding and boost your confidence. These problems are designed to challenge you and help you apply what you've learned in a variety of situations.

Here are a few practice problems for you to try:

  1. Solve for x: 18 = 2(x + 3)
  2. Solve for y: 30 = 5(y - 2)
  3. Solve for a: 42 = 6(a + 1)
  4. Solve for b: 15 = 3(b - 4)
  5. Solve for m: 28 = 4(m + 2)

For each of these problems, follow the same steps we used earlier: distribute, isolate the term with the variable, and then solve for the variable. Remember to double-check your work and verify your solutions by plugging them back into the original equations. This will not only confirm your answers but also reinforce your understanding of the process. Don't be afraid to make mistakes; they're a natural part of the learning process. The key is to learn from those mistakes and keep practicing. If you get stuck on a problem, go back and review the steps we discussed earlier, or try breaking the problem down into smaller, more manageable steps. Algebra is like a puzzle, and each problem you solve is another piece of the puzzle falling into place. So, grab a pencil and paper, and let's get practicing! You've got this!

Conclusion

So, there you have it, guys! We've successfully navigated the equation 24 = 3(n - 5), solved for 'n', and even explored some common mistakes to avoid. More importantly, we've equipped ourselves with a step-by-step approach that can be applied to a wide range of algebraic equations. Remember, solving for a variable is like detective work; it's about carefully unwrapping the equation, following the clues (the operations), and methodically isolating the unknown. And just like any skill, mastering algebra takes practice and patience. But with each equation you solve, you're building your confidence and sharpening your problem-solving abilities.

Algebra is more than just numbers and symbols; it's a powerful tool for understanding and modeling the world around us. From calculating finances to designing structures, algebraic principles are at play in countless aspects of our lives. By mastering these fundamental concepts, you're not just acing your math tests; you're also developing valuable critical thinking skills that will serve you well in any field. So, keep practicing, keep exploring, and never be afraid to ask questions. The world of mathematics is vast and fascinating, and there's always something new to learn. And who knows, maybe one day you'll be the one helping others unravel the mysteries of algebra! Keep up the great work, and happy solving!