Solving Equations A Step-by-Step Guide To N - 3 = 5(n - 3)

Hey guys! Let's dive into solving the equation n - 3 = 5(n - 3). Don't worry, it might look a bit tricky at first, but we'll break it down step by step so it's super easy to understand. Math can be fun, especially when you crack the code to solve these problems! We're going to use some basic algebraic principles, and by the end of this article, you'll be able to tackle similar equations with confidence. So, let's get started and unravel the mystery behind this equation!

Understanding the Basics of Algebraic Equations

Before we jump right into solving our specific equation, let's make sure we're all on the same page with the basics. Algebraic equations are like puzzles where we need to find the value of a variable (in this case, 'n') that makes the equation true. Think of it as a balancing act – what you do on one side of the equation, you must also do on the other side to keep things balanced. This principle is super important for solving any algebraic equation. Variables are simply symbols (usually letters) that represent unknown numbers. Our goal is to isolate the variable on one side of the equation, so we can clearly see its value. This often involves performing operations like addition, subtraction, multiplication, and division. Remember, the key is to perform the same operation on both sides to maintain the balance. When we talk about solving an equation, we're essentially finding the number that, when substituted for the variable, makes both sides of the equation equal. For instance, if we find that n = 8, we're saying that if we replace 'n' with 8 in the original equation, both sides will have the same value. This foundation is crucial for tackling more complex equations, and it's the backbone of algebra. With these basics in mind, we're well-equipped to solve n - 3 = 5(n - 3). So, let’s move on to the specific steps involved in solving our equation and see how these principles come into play. Are you ready to dive deeper and become an equation-solving pro? Let's do it!

Step 1: Distribute on the Right Side

The first thing we need to do with our equation, n - 3 = 5(n - 3), is to simplify the right side. Notice that we have 5 multiplied by the expression (n - 3). To get rid of these parentheses, we need to distribute the 5 across both terms inside. This means we're going to multiply 5 by 'n' and 5 by '-3'. When we multiply 5 by 'n', we get 5n. And when we multiply 5 by -3, we get -15. So, after distributing, the right side of our equation becomes 5n - 15. Now, let’s rewrite the entire equation with this simplification: n - 3 = 5n - 15. This step is super important because it helps us get rid of the parentheses, making the equation much easier to work with. It's like clearing away the clutter so we can see the problem more clearly. Distribution is a fundamental technique in algebra, and you'll use it all the time when solving equations. Make sure you understand this step completely before moving on. Sometimes, distribution might involve negative numbers or more complex expressions inside the parentheses, but the principle remains the same: multiply the term outside the parentheses by each term inside. Now that we've successfully distributed and simplified our equation, we're one step closer to finding the value of 'n'. Our next step will involve rearranging the terms to get all the 'n' terms on one side and the constants on the other. So, let's move on and see how we can do that!

Step 2: Rearrange the Equation

Alright, now that we've distributed and have the equation n - 3 = 5n - 15, it's time to rearrange things a bit. Our goal here is to get all the terms with 'n' on one side of the equation and all the constant terms (the numbers) on the other side. This makes it much easier to isolate 'n' and find its value. Let’s start by moving the 'n' term from the left side to the right side. We can do this by subtracting 'n' from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced! So, if we subtract 'n' from both sides, we get: n - 3 - n = 5n - 15 - n. On the left side, 'n' and '-n' cancel each other out, leaving us with just -3. On the right side, 5n - n simplifies to 4n. So, our equation now looks like this: -3 = 4n - 15. Great! We've moved the 'n' term to the right side. Now, let’s move the constant term (-15) from the right side to the left side. We can do this by adding 15 to both sides of the equation: -3 + 15 = 4n - 15 + 15. On the left side, -3 + 15 equals 12. On the right side, -15 and +15 cancel each other out, leaving us with just 4n. So, our equation is now: 12 = 4n. See how much simpler it looks now? By rearranging the terms, we've managed to isolate the 'n' term on one side and the constants on the other. This is a crucial step in solving equations, and you'll use this technique all the time. Next up, we'll finally isolate 'n' completely to find its value. So, let’s move on to the final step!

Step 3: Isolate 'n' and Solve

Okay, we're in the home stretch! We've simplified and rearranged our equation, and now we have 12 = 4n. Our final goal is to isolate 'n' completely so we can see its value. To do this, we need to get rid of the 4 that's multiplying 'n'. How do we do that? We use the inverse operation – in this case, division. If 4 is multiplying 'n', we need to divide both sides of the equation by 4. Remember, we have to do the same thing to both sides to keep the equation balanced. So, let's divide both sides by 4: 12 / 4 = (4n) / 4. On the left side, 12 divided by 4 is 3. On the right side, 4n divided by 4 simplifies to just 'n'. So, our equation now looks like this: 3 = n. And there you have it! We've solved for 'n'. This means that the value of 'n' that makes the original equation true is 3. To double-check our answer, we can substitute 'n' with 3 in the original equation and see if both sides are equal. Let’s do that quickly: Original equation: n - 3 = 5(n - 3) Substitute n = 3: 3 - 3 = 5(3 - 3) Simplify: 0 = 5(0) 0 = 0. It works! Both sides are equal, so we know our solution is correct. Isolating the variable is the final step in solving most algebraic equations. It involves using inverse operations (like division when there's multiplication, or addition when there's subtraction) to get the variable all by itself on one side of the equation. With practice, this step becomes second nature. Congratulations! You've successfully solved the equation n - 3 = 5(n - 3). You've taken it step by step, from distributing to rearranging, and finally to isolating the variable. You're well on your way to becoming an equation-solving expert. Now, let's wrap things up with a quick summary and some final thoughts.

Summary and Final Thoughts

Wow, guys, we did it! We successfully solved the equation n - 3 = 5(n - 3) by breaking it down into manageable steps. Let's quickly recap what we did: First, we distributed the 5 on the right side of the equation to get rid of the parentheses. This gave us n - 3 = 5n - 15. Then, we rearranged the equation to get all the 'n' terms on one side and the constants on the other. We did this by subtracting 'n' from both sides and adding 15 to both sides, which gave us 12 = 4n. Finally, we isolated 'n' by dividing both sides by 4, which gave us our solution: n = 3. We even checked our answer by substituting it back into the original equation and confirming that both sides were equal. Solving algebraic equations is a fundamental skill in math, and it’s something you’ll use in many different areas, from science and engineering to everyday problem-solving. The key is to take it step by step, stay organized, and remember the basic principles of algebra. Always remember to perform the same operations on both sides of the equation to maintain balance, and use inverse operations to isolate the variable. Practice makes perfect, so don't be afraid to tackle more equations and build your confidence. The more you practice, the easier it will become to recognize patterns and apply the right techniques. And remember, math can be fun! It’s like solving a puzzle, and every equation you solve is a small victory. So, keep practicing, keep learning, and keep enjoying the process. You've got this! If you ever get stuck, remember to break the problem down into smaller steps, and don't hesitate to ask for help. There are tons of resources available, from textbooks and online tutorials to teachers and classmates who can lend a hand. Keep up the great work, and happy equation-solving! Now you have the skills to solve similar problems. Keep practicing, and you'll become an algebra ace in no time!