Solving 3.5n + 6.4 = 42.5: The First Step Explained

Hey guys! Ever find yourself staring at an equation and wondering where to even begin? Don't worry, we've all been there! Today, we're going to break down the first step in solving the equation 3.5n + 6.4 = 42.5. Math can seem intimidating, but with a clear understanding of the fundamentals, you'll be tackling these problems like a pro in no time. So, let's dive in and make math a little less mysterious.

Understanding the Equation: 3.5n + 6.4 = 42.5

Before we jump into the solution, let's quickly understand what this equation represents. We have a variable, n, which is being multiplied by 3.5. Then, we're adding 6.4 to that result, and the whole thing equals 42.5. Our goal is to isolate n on one side of the equation to find its value. Think of it like peeling away layers to get to the core. To solve for n, we need to undo the operations that are being performed on it, but we have to do it in the right order. This is where the concept of inverse operations comes into play. Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced – like a seesaw! So, with this equation in mind, the key is to identify what's directly impacting 'n' and address those operations strategically. We'll get into the specifics in the next section, but for now, just remember the big picture: isolate n by carefully undoing the operations.

Identifying the First Step: Why Subtract 6.4?

The question asks for the first step. Looking at the equation, we see two operations affecting n: multiplication by 3.5 and addition of 6.4. The order of operations (PEMDAS/BODMAS) tells us that we typically handle multiplication before addition. However, when we're solving an equation, we essentially work backward through the order of operations. This is a crucial concept for anyone tackling algebraic equations. When we say "work backward," we mean we undo the operations in the reverse order they would be performed. So, instead of multiplying first, we'll address addition and subtraction first. This is because we want to isolate the term with n in it before we deal with the multiplication. By focusing on the addition/subtraction first, we create a simpler equation that's easier to manage. This strategic approach is what makes solving equations efficient and less prone to errors. So, which operation should we undo first? Since 6.4 is being added to 3.5n, the inverse operation is subtraction. Therefore, the first step is to subtract 6.4 from both sides of the equation. Why both sides? Because we need to maintain the balance of the equation, ensuring that both sides remain equal. This principle of maintaining balance is fundamental to solving equations and will guide us through the rest of the process. Think of it as a golden rule for algebra – a rule you definitely want to remember!

Step-by-Step Explanation: Subtracting 6.4 from Both Sides

So, let's get into the nitty-gritty. The initial equation is 3.5n + 6.4 = 42.5. Our mission for the first step, as we've established, is to subtract 6.4 from both sides. This maintains the equation's balance and moves us closer to isolating n. Let's write that down: 3.5n + 6.4 - 6.4 = 42.5 - 6.4. See how we've applied the same operation to both the left and right sides? Now, let's simplify. On the left side, +6.4 and -6.4 cancel each other out, leaving us with just 3.5n. On the right side, 42.5 minus 6.4 equals 36.1. So, our equation now looks like this: 3.5n = 36.1. We've successfully completed the first step! Notice how much simpler the equation looks already. By subtracting 6.4 from both sides, we've cleared away a major hurdle and brought ourselves closer to the solution. This process of simplifying equations one step at a time is key to mastering algebra. Remember, each step is a strategic move that brings you closer to your goal – in this case, finding the value of n.

Why Not Other Operations First?

You might be wondering, "Why couldn't we divide or multiply first?" Great question! It's crucial to understand why we choose subtraction as the initial move. If we were to divide or multiply both sides of the original equation (3.5n + 6.4 = 42.5) right away, we'd be making things more complicated, not less. Imagine dividing both sides by 3.5. We'd end up with fractions and a more complex expression on the left side. Our goal is always to simplify, and dealing with that division upfront would take us in the opposite direction. Similarly, multiplying both sides by some number wouldn't directly help us isolate n. In fact, it would just make all the numbers bigger and the equation harder to manage. The key is to undo the operations in the reverse order of PEMDAS/BODMAS, as we discussed earlier. This means addressing addition and subtraction before multiplication and division. By subtracting 6.4 first, we neatly eliminate the constant term on the left side, paving the way for the next step in solving for n. It's all about choosing the most strategic path to simplification, and in this case, subtraction is the clear winner. So, next time you're faced with a similar equation, remember the importance of prioritizing addition and subtraction early on.

Next Steps: Isolating 'n' Completely

Okay, so we've nailed the first step: subtracting 6.4 from both sides. We've arrived at the simplified equation 3.5n = 36.1. But we're not quite done yet! Remember, our ultimate goal is to isolate n completely, meaning we want to get n all by itself on one side of the equation. So, what's the next logical step? Looking at the equation, we see that n is currently being multiplied by 3.5. To undo this multiplication, we need to perform the inverse operation, which is division. That's right, we're going to divide both sides of the equation by 3.5. This will cancel out the 3.5 on the left side, leaving us with just n. And whatever value we get on the right side after the division will be the solution for n! This is how we gradually peel away the layers of the equation until we reveal the value of our variable. Keep an eye out for the full solution in a future discussion, where we'll tackle this division and fully solve for n. For now, you've mastered the crucial first step and understand why it's so important. You're well on your way to becoming an equation-solving superstar!

Conclusion: Mastering the First Step

So, to recap, the first step in solving the equation 3.5n + 6.4 = 42.5 is to subtract 6.4 from both sides. This is because we need to undo the addition before we can tackle the multiplication. Remember, solving equations is like working backward through the order of operations. By subtracting 6.4, we simplify the equation and get closer to isolating n. This initial move is a crucial step in the overall process, setting the stage for the remaining operations. By understanding why we choose subtraction as the first step, you're not just memorizing a rule; you're grasping the underlying logic of equation solving. This deeper understanding will empower you to tackle a wider range of algebraic problems with confidence. So, congratulations on mastering this fundamental concept! Keep practicing, and you'll find that even the most daunting equations become manageable when you break them down step by step. Now you are equipped to confidently approach similar problems and lead the way in your math journey. Keep up the great work! This foundational step will significantly bolster your ability to tackle more complex equations in the future.