Solving Linear Equations A Step-by-Step Guide To 3.4 + 2(9.7 - 4.8x) = 612

Hey everyone! Today, we're diving into the world of linear equations with a specific problem that might seem a bit tricky at first glance: 3.4 + 2(9.7 - 4.8x) = 612. Don't worry, though! We're going to break it down step by step, so you'll be solving equations like a pro in no time. Let's get started!

Understanding the Equation: A Foundation for Success

Before we jump into the actual solving process, it's crucial to understand what a linear equation is and the different components involved. This foundational knowledge will make the steps we take later much clearer. So, what exactly is a linear equation? At its core, a linear equation is a mathematical statement that shows the equality between two expressions. These expressions involve variables (usually represented by letters like x, y, or z) raised to the power of one, and constants (numbers). The goal is to find the value of the variable that makes the equation true. In our case, the variable is x, and we need to find the specific value of x that satisfies the equation 3.4 + 2(9.7 - 4.8x) = 612. Understanding this goal is the first step in conquering any equation. Now, let's break down the components of our specific equation. We have constants like 3.4 and 612, which are fixed numbers. We also have a term inside parentheses, (9.7 - 4.8x), which involves both a constant (9.7) and a variable term (-4.8x). The number 2 outside the parentheses is a coefficient, meaning it will be multiplied by everything inside the parentheses. This operation is crucial and is called the distributive property, which we'll discuss in more detail later. Recognizing these components – constants, variables, coefficients, and terms – is like having a map before embarking on a journey. It helps us navigate the equation systematically and avoid common pitfalls. Without this understanding, we might try to combine terms that shouldn't be combined or miss crucial steps. So, before we start manipulating the equation, take a moment to appreciate its structure. Understanding the pieces is the key to putting the puzzle together. Remember, solving linear equations is not just about following steps blindly; it's about understanding why we take those steps. This understanding will empower you to solve a wide range of equations, not just this specific one. Linear equations are the building blocks of algebra, and mastering them opens doors to more advanced mathematical concepts. So, let's build a strong foundation together!

Step-by-Step Solution: Cracking the Code

Now, let's get down to the nitty-gritty and solve the equation 3.4 + 2(9.7 - 4.8x) = 612 step by step. We'll break down each action and explain the reasoning behind it, so you're not just following instructions but truly understanding the process. This is where the magic happens! The first key step in solving this equation, and many others like it, is to simplify the expression. This involves getting rid of parentheses and combining like terms. In our equation, we have the term 2(9.7 - 4.8x). This is where the distributive property comes into play. The distributive property states that a(b + c) = ab + ac. In simpler terms, it means we need to multiply the number outside the parentheses (2 in our case) by each term inside the parentheses (9.7 and -4.8x). So, let's do that: 2 * 9.7 = 19.4 and 2 * -4.8x = -9.6x. This transforms our equation to: 3.4 + 19.4 - 9.6x = 612. See how much cleaner it looks already? Now, we have like terms on the left side of the equation: 3.4 and 19.4. Like terms are terms that have the same variable raised to the same power (or, in this case, are just constants). We can combine them by simply adding them together: 3.4 + 19.4 = 22.8. Our equation now looks even simpler: 22.8 - 9.6x = 612. We've successfully simplified the left side of the equation, making it much easier to work with. The next goal is to isolate the variable term, which is -9.6x in our case. This means we want to get -9.6x by itself on one side of the equation. To do this, we need to get rid of the 22.8. Since 22.8 is being added to -9.6x, we perform the opposite operation: subtraction. We subtract 22.8 from both sides of the equation to maintain the balance. Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation true. So, we have: 22.8 - 9.6x - 22.8 = 612 - 22.8. This simplifies to: -9.6x = 589.2. We're almost there! We now have the variable term isolated on the left side. The final step is to get x by itself. Currently, x is being multiplied by -9.6. To undo this multiplication, we perform the opposite operation: division. We divide both sides of the equation by -9.6: -9.6x / -9.6 = 589.2 / -9.6. This gives us: x = -61.375. And there you have it! We've successfully solved the equation. The value of x that makes the equation true is -61.375. Remember, each step we took was designed to simplify the equation and isolate the variable. By understanding the reasoning behind each step, you can apply these techniques to solve a wide variety of linear equations. Practice makes perfect, so keep working at it, and you'll become a master equation solver in no time!

Possible Steps: Analyzing the Options

Okay, guys, now that we've solved the equation 3.4 + 2(9.7 - 4.8x) = 612, let's look at the options provided and see which steps were actually involved in our solution. This is a great way to reinforce our understanding of the process. We'll go through each option one by one and explain why it was or wasn't a step we took. This will not only help us answer the question correctly but also solidify our grasp of solving linear equations. The first option is A. Add 3.4 and 2. Did we add 3.4 and 2 at any point in our solution? Nope! Remember, we need to follow the order of operations (PEMDAS/BODMAS), which means we deal with parentheses first. The 2 is being multiplied by the expression inside the parentheses, so we can't just add it to 3.4. This option is incorrect. Next up, we have B. Distribute 2 to 9.7 and -4.8x. This sounds familiar, right? This is exactly what we did using the distributive property! We multiplied 2 by both 9.7 and -4.8x to eliminate the parentheses. So, this option is definitely a step we took. Let's move on to C. Combine 3.4 and 19.4. After distributing the 2, we had the terms 3.4 and 19.4 in our equation. These are like terms (both constants), so we combined them by adding them together. This option is also a step we took. Now, let's consider D. Divide both sides by 22.8. While we did divide both sides of the equation at one point, we didn't divide by 22.8. We divided by -9.6, which was the coefficient of x after we isolated the variable term. Dividing by 22.8 wouldn't have helped us isolate x. So, this option is incorrect. By carefully analyzing each option and comparing it to the steps we actually took, we can see which actions were crucial to solving the equation. This process of reflection is just as important as the solving itself. It helps us understand the underlying principles and apply them to other problems. Remember, math isn't just about getting the right answer; it's about understanding the process and reasoning behind it. By breaking down the problem and analyzing the options, we've not only answered the question but also deepened our understanding of linear equations. So, let's keep practicing and exploring the world of math together!

Final Answer: Identifying the Correct Steps

Alright, guys, after carefully dissecting each step in solving the equation 3.4 + 2(9.7 - 4.8x) = 612 and analyzing the options, we're ready to pinpoint the correct steps. This is the moment of truth! We've walked through the entire solution process, from understanding the equation's structure to isolating the variable. We've used the distributive property, combined like terms, and performed inverse operations to arrive at our answer. Now, it's time to match our actions with the given options. Remember, the key is to identify the steps that were actually involved in our solution, not just steps that might seem related. Let's recap our findings. We determined that Option A, Add 3.4 and 2, was incorrect. We couldn't add these numbers directly because of the order of operations (PEMDAS/BODMAS). We had to deal with the parentheses first. Option B, Distribute 2 to 9.7 and -4.8x, was a crucial step. This is where we applied the distributive property to eliminate the parentheses and simplify the equation. Option C, Combine 3.4 and 19.4, was also a step we took. After distributing, we had these like terms (constants) that we could combine to further simplify the equation. Finally, Option D, Divide both sides by 22.8, was incorrect. We divided both sides of the equation, but we divided by -9.6, the coefficient of x, not 22.8. So, based on our analysis, the correct steps involved in solving the equation are B. Distribute 2 to 9.7 and -4.8x and C. Combine 3.4 and 19.4. These were the key actions that allowed us to simplify the equation and move closer to isolating the variable x. Identifying these steps reinforces our understanding of the solution process. It's not just about getting the numerical answer; it's about recognizing the specific techniques and principles we applied along the way. By understanding these principles, we can tackle other linear equations with confidence. Remember, solving equations is like building a puzzle. Each step is a piece, and understanding how the pieces fit together is the key to success. We've successfully identified the correct pieces in this puzzle, and we're ready to move on to more challenging problems!

I hope this detailed explanation has helped you understand how to solve the linear equation 3.4 + 2(9.7 - 4.8x) = 612 and identify the correct steps involved. Keep practicing, and you'll become a master of linear equations!