Solving Linear Equations: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever get stuck on a math problem that looks like a jumbled mess of numbers and letters? Don't sweat it! Today, we're going to break down a common type of equation – a linear equation – and show you how to solve it like a pro. We'll tackle the equation $-\frac{1}{3}x + 12 = -2x + 2$ step-by-step. So, grab your pencils, and let's dive in!

Understanding the Equation

Before we jump into solving, let's make sure we understand what we're looking at. The equation $-\frac{1}{3}x + 12 = -2x + 2$ is a linear equation. But what exactly is a linear equation, you ask? Well, a linear equation is basically an equation where the highest power of the variable (in this case, 'x') is 1. Think of it as a straight line if you were to graph it. Our goal here is to find the value of 'x' that makes the equation true. In other words, we want to find the number that, when we substitute it for 'x', will make both sides of the equation equal. Solving equations like this is a foundational skill in algebra, and mastering it opens the door to tackling more complex mathematical problems. Understanding the structure of the equation is the first step. We have variables ('x'), constants (numbers like 12 and 2), and coefficients (the numbers multiplying the variables, like $-\frac{1}{3}$ and -2). Our mission, should we choose to accept it (and we do!), is to isolate 'x' on one side of the equation. This means getting 'x' all by itself, so we can see what value it truly holds. This might sound like a daunting task, but fear not! We're going to break it down into manageable steps, each building upon the last, until we arrive at the solution. Remember, the key to success in math (and in life!) is to approach challenges methodically. So, let's get started and unravel this equation, one step at a time. You'll see that with a little patience and the right techniques, even the most seemingly complex problems can be conquered. And who knows, you might even find you enjoy the process!

Step 1: Clearing the Fraction

Okay, first things first, let's deal with that fraction – $-\frac{1}{3}$. Fractions can sometimes make equations look intimidating, but trust me, we can get rid of it easily. The trick is to multiply both sides of the equation by the denominator of the fraction, which in this case is 3. Why do we do this? Because multiplying by the denominator will cancel out the fraction, making our lives much simpler. It's like magic, but it's actually just good old math! So, we multiply both sides of the equation $-\frac{1}{3}x + 12 = -2x + 2$ by 3. Remember, what we do to one side of the equation, we must do to the other side to keep things balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, let's do the multiplication: 3 * $(-\frac{1}{3}x + 12)$ = 3 * $(-2x + 2)$. Now, we need to distribute the 3 to each term inside the parentheses on both sides. This means we multiply 3 by $-\frac{1}{3}x$ and 3 by 12 on the left side, and 3 by -2x and 3 by 2 on the right side. Let's break it down: 3 * $-\frac{1}{3}x$ becomes -x (the 3s cancel out), 3 * 12 becomes 36, 3 * -2x becomes -6x, and 3 * 2 becomes 6. So, after multiplying and simplifying, our equation now looks like this: -x + 36 = -6x + 6. See? Much cleaner already! The fraction is gone, and we're left with a simpler equation to work with. This is a crucial step in solving many equations, and it's a technique you'll use again and again in your mathematical adventures. So, remember, when you see a fraction lurking in your equation, don't panic! Just multiply both sides by the denominator, and watch it disappear. You've successfully cleared the first hurdle! Now, let's move on to the next step and continue our quest to isolate 'x'.

Step 2: Grouping the 'x' Terms

Alright, we've gotten rid of the fraction, which is a major win! Now, let's focus on getting all the 'x' terms on one side of the equation. This is like gathering all your friends together in one place – we want to bring all the 'x's to the same side so we can work with them more easily. Looking at our current equation, -x + 36 = -6x + 6, we have 'x' terms on both sides: -x on the left and -6x on the right. To group them together, we can add 6x to both sides of the equation. Why add 6x? Because adding the opposite of a term will cancel it out. So, adding 6x to -6x will eliminate the 'x' term from the right side, moving it over to the left side. Remember, we always do the same thing to both sides to maintain the balance. So, let's add 6x to both sides: -x + 36 + 6x = -6x + 6 + 6x. Now, let's simplify. On the left side, we have -x + 6x, which combines to 5x. So, the left side becomes 5x + 36. On the right side, -6x and +6x cancel each other out, leaving us with just 6. So, our equation now looks like this: 5x + 36 = 6. We're making great progress! All the 'x' terms are now on the left side, and we're one step closer to isolating 'x'. Grouping like terms is a fundamental strategy in solving equations. It helps us simplify the equation and bring us closer to our goal. Think of it as organizing your workspace – putting all the similar tools together makes it easier to find what you need. In this case, we've organized the 'x' terms, making the next steps much smoother. So, remember, when you have 'x' terms scattered on both sides of the equation, gather them together by adding or subtracting the appropriate term from both sides. You're doing awesome! Let's move on to the next step and continue our journey to solve for 'x'.

Step 3: Isolating the 'x' Term

We're on the home stretch now, guys! We've successfully grouped the 'x' terms on one side of the equation. Our equation currently looks like this: 5x + 36 = 6. Now, we need to isolate the 'x' term completely. This means getting the 5x by itself on the left side. To do this, we need to get rid of the +36. How do we do that? By subtracting 36 from both sides of the equation! Just like before, we're using the principle of doing the same thing to both sides to keep the equation balanced. Subtracting 36 is the opposite operation of adding 36, so it will effectively cancel out the 36 on the left side. Let's do it: 5x + 36 - 36 = 6 - 36. Now, let's simplify. On the left side, +36 and -36 cancel each other out, leaving us with just 5x. On the right side, 6 - 36 equals -30. So, our equation now looks like this: 5x = -30. We're so close! The 'x' term is almost completely isolated. We just have one more little step to take. Isolating the variable is a crucial step in solving any equation. It's like peeling away the layers of an onion – we're removing all the extra bits and pieces until we get to the core, which is the variable itself. By subtracting 36 from both sides, we've peeled away the constant term, bringing us closer to the final solution. Remember, the key is to use inverse operations. If there's addition, we subtract. If there's subtraction, we add. And so on. You're doing a fantastic job! Let's finish this equation strong and find the value of 'x'. We're almost there!

Step 4: Solving for 'x'

Okay, the moment we've been working towards! We're at the final step of solving for 'x'. Our equation now stands as 5x = -30. This means 5 times 'x' is equal to -30. So, how do we find out what 'x' is? We need to undo the multiplication. And how do we undo multiplication? By dividing! We're going to divide both sides of the equation by 5. This will isolate 'x' on the left side and give us its value. Let's do it: 5x / 5 = -30 / 5. Now, let's simplify. On the left side, 5x divided by 5 is simply 'x'. On the right side, -30 divided by 5 is -6. So, our equation now looks like this: x = -6. And there we have it! We've solved for 'x'. The value of 'x' that makes the original equation true is -6. You did it! You successfully navigated the equation and found the solution. Solving for 'x' is the ultimate goal in this type of problem. It's like finding the missing piece of a puzzle. By dividing both sides by the coefficient of 'x', we've uncovered the value that satisfies the equation. This is a skill that will serve you well in all your future mathematical endeavors. Remember, the key is to understand the operations involved and use inverse operations to isolate the variable. You've shown incredible problem-solving skills! Let's recap our journey and celebrate our achievement.

Conclusion: The Solution and Key Takeaways

Woohoo! We've reached the end of our equation-solving adventure. We started with the equation $-\frac{1}{3}x + 12 = -2x + 2$ and, through a series of careful steps, we discovered that x = -6. So, the correct answer is B. x = -6. But more than just finding the answer, we've learned some valuable techniques for solving linear equations. Let's recap the key steps we took:

1. Clearing the Fraction: We multiplied both sides of the equation by the denominator of the fraction to eliminate it.
2. Grouping the 'x' Terms: We added or subtracted 'x' terms to get all the variables on one side of the equation.
3. Isolating the 'x' Term: We added or subtracted constants to get the 'x' term by itself.
4. Solving for 'x': We divided both sides of the equation by the coefficient of 'x' to find its value.

These steps can be applied to solve a wide variety of linear equations. Remember, the key is to stay organized, perform the same operations on both sides of the equation, and use inverse operations to isolate the variable. Solving equations is like building a strong foundation in math. The more you practice, the more confident you'll become, and the better you'll be able to tackle more complex problems. So, keep practicing, keep exploring, and never stop learning! You've got this! And remember, if you ever get stuck, don't hesitate to break down the problem into smaller steps, review the basic principles, and ask for help if you need it. The world of mathematics is vast and exciting, and we're all in this learning journey together. So, let's celebrate our success today and look forward to the next mathematical challenge!

Great job, everyone! You've conquered this equation, and you're well on your way to becoming math masters!