Solving For X: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever stared at an equation and felt a little lost? Don't worry, we've all been there! Today, we're diving into the equation $\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3 $ and breaking down how to solve for x. This isn't just about getting an answer; it's about understanding the process, so you can tackle similar problems with confidence. So, grab your pencils (or your favorite digital stylus!), and let's get started. We'll go step-by-step, making sure everything is clear, even if math isn't your favorite subject. The goal is to make math approachable and, dare we say, even a little fun! Let's get right into the heart of the matter! This equation might look a bit intimidating at first glance, with all those fractions and parentheses, but trust me, it's totally manageable. We'll use a clear, logical process. The key is to take it one step at a time, and always double-check your work along the way. Are you ready? Let's begin! We are going to go over how to solve for x in the equation $\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3 $ using a step by step guide. First of all, we need to understand the equation that we will be working with and after that, we can start with the step by step guide to obtain the result. Math is awesome, and you can solve problems with this step-by-step approach!

Step-by-Step Guide to Solve for X

Step 1: Distribute the Fractions (Opening the Parentheses)

Alright, guys, our first move is to get rid of those pesky parentheses. This is where we use the distributive property. It's like sharing – we need to multiply the number outside the parentheses by each term inside the parentheses. So, on the left side of the equation, we're multiplying $\frac{2}{3}$ by both $\frac{1}{2}x$ and $12$. And on the right side, we multiply $\frac{1}{2}$ by both $\frac{1}{3}x$ and $14$. This is a crucial step because it simplifies the equation and lets us combine like terms later on. Remember, the distributive property states that $a(b + c) = ab + ac$. Keep in mind the distributive property is one of the most fundamental concepts in algebra, and it's essential for simplifying expressions and solving equations. Getting it right here makes everything else a lot smoother. So, let's carefully apply this to our equation. This is going to be the most important step of all, so keep in mind all the operations and do them in the proper order. It's all about making sure we multiply each term correctly. Are you with me? Let's break it down: $\frac{2}{3} * \frac{1}{2}x = \frac{1}{3}x$ and $\frac{2}{3} * 12 = 8$. Similarly, $\frac{1}{2} * \frac{1}{3}x = \frac{1}{6}x$ and $\frac{1}{2} * 14 = 7$. Now, we can rewrite our equation as $\frac{1}{3}x + 8 = \frac{1}{6}x + 7 - 3$. Make sure that you didn't skip any step, because if you did, it's possible you will get a wrong result. Keep going with me, we are close to the result!

Step 2: Simplify the Equation (Combine Like Terms)

Now that we've distributed, it's time to clean things up a bit. We'll combine any like terms on each side of the equation. This involves simplifying the constants. Remember, constants are just numbers that stand alone. Our equation currently looks like this: $\frac{1}{3}x + 8 = \frac{1}{6}x + 7 - 3$. On the right side, we have $7 - 3$, which simplifies to $4$. So, our equation becomes $\frac{1}{3}x + 8 = \frac{1}{6}x + 4$. Combining like terms is all about making the equation easier to work with. It's like tidying up a room before you start organizing – it gives you a clearer view of what you're dealing with. Just take a look at the equation and you'll realize it's easier to continue now! This step might seem simple, but it's important because it prepares us for the next steps where we'll isolate x. Don't rush; take your time to make sure you've combined the terms correctly. Are you following along? Remember, you can always pause and go back if you need to. We're in no rush. The goal is understanding, not speed. Great job so far, guys! You're doing awesome!

Step 3: Isolate the Variable (Get the x's Together)

Here comes the fun part: isolating x. Our goal is to get all the terms containing x on one side of the equation and all the constants on the other side. This is like herding cats – you want all the x's together. To do this, we need to move the $\frac{1}{6}x$ from the right side to the left side. We do this by subtracting $\frac{1}{6}x$ from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced. This is a fundamental rule in algebra – it's like a seesaw; to keep it level, you have to add or remove the same weight from both sides. When we subtract $\frac{1}{6}x$ from both sides, we get: $\frac{1}{3}x - \frac{1}{6}x + 8 = 4$. Next, we need to subtract 8 from both sides to move the constant terms to the right side. This gives us: $\frac{1}{3}x - \frac{1}{6}x = 4 - 8$. Now we are closer to the result! Just one more step and we can say that we've reached the result for $x$.

Step 4: Solve for x (Final Calculation)

Alright, we're almost there! Now that we've isolated the x terms, it's time to solve for x. First, let's simplify the left side of the equation, which is $\frac{1}{3}x - \frac{1}{6}x$. To subtract these fractions, we need a common denominator, which in this case is 6. So, we rewrite $\frac{1}{3}x$ as $\frac{2}{6}x$. Now we can subtract: $\frac{2}{6}x - \frac{1}{6}x = \frac{1}{6}x$. On the right side, we simplify $4 - 8$, which equals $-4$. Our equation now looks like this: $\frac{1}{6}x = -4$. To solve for x, we need to get rid of that pesky fraction. We do this by multiplying both sides of the equation by the reciprocal of $\frac{1}{6}$, which is $6$. Multiplying both sides by 6, we get: $6 * \frac{1}{6}x = -4 * 6$. This simplifies to $x = -24$. Congratulations, guys! We've found the solution! We have successfully solved for x and found its value. Now you know how to solve for x.

Conclusion: The Answer Revealed!

And there you have it, Plastik Magazine readers! The value of x in the equation $\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3 $ is -24. We've walked through each step, from distributing and combining like terms to isolating the variable and solving for x. Remember, the key is to take it slow, be careful with your calculations, and double-check your work. Math can seem tough, but with a clear process, it becomes much more manageable. Keep practicing, and you'll find yourself solving equations with ease. Keep in mind all the steps in order and you will be able to solve any kind of problem. Now you can solve this kind of equation. You've got this, guys! Keep up the awesome work!