Solving For X: A Step-by-Step Guide To The Equation

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Let's dive into solving this equation together! We're going to break it down step-by-step so you can see exactly how to find the value of x. If you've ever felt a little lost when faced with equations like this, don't worry, we've got you covered. Grab your pencil and paper, and let's get started!

The Equation We're Tackling

The equation we need to solve is:

(2/3)((1/2)x + 12) = (1/2)((1/3)x + 14) - 3

This might look a little intimidating at first glance, but we're going to simplify it piece by piece. Remember, the key to solving equations is to isolate the variable (in this case, x) on one side of the equation.

Step 1: Distribute the Fractions

The first thing we need to do is get rid of those parentheses. We do this by distributing the fractions on both sides of the equation.

On the left side, we'll distribute the 2/3:

(2/3) * (1/2)x + (2/3) * 12

This simplifies to:

(1/3)x + 8

Now, let's do the same for the right side. We'll distribute the 1/2:

(1/2) * (1/3)x + (1/2) * 14

Which simplifies to:

(1/6)x + 7

Don't forget the - 3 at the end! So, our equation now looks like this:

(1/3)x + 8 = (1/6)x + 7 - 3

See? It's already looking a bit cleaner.

Step 2: Combine Like Terms

Before we start moving terms around, let's simplify the right side a little further by combining the constant terms, 7 and -3:

7 - 3 = 4

So, our equation becomes:

(1/3)x + 8 = (1/6)x + 4

Now we're ready to start isolating x.

Step 3: Move x Terms to One Side

Our goal is to get all the x terms on one side of the equation and all the constant terms on the other. Let's subtract (1/6)x from both sides. This will move the x term from the right side to the left:

(1/3)x - (1/6)x + 8 = (1/6)x - (1/6)x + 4

To subtract the fractions, we need a common denominator. The least common denominator for 3 and 6 is 6. So, we'll rewrite (1/3)x as (2/6)x:

(2/6)x - (1/6)x + 8 = 4

Now we can subtract:

(1/6)x + 8 = 4

Step 4: Move Constant Terms to the Other Side

Next, we need to move the constant term, 8, from the left side to the right side. We do this by subtracting 8 from both sides:

(1/6)x + 8 - 8 = 4 - 8

This simplifies to:

(1/6)x = -4

We're getting closer!

Step 5: Isolate x

Finally, we need to get x all by itself. Right now, x is being multiplied by 1/6. To undo this, we'll multiply both sides of the equation by the reciprocal of 1/6, which is 6:

6 * (1/6)x = 6 * -4

This simplifies to:

x = -24

The Solution

And there you have it! We've solved for x. The value of x in the equation is -24.

So, the correct answer is:

A. -24

Breaking Down Each Step for Clarity

Let's recap each step to make sure everything is crystal clear. Understanding the why behind each step is just as important as knowing the how.

1. Distribute the Fractions

  • Why? Distributing eliminates the parentheses, making the equation easier to work with. It's like unwrapping a gift – you need to get inside to see what's there!
  • How? Multiply the term outside the parentheses by each term inside the parentheses. Remember the distributive property: a(b + c) = ab + ac.

2. Combine Like Terms

  • Why? Combining like terms simplifies the equation, making it less cluttered and easier to manage. Think of it as organizing your workspace before tackling a big project.
  • How? Identify terms that have the same variable (like x terms) or are constants (just numbers) and add or subtract them accordingly.

3. Move x Terms to One Side

  • Why? We want to isolate x, so we need to get all the x terms together on one side of the equation. It's like gathering all your tools in one place before starting a task.
  • How? Use inverse operations (addition or subtraction) to move x terms from one side to the other. Remember, whatever you do to one side of the equation, you must do to the other to keep it balanced.

4. Move Constant Terms to the Other Side

  • Why? Now that the x terms are on one side, we need to get the constant terms to the other side. This is like separating the ingredients you need from the tools you'll use.
  • How? Use inverse operations (addition or subtraction) to move constant terms from one side to the other, keeping the equation balanced.

5. Isolate x

  • Why? This is the final step! We want to find the value of x, so we need to get it all by itself on one side of the equation. It's like finding the missing piece of a puzzle.
  • How? If x is being multiplied by a number, divide both sides of the equation by that number. If x is being divided by a number, multiply both sides of the equation by that number. Again, keep the equation balanced.

Common Mistakes to Avoid

Solving equations is like following a recipe – you need to be precise to get the right result. Here are some common pitfalls to watch out for:

  • Forgetting to Distribute: Make sure you multiply the term outside the parentheses by every term inside the parentheses. It's easy to miss one, especially if there are multiple terms.
  • Not Combining Like Terms Correctly: Be careful with your signs (positive and negative) when adding or subtracting like terms. A small mistake here can throw off the whole solution.
  • Not Keeping the Equation Balanced: Whatever operation you perform on one side of the equation, you must do the same operation on the other side. This is crucial for maintaining equality.
  • Incorrectly Using Inverse Operations: Remember, to undo addition, you subtract; to undo subtraction, you add; to undo multiplication, you divide; and to undo division, you multiply.
  • Rushing Through the Steps: It's tempting to try to solve the equation quickly, but taking your time and carefully checking each step will help you avoid errors.

Practice Makes Perfect

The best way to master solving equations is to practice! Try working through similar problems on your own. The more you practice, the more comfortable and confident you'll become. You got this!

Real-World Applications

Now, you might be thinking, "Okay, this is great, but when am I ever going to use this in real life?" Well, you'd be surprised!

Solving equations is a fundamental skill that comes up in many different areas:

  • Science: Calculating the trajectory of a projectile, determining the rate of a chemical reaction, or analyzing electrical circuits all involve solving equations.
  • Engineering: Designing structures, building machines, and developing software often require solving complex equations.
  • Finance: Calculating interest rates, managing budgets, and making investment decisions involve mathematical equations.
  • Everyday Life: Even simple tasks like calculating the tip at a restaurant, figuring out how much paint you need for a room, or determining the sale price of an item can involve basic algebra.

So, while it might seem abstract now, the skills you're learning in algebra will be valuable in many different aspects of your life.

More Tips for Success

Here are a few more tips to help you succeed in solving equations:

  • Show Your Work: Write down each step clearly and neatly. This will make it easier to follow your thought process and catch any mistakes.
  • Check Your Answer: Once you've found a solution, plug it back into the original equation to make sure it works. This is a great way to verify your answer and catch any errors.
  • Use Online Resources: There are many websites and apps that can help you practice solving equations and check your work.
  • Ask for Help: Don't be afraid to ask your teacher, classmates, or a tutor for help if you're struggling. Everyone learns at their own pace, and there's no shame in seeking assistance.

Solving equations might seem challenging at first, but with practice and persistence, you can master this important skill. Remember to break down the problem into smaller steps, stay organized, and don't give up! You're well on your way to becoming an equation-solving pro! Guys, remember that math is like a puzzle, and each equation is a new challenge to conquer. Keep practicing, and you'll become a math whiz in no time!