Solving The Equation: (3x + 1)/4 - (1 - 2x)/6 = (3x + 2)/8

by Andrew McMorgan 59 views

Hey Plastik Magazine readers! Today, we're diving into a fun little mathematical problem. We're going to break down how to solve the equation (3x + 1)/4 - (1 - 2x)/6 = (3x + 2)/8. Don't worry if it looks intimidating at first; we'll take it step by step and make sure everyone's on board. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what we're looking at. The equation (3x + 1)/4 - (1 - 2x)/6 = (3x + 2)/8 involves fractions, variables, and a mix of addition and subtraction. Our goal is to find the value of 'x' that makes this equation true. Sounds like a mission, right? But trust me, it's totally achievable!

When you first see an equation like this, it might seem daunting. Fractions often make things look more complex than they are. But remember, mathematics is all about breaking down problems into smaller, manageable steps. We'll tackle this equation by first eliminating the fractions, which will make the equation much easier to work with. We'll then simplify and isolate 'x' to find our solution. So, let's dive in and see how it's done!

The Importance of Order of Operations

Just a quick reminder, guys, about the order of operations! You might have heard of it as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order is crucial when solving any equation. In our case, we'll focus on simplifying each fraction and then combining like terms. Keeping this order in mind will help us avoid common mistakes and keep our solution on track. Understanding the structure of the equation and the role each term plays is the first step toward conquering it.

Step-by-Step Solution

Okay, let's get our hands dirty and walk through the solution step-by-step. We'll break it down into easy-to-follow actions. By the end, you'll feel like equation-solving pros!

1. Finding the Least Common Multiple (LCM)

The first thing we need to do is get rid of those pesky fractions. To do that, we'll find the Least Common Multiple (LCM) of the denominators (4, 6, and 8). The LCM is the smallest number that all these denominators can divide into evenly. So, what's the LCM of 4, 6, and 8? It's 24! This means we'll multiply both sides of the equation by 24 to eliminate the fractions. Finding the LCM is a crucial step as it simplifies the equation and allows us to work with whole numbers instead of fractions. This not only makes the calculations easier but also reduces the chance of making errors.

2. Multiplying Both Sides by the LCM

Now that we know the LCM is 24, we're going to multiply both sides of the equation by it. This is a crucial step because it clears the fractions and transforms our equation into something much simpler to manage. When we multiply each term by 24, we're essentially scaling up the equation while maintaining the balance. Remember, whatever we do to one side of the equation, we must do to the other to keep it equal. This is a fundamental principle in mathematics, and it's what allows us to manipulate equations to solve for our variable. The multiplication step is where the equation starts to become less intimidating, and we can see the path to the solution more clearly.

So, here's what it looks like:

24 * [(3x + 1)/4] - 24 * [(1 - 2x)/6] = 24 * [(3x + 2)/8]

3. Simplifying the Equation

After multiplying by the LCM, we can simplify each term. Let's break it down:

  • 24 * [(3x + 1)/4] becomes 6(3x + 1)
  • 24 * [(1 - 2x)/6] becomes 4(1 - 2x)
  • 24 * [(3x + 2)/8] becomes 3(3x + 2)

So, our equation now looks like this:

6(3x + 1) - 4(1 - 2x) = 3(3x + 2)

See? Much cleaner already! Simplifying the equation involves reducing the fractions and applying the distributive property, which is a key technique in mathematical problem-solving. This process makes the equation more accessible and allows us to combine like terms later on, bringing us closer to isolating 'x' and finding its value. Each simplification step is a move in the right direction, making the solution clearer and more achievable.

4. Distributing and Expanding

Now, let's distribute the numbers outside the parentheses:

  • 6(3x + 1) = 18x + 6
  • -4(1 - 2x) = -4 + 8x
  • 3(3x + 2) = 9x + 6

Our equation is now:

18x + 6 - 4 + 8x = 9x + 6

The distribution step is crucial because it removes the parentheses, allowing us to combine like terms. This is where we start to see the equation in its simplest form, with all the 'x' terms and constant terms clearly laid out. Expanding the expressions makes the next step, combining like terms, much easier and more straightforward. It's like organizing your workspace before tackling the main task – it sets you up for success!

5. Combining Like Terms

Let's combine like terms on the left side of the equation:

18x + 8x + 6 - 4 = 26x + 2

So, our equation becomes:

26x + 2 = 9x + 6

Combining like terms is a fundamental step in solving equations. It simplifies the equation by grouping similar terms together, making it easier to isolate the variable. In our case, we combined the 'x' terms and the constant terms on the left side. This process reduces the complexity of the equation and brings us closer to a solution. It's like putting all the pieces of a puzzle together – we're organizing the information to reveal the bigger picture.

6. Isolating the Variable

Now, we need to get all the 'x' terms on one side and the constants on the other. Let's subtract 9x from both sides:

26x - 9x + 2 = 9x - 9x + 6

17x + 2 = 6

Then, subtract 2 from both sides:

17x + 2 - 2 = 6 - 2

17x = 4

Isolating the variable is a key step in solving any equation. It involves moving all the terms containing the variable to one side and all the constant terms to the other. This is done by performing the same operation on both sides of the equation, maintaining the balance. In our case, we subtracted 9x from both sides and then subtracted 2 from both sides to isolate the 'x' term. This process simplifies the equation and brings us closer to the final solution.

7. Solving for x

Finally, to solve for x, we'll divide both sides by 17:

17x / 17 = 4 / 17

x = 4/17

There you have it! The solution to the equation is x = 4/17. Solving for 'x' is the final step in our journey. It involves isolating 'x' and then performing the necessary operation to find its value. In our case, we divided both sides of the equation by 17 to get x = 4/17. This is the value of 'x' that makes the original equation true. It's like finding the missing piece of the puzzle – we've successfully solved the equation!

Conclusion

And that's a wrap, guys! We successfully solved the equation (3x + 1)/4 - (1 - 2x)/6 = (3x + 2)/8, and we found that x = 4/17. High five! Solving equations like this might seem tough at first, but with a step-by-step approach, it becomes much more manageable. Remember, the key is to break down the problem, eliminate fractions, simplify, and isolate the variable. You've got this!

So, the next time you encounter an equation that looks a bit scary, don't sweat it. Just remember the steps we've covered, and you'll be solving like a pro in no time. Keep practicing, and mathematics will become your playground. Until next time, keep those mathematical gears turning, and stay curious!

We hope you found this breakdown helpful and easy to follow. Remember, mathematics is a skill that gets better with practice, so keep at it! And who knows? Maybe you'll even start to enjoy these equation-solving adventures. Keep shining, math whizzes!