Solving (3x)/(3x-1) + 2/x = 1: A Step-by-Step Guide

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Hey guys! Today, we're going to dive into solving a fun little equation: (3x)/(3x-1) + 2/x = 1. Don't worry, it's not as scary as it looks! We'll break it down step by step so you can follow along easily. Our goal is to find the value of 'x' that makes this equation true, and we're going to express our answer as a simplified fraction. So, grab your pencils and let's get started!

Step 1: Clearing the Fractions

The first thing we want to do is get rid of those pesky fractions. To do this, we need to find the least common denominator (LCD) of all the fractions in the equation. In our case, the denominators are (3x - 1) and x. Therefore, the LCD is simply x(3x - 1). Now, we're going to multiply every term in the equation by this LCD. This will clear out the fractions and leave us with a more manageable equation.

Here's how it looks:

x(3x - 1) * (3x)/(3x - 1) + x(3x - 1) * (2/x) = x(3x - 1) * 1

When we simplify this, we get:

3x * x + 2 * (3x - 1) = x(3x - 1)

Which further simplifies to:

3x^2 + 6x - 2 = 3x^2 - x

Step 2: Simplifying and Rearranging the Equation

Now that we've cleared the fractions, let's simplify and rearrange the equation. Our goal is to get all the terms on one side of the equation and set it equal to zero. Looking at our current equation, we have:

3x^2 + 6x - 2 = 3x^2 - x

Notice that we have 3x^2 on both sides of the equation. We can subtract 3x^2 from both sides, which will eliminate these terms. This leaves us with:

6x - 2 = -x

Now, let's add x to both sides to get all the 'x' terms on one side:

6x + x - 2 = 0

This simplifies to:

7x - 2 = 0

Step 3: Isolating 'x'

Alright, we're in the home stretch! Now we need to isolate 'x' to find its value. We have the equation:

7x - 2 = 0

First, let's add 2 to both sides of the equation:

7x = 2

Now, to get 'x' by itself, we'll divide both sides by 7:

x = 2/7

Step 4: Checking the Solution

It's always a good idea to check our solution to make sure it's correct and doesn't lead to any undefined terms in the original equation. Our solution is x = 2/7. Let's plug this value back into the original equation:

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

Substituting x = 2/7, we get:

(3 * (2/7)) / (3 * (2/7) - 1) + 2 / (2/7) = 1

Let's simplify this:

(6/7) / (6/7 - 1) + 2 * (7/2) = 1

(6/7) / (6/7 - 7/7) + 7 = 1

(6/7) / (-1/7) + 7 = 1

(6/7) * (-7/1) + 7 = 1

-6 + 7 = 1

1 = 1

Our solution checks out! So, x = 2/7 is indeed the correct answer.

Common Mistakes to Avoid

  • Forgetting to Distribute: When multiplying by the LCD, make sure you distribute it to every term in the equation.
  • Incorrectly Simplifying Fractions: Double-check your work when simplifying fractions to avoid errors.
  • Not Checking the Solution: Always check your solution in the original equation to make sure it's valid.
  • Arithmetic Errors: Watch out for simple arithmetic mistakes, especially when dealing with fractions and negative numbers.

Alternative Methods

While clearing fractions is a common and effective method, there are other approaches you could take:

  • Combining Fractions First: You could start by combining the fractions on the left side of the equation into a single fraction. Then, cross-multiply to solve for 'x.'
  • Substitution (Sometimes): In some cases, you might be able to use substitution to simplify the equation. However, this method isn't as straightforward for this particular equation.

Real-World Applications

While solving equations like this might seem purely academic, they actually have real-world applications in various fields:

  • Physics: These types of equations can arise when dealing with electrical circuits, optics, and mechanics.
  • Engineering: Engineers use similar equations in designing structures, analyzing fluid flow, and modeling systems.
  • Economics: Economists use equations to model supply and demand, analyze market trends, and make predictions.
  • Computer Science: These concepts are fundamental in algorithm design and optimization.

Conclusion

So, there you have it! We've successfully solved the equation (3x)/(3x-1) + 2/x = 1 and found that x = 2/7. Remember the key steps: clear the fractions, simplify, isolate 'x,' and check your solution. With practice, you'll become a pro at solving these types of equations. Keep up the great work, and happy solving!

Remember, guys, math isn't about memorizing formulas; it's about understanding the process and applying it logically. So, keep practicing, and don't be afraid to ask for help when you need it. You got this!

Key takeaways: Always double-check your work, practice makes perfect, and understanding the underlying concepts is crucial. Good luck, and have fun with math! And remember, expressing the answer as a simplified fraction is essential for clarity and precision. This ensures that the solution is in its most reduced form, making it easier to interpret and use in further calculations.