Solving The Equation: (x-1)/3 - (x+3)/9 = (x-5)/2 + 1/6

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Hey guys! Let's dive into solving this equation together. It looks a bit complex at first glance, but don't worry, we'll break it down step by step. Our main goal here is to find the value of 'x' that makes the equation true. So, grab your pencils and let's get started!

Understanding the Equation

The equation we're tackling is: (x-1)/3 - (x+3)/9 = (x-5)/2 + 1/6. This is a linear equation, meaning the highest power of 'x' is 1. To solve it, we'll need to get rid of the fractions, combine like terms, and isolate 'x' on one side of the equation. Remember, the key is to perform the same operations on both sides to maintain the balance. We're essentially playing a mathematical game of seesaw, ensuring both sides remain equal.

Breaking Down the Components

Before we jump into the solution, let's understand each part of the equation:

  • (x-1)/3: This term represents 'x' minus 1, all divided by 3.
  • -(x+3)/9: Here, we have 'x' plus 3, divided by 9, and then subtracted from the previous term. The negative sign is crucial, so we'll need to distribute it carefully later on.
  • (x-5)/2: This is 'x' minus 5, divided by 2.
  • 1/6: A simple fraction, one-sixth, added to the right side of the equation.

Now that we understand the equation's components, we can move on to the exciting part: solving it!

Step-by-Step Solution

Okay, let’s get our hands dirty and solve this equation. We'll go through each step meticulously, so you can follow along easily. Trust me, it's like following a recipe – just follow the steps, and you'll get the right result!

1. Eliminate the Fractions

Fractions can be a bit of a headache, so our first step is to get rid of them. To do this, we need to find the least common multiple (LCM) of the denominators: 3, 9, 2, and 6. The LCM is the smallest number that all these denominators can divide into evenly. In this case, the LCM is 18. Think of it as finding a common ground for all the fractions.

Now, we'll multiply both sides of the equation by 18. This might seem like a big step, but it's a game-changer. It clears out those pesky fractions and makes the equation much easier to handle. Here’s how it looks:

18 * [(x-1)/3 - (x+3)/9] = 18 * [(x-5)/2 + 1/6]

Distribute the 18 to each term:

(18 * (x-1))/3 - (18 * (x+3))/9 = (18 * (x-5))/2 + (18 * 1)/6

Now, simplify each term:

6(x-1) - 2(x+3) = 9(x-5) + 3

See? Much cleaner already! We've successfully eliminated the fractions, and the equation looks way less intimidating.

2. Distribute and Simplify

Next up, we'll distribute the numbers outside the parentheses to the terms inside. This means multiplying the 6 with (x-1), the -2 with (x+3), and the 9 with (x-5). It's like unwrapping a gift, we're revealing the terms inside the parentheses.

Here’s how the distribution looks:

6x - 6 - 2x - 6 = 9x - 45 + 3

Now, we'll combine the like terms on each side of the equation. Like terms are those that have the same variable (in this case, 'x') or are constants (just numbers). Think of it as sorting your socks – pairing up the matching ones.

Combine the 'x' terms and the constants:

(6x - 2x) + (-6 - 6) = 9x + (-45 + 3)

Simplify:

4x - 12 = 9x - 42

Great! We've simplified both sides of the equation. Now, we're getting closer to isolating 'x'.

3. Isolate the Variable

Our goal now is to get all the 'x' terms on one side of the equation and all the constants on the other side. To do this, we'll perform some algebraic maneuvers. Imagine we're herding sheep, getting all the 'x' sheep into one pen and the number sheep into another.

Let's start by subtracting 4x from both sides. This will move the 'x' term from the left side to the right side:

4x - 12 - 4x = 9x - 42 - 4x

Simplify:

-12 = 5x - 42

Next, we'll add 42 to both sides to move the constant term from the right side to the left side:

-12 + 42 = 5x - 42 + 42

Simplify:

30 = 5x

We're almost there! Now, we have 'x' on one side, but it's being multiplied by 5. To isolate 'x' completely, we need to do one more step.

4. Solve for x

To get 'x' by itself, we'll divide both sides of the equation by 5. This is like splitting the loot equally among five pirates.

30 / 5 = (5x) / 5

Simplify:

6 = x

Or, to put it more clearly:

x = 6

And there you have it! We've solved the equation. The value of 'x' that makes the equation true is 6. We've cracked the code! But, just to be sure, let's verify our solution.

Verifying the Solution

It's always a good idea to check your answer, especially in math. To verify our solution, we'll substitute x = 6 back into the original equation and see if both sides are equal. Think of it as double-checking your work before submitting it.

Original equation:

(x-1)/3 - (x+3)/9 = (x-5)/2 + 1/6

Substitute x = 6:

(6-1)/3 - (6+3)/9 = (6-5)/2 + 1/6

Simplify each term:

5/3 - 9/9 = 1/2 + 1/6

5/3 - 1 = 1/2 + 1/6

Convert the fractions to have a common denominator (6):

(52)/(32) - (16)/6 = (13)/(2*3) + 1/6

10/6 - 6/6 = 3/6 + 1/6

4/6 = 4/6

Both sides are equal! This confirms that our solution, x = 6, is correct. We nailed it! High five!

Conclusion

So, guys, we've successfully solved the equation (x-1)/3 - (x+3)/9 = (x-5)/2 + 1/6. The value of x that satisfies the equation is 6. We did it by eliminating fractions, distributing and simplifying terms, isolating the variable, and finally, solving for x. And, we didn’t forget to verify our solution to make sure we were spot on.

Remember, solving equations is like building a puzzle – each step fits together to reveal the final answer. The key is to take it one step at a time, stay organized, and double-check your work. You've got this! Keep practicing, and you'll become a math whiz in no time. Keep up the great work, and happy solving!