Solving For X: 1/6(x + 3/4) = -13/24 Explained

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Solving for x in the Equation 1/6(x + 3/4) = -13/24

Hey guys! Let's dive into solving this equation step-by-step. We've got the equation 1/6(x + 3/4) = -13/24, and our mission is to find the value of x that makes this equation true. Trust me, it's not as scary as it looks! We'll break it down, and you'll be a pro in no time. So, grab your pencils, and let's get started!

Understanding the Equation

First off, let’s make sure we understand what the equation is telling us. We have a fraction, 1/6, multiplied by the sum of x and another fraction, 3/4. The result of this multiplication should equal -13/24. Our main goal here is to isolate x on one side of the equation. To do that, we'll need to undo all the operations that are affecting x. This involves using inverse operations, like subtracting if we see addition, or dividing if we see multiplication. Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced. Think of it like a scale – we need to keep both sides weighing the same. This principle is fundamental in algebra and will serve you well in solving more complex equations later on.

Step 1: Distribute the 1/6

The first thing we need to do is get rid of those parentheses. We can do this by distributing the 1/6 across the terms inside the parentheses. This means we multiply 1/6 by both x and 3/4. When we multiply 1/6 by x, we simply get (1/6)*x, which can also be written as x/6. Next, we multiply 1/6 by 3/4. To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (1/6) * (3/4) = (1 * 3) / (6 * 4) = 3/24. Now, we can simplify 3/24 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, 3/24 simplifies to 1/8. After distributing, our equation looks like this: x/6 + 1/8 = -13/24. See, we're already making progress!

Step 2: Eliminate the Fractions

Fractions can sometimes make things look more complicated than they are. So, let's get rid of them! To do this, we'll find the least common multiple (LCM) of the denominators: 6, 8, and 24. The LCM is the smallest number that all the denominators can divide into evenly. Let's list the multiples of each number: Multiples of 6: 6, 12, 18, 24, 30, ... Multiples of 8: 8, 16, 24, 32, ... Multiples of 24: 24, 48, ... We can see that the LCM of 6, 8, and 24 is 24. Now, we'll multiply every term in the equation by 24. This includes the x/6 term, the 1/8 term, and the -13/24 term. Multiplying x/6 by 24 gives us (24 * x)/6 = 4x. Multiplying 1/8 by 24 gives us 24/8 = 3. Multiplying -13/24 by 24 gives us -13. So, after multiplying every term by 24, our equation becomes: 4x + 3 = -13. Wow, much cleaner, right?

Step 3: Isolate the Term with x

Now, we want to isolate the term with x, which is 4x. To do this, we need to get rid of the +3 on the left side of the equation. We can do this by subtracting 3 from both sides of the equation. Remember, we need to keep the equation balanced! Subtracting 3 from both sides gives us: 4x + 3 - 3 = -13 - 3. This simplifies to 4x = -16. We're getting so close!

Step 4: Solve for x

Finally, we need to solve for x. We have 4x = -16. To isolate x, we need to undo the multiplication. We can do this by dividing both sides of the equation by 4. So, (4x)/4 = -16/4. This simplifies to x = -4. And there we have it! We've found the value of x.

The Solution

So, the value of x that satisfies the equation 1/6(x + 3/4) = -13/24 is x = -4. Awesome job, guys! You've successfully solved for x. To make sure we've got the right answer, we can plug x = -4 back into the original equation and see if it holds true. This is always a good practice to verify your solution. Let's do that now.

Verifying the Solution

To verify our solution, we'll substitute x = -4 into the original equation: 1/6(x + 3/4) = -13/24. Replace x with -4: 1/6(-4 + 3/4) = -13/24. Now, we need to simplify the expression inside the parentheses first. To add -4 and 3/4, we need to convert -4 into a fraction with a denominator of 4. We can write -4 as -16/4. So, -4 + 3/4 = -16/4 + 3/4 = -13/4. Now our equation looks like this: 1/6(-13/4) = -13/24. Next, we multiply 1/6 by -13/4. To multiply fractions, we multiply the numerators and the denominators: (1 * -13) / (6 * 4) = -13/24. So, we have -13/24 = -13/24. This is true! Our solution is correct.

Key Takeaways

Let's recap the key steps we took to solve this equation:

  1. Distribute: We distributed the 1/6 across the terms inside the parentheses.
  2. Eliminate Fractions: We found the least common multiple (LCM) of the denominators and multiplied every term in the equation by it.
  3. Isolate the Term with x: We used inverse operations to get the term with x by itself on one side of the equation.
  4. Solve for x: We divided both sides of the equation by the coefficient of x to find the value of x.
  5. Verify the Solution: We plugged our solution back into the original equation to make sure it was correct.

These steps are a solid foundation for solving all sorts of algebraic equations. Remember, practice makes perfect! The more you work through problems like this, the more confident you'll become. Keep at it, and you'll be solving equations like a pro in no time.

Conclusion

We successfully solved the equation 1/6(x + 3/4) = -13/24 and found that x = -4. We also verified our solution by plugging it back into the original equation. You guys did an amazing job following along! Keep practicing these steps, and you'll be well on your way to mastering algebraic equations. Remember, math is like a puzzle – it might seem tricky at first, but with a little effort and the right tools, you can solve anything. So keep up the great work, and never stop learning!