Unraveling The Equation: Finding The Value Of X

Hey guys! Let's dive into a cool math problem where we need to find the value of x. The equation looks a bit intimidating at first glance, but trust me, we can totally break it down step by step. We're going to solve for x in this equation:

$\frac{3x+14}{4} -\frac{5x+12}{5} -\frac{x+38}{10} =\frac{3}{4} -\frac{2x+13}{5} -\frac{11}{20}$

It might seem like a lot to handle, but don't sweat it. We'll use some basic algebraic techniques to simplify things and get to our answer. Remember, the goal is to isolate x on one side of the equation. Ready? Let's get started!

Step-by-Step Solution: Unveiling the Mystery of x

Alright, first things first. To solve this equation, our main aim is to get rid of those pesky fractions. The easiest way to do that is to find the least common multiple (LCM) of the denominators (4, 5, and 10, and 20). The LCM of 4, 5, 10, and 20 is 20. This means if we multiply every single term in the equation by 20, we'll clear out those fractions like magic. So, let’s do it. Multiplying each term by 20 gives us:

$20 * \frac{3x+14}{4} - 20 * \frac{5x+12}{5} - 20 * \frac{x+38}{10} = 20 * \frac{3}{4} - 20 * \frac{2x+13}{5} - 20 * \frac{11}{20}$

This looks a bit messy still, but we can simplify by multiplying. Let's start simplifying each term:

• $20 * \frac{3x+14}{4} = 5(3x+14)$
• $20 * \frac{5x+12}{5} = 4(5x+12)$
• $20 * \frac{x+38}{10} = 2(x+38)$
• $20 * \frac{3}{4} = 15$
• $20 * \frac{2x+13}{5} = 4(2x+13)$
• $20 * \frac{11}{20} = 11$

Substituting these values back into our equation, it becomes:

$5(3x+14) - 4(5x+12) - 2(x+38) = 15 - 4(2x+13) - 11$

Now, we are getting somewhere! The fractions are gone, and we can start expanding the brackets. Remember to carefully multiply each term inside the brackets by the number outside. Let's do the expansion:

Expanding and Simplifying: The Heart of the Matter

Expanding the brackets gives us:

$15x + 70 - 20x - 48 - 2x - 76 = 15 - 8x - 52 - 11$

Now, let's group the like terms together on both sides of the equation. Combine the x terms and the constant terms separately. This will help to make the equation a lot easier to manage. Combining like terms:

On the left side: $15x - 20x - 2x = -7x$ and $70 - 48 - 76 = -54$

On the right side: $-8x$ and $15 - 52 - 11 = -48$

So, our equation simplifies to:

$-7x - 54 = -8x - 48$

Next, we need to get all the x terms on one side of the equation and the constants on the other side. This is where it starts to get even easier. It's like sorting things out and making everything neat and tidy. We can add $8x$ to both sides and add $54$ to both sides to isolate the x terms:

Adding $8x$ to both sides: $-7x + 8x - 54 = -8x + 8x - 48$ which simplifies to $x - 54 = -48$

Adding $54$ to both sides: $x - 54 + 54 = -48 + 54$

Which simplifies to $x = 6$. Woohoo! We've found the value of x. The value of x that satisfies this equation is 6. By following these steps, we've successfully solved for x and understood the process. Wasn't that fun?

Checking Your Work: Ensuring Accuracy

It’s always a good practice to check if our answer is correct. We can do this by substituting the value of x back into the original equation and seeing if both sides are equal. This way, we can be sure that our solution is correct. Let's substitute $x = 6$ into the original equation:

$\frac{3(6)+14}{4} -\frac{5(6)+12}{5} -\frac{(6)+38}{10} =\frac{3}{4} -\frac{2(6)+13}{5} -\frac{11}{20}$

Let's simplify each fraction individually:

• $\frac{3(6)+14}{4} = \frac{18+14}{4} = \frac{32}{4} = 8$
• $\frac{5(6)+12}{5} = \frac{30+12}{5} = \frac{42}{5}$
• $\frac{(6)+38}{10} = \frac{44}{10} = \frac{22}{5}$
• $\frac{3}{4}$
• $\frac{2(6)+13}{5} = \frac{12+13}{5} = \frac{25}{5} = 5$
• $\frac{11}{20}$

So, substituting these values back in:

$8 - \frac{42}{5} - \frac{22}{5} = \frac{3}{4} - 5 - \frac{11}{20}$

Now, let's simplify each side:

• Left side: $8 - \frac{42}{5} - \frac{22}{5} = 8 - \frac{64}{5} = \frac{40}{5} - \frac{64}{5} = -\frac{24}{5}$
• Right side: $\frac{3}{4} - 5 - \frac{11}{20} = \frac{15}{20} - \frac{100}{20} - \frac{11}{20} = \frac{15 - 100 - 11}{20} = \frac{-96}{20} = -\frac{24}{5}$

Since both sides are equal ($-\frac{24}{5} = -\frac{24}{5}$), our solution is correct! This confirmation step is super important, as it helps prevent any calculation errors. It’s like double-checking your work to make sure you've got it right. Good job, guys!

Key Takeaways and Tips for Similar Problems

Alright, what did we learn today, and how can we apply this knowledge to other problems? Let's summarize the key takeaways:

1. Fraction Busters: The first step in solving equations with fractions is to eliminate those fractions by multiplying every term by the least common multiple (LCM) of the denominators. This simplifies the equation significantly. This is your first weapon of choice!
2. Expand and Conquer: Next, expand any brackets by multiplying the terms inside by the number outside. Careful with the signs here! That’s where the mistakes often happen.
3. Gather Your Troops: Combine like terms – the x terms with other x terms, and the constant numbers with other constant numbers. This makes the equation much cleaner and easier to manage. Think of it like organizing your desk to make sure everything has its place.
4. Isolate and Solve: Isolate the variable (x in this case) on one side of the equation. This involves adding or subtracting terms from both sides until you have x all by itself. This is the ultimate goal!
5. Always Check: Always check your solution by substituting the value back into the original equation. This is the best way to ensure that your answer is correct. This is like the final proof.

General Tips:

• Practice Makes Perfect: The more problems you solve, the easier it becomes. The more you solve the better your understanding.
• Stay Organized: Write each step clearly and neatly. This helps you avoid mistakes and makes it easier to review your work.
• Don’t Be Afraid to Ask for Help: If you get stuck, don’t hesitate to ask a friend, teacher, or use online resources for help. There are tons of resources out there, like videos, and forums, that can give you a hand.

By following these steps and tips, you'll be well-equipped to tackle similar equations with confidence. Remember, math is like a puzzle – and solving it is so satisfying. Keep practicing, keep learning, and you'll get better and better. You got this, guys! And congratulations on solving another math problem!