Solving For X: (2/3)(x + 5/4) = 25/6 - Find The Value!

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Hey guys! Today, we're diving into a fun math problem where we need to find the value of x that makes the equation (2/3)(x + 5/4) = 25/6 true. Don't worry, it's not as scary as it looks! We'll break it down step by step so you can easily follow along. So grab your pencils, and let's get started!

Understanding the Equation

Before we jump into solving, let's take a closer look at the equation we're dealing with:

(2/3)(x + 5/4) = 25/6

This equation involves fractions, parentheses, and our mystery variable, x. Our goal is to isolate x on one side of the equation so we can figure out its value. To do this, we'll use some basic algebraic principles. Think of it like peeling an onion – we'll carefully remove each layer until we get to the core, which is x.

The first thing we need to do is get rid of those parentheses. Remember the distributive property? It's our best friend here! The distributive property states that a(b + c) = ab + ac. This means we need to multiply the (2/3) outside the parentheses by both terms inside the parentheses.

So, let's apply this property to our equation:

(2/3) * x + (2/3) * (5/4) = 25/6

Now we have:

(2/3)x + 10/12 = 25/6

Notice that we multiplied (2/3) by x to get (2/3)x, and then we multiplied (2/3) by (5/4). To multiply fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers): (2 * 5) / (3 * 4) = 10/12.

Simplifying the Fractions

Before we move on, let's simplify the fraction 10/12. Both 10 and 12 are divisible by 2, so we can reduce the fraction:

10/12 = (10 ÷ 2) / (12 ÷ 2) = 5/6

Now our equation looks a bit cleaner:

(2/3)x + 5/6 = 25/6

See? Math isn't so bad when you break it down into smaller, manageable steps. Now that we've simplified the equation, we're one step closer to finding the value of x.

Isolating x: Step-by-Step

Okay, now that we've simplified our equation to (2/3)x + 5/6 = 25/6, it’s time to isolate x. Remember, isolating x means getting it all by itself on one side of the equation. To do this, we need to undo the operations that are being performed on x.

The first operation we need to tackle is the addition of 5/6. To undo addition, we use subtraction. We're going to subtract 5/6 from both sides of the equation. Why both sides? Because whatever we do to one side of an equation, we must do to the other to keep it balanced. Think of it like a seesaw – if you take weight off one side, you need to take the same weight off the other to keep it level.

So, let's subtract 5/6 from both sides:

(2/3)x + 5/6 - 5/6 = 25/6 - 5/6

On the left side, 5/6 - 5/6 cancels out, leaving us with just (2/3)x. On the right side, we subtract the fractions. Since they have the same denominator (6), we can simply subtract the numerators:

25/6 - 5/6 = (25 - 5) / 6 = 20/6

Now our equation looks like this:

(2/3)x = 20/6

We're getting closer! Now we have x multiplied by a fraction (2/3). To undo multiplication, we use division. But here's a little trick: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 2/3 is 3/2 (we just flipped the numerator and denominator).

So, let's multiply both sides of the equation by 3/2:

(3/2) * (2/3)x = (3/2) * (20/6)

On the left side, (3/2) * (2/3) cancels out, leaving us with just x. On the right side, we multiply the fractions:

(3/2) * (20/6) = (3 * 20) / (2 * 6) = 60/12

Now we have:

x = 60/12

Final Simplification

We're almost there! The last step is to simplify the fraction 60/12. Both 60 and 12 are divisible by 12, so let's divide them:

60/12 = (60 ÷ 12) / (12 ÷ 12) = 5/1 = 5

So, we've finally found the value of x:

x = 5

Verifying the Solution

Before we celebrate, it's always a good idea to check our answer. We can do this by plugging the value we found for x (which is 5) back into the original equation and seeing if it holds true.

Our original equation was:

(2/3)(x + 5/4) = 25/6

Let's substitute x with 5:

(2/3)(5 + 5/4) = 25/6

First, we need to deal with the parentheses. To add 5 and 5/4, we need a common denominator. We can rewrite 5 as 20/4 (since 5 * 4 = 20, and 20/4 = 5):

(2/3)(20/4 + 5/4) = 25/6

Now we can add the fractions inside the parentheses:

(2/3)(25/4) = 25/6

Next, we multiply (2/3) by (25/4):

(2 * 25) / (3 * 4) = 50/12

So we have:

50/12 = 25/6

Wait a minute! 50/12 doesn't look like 25/6. But remember, we can simplify fractions. Both 50 and 12 are divisible by 2:

50/12 = (50 ÷ 2) / (12 ÷ 2) = 25/6

Now we have:

25/6 = 25/6

Yay! The equation holds true. This means our solution for x is correct.

Conclusion: x = 5

Alright, guys, we did it! We successfully solved the equation (2/3)(x + 5/4) = 25/6 and found that the value of x is 5. We tackled fractions, used the distributive property, isolated x, and even verified our answer. Math might seem tricky sometimes, but by breaking it down into steps and understanding the basic principles, you can conquer any equation that comes your way. Keep practicing, and you'll become a math whiz in no time!

Remember the key takeaways:

  • Distribute: Multiply the term outside the parentheses by each term inside.
  • Isolate: Use inverse operations (addition/subtraction, multiplication/division) to get x by itself.
  • Simplify: Reduce fractions to their simplest form.
  • Verify: Plug your solution back into the original equation to check your work.

Keep up the awesome work, and I'll catch you in the next math adventure!