Solving Equations Finding The Value Of X

Hey math enthusiasts! Ever feel like you're decoding a secret message when you're solving equations? Well, today, we're going to crack the code of a particularly interesting equation together. We'll break down each step, making sure you not only get the answer but also understand the why behind it. Let's dive in!

The Challenge: Deciphering the Equation

Our mission, should we choose to accept it (and we do!), is to find the value of x that makes the following equation true:

$$\frac{3 \pi-6}{3}=\frac{7 \pi-9}{6}$$

This might look a bit intimidating at first glance, with its fractions and that mysterious π symbol. But don't worry, guys! We'll tackle it piece by piece. The options presented to us are:

A. -11 B. -9 C. -3 D. 3

So, one of these numbers is the key that unlocks the equation. Let's find out which one it is!

Step 1: Simplifying the Fractions – Our First Move

The golden rule of equation solving? Make things simpler! Our first target is those fractions. To get rid of them, we'll use a clever trick: multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In our case, the denominators are 3 and 6. The LCM of 3 and 6 is 6. So, we'll multiply both sides of the equation by 6. This ensures we maintain the balance of the equation while eliminating the fractions.

So, what happens when we multiply both sides by 6? On the left side, we have 6 * ( (3π - 6) / 3 ). The 6 and the 3 can simplify, leaving us with 2 * (3π - 6). On the right side, we have 6 * ( (7π - 9) / 6 ). Here, the 6s cancel each other out completely, leaving us with just (7π - 9). See? Much cleaner already!

Remember, the key here is maintaining balance. Whatever we do to one side of the equation, we must do to the other side. This ensures that the equation remains true.

Step 2: Distribute and Conquer – Unveiling the Terms

Now that we've cleared the fractional hurdle, let's distribute the 2 on the left side of the equation. This means multiplying the 2 by each term inside the parentheses: 2 * (3π) and 2 * (-6). This gives us 6π - 12. On the right side, we still have our friend (7π - 9), patiently waiting.

So, our equation now looks like this:

6π - 12 = 7π - 9

We're getting closer! Now, we need to gather like terms – the π terms on one side and the constant terms on the other. This is like sorting socks after laundry day – putting similar items together.

Step 3: Gathering Like Terms – The Art of Sorting

To get all the π terms on one side, let's subtract 6π from both sides of the equation. This will eliminate the 6π term on the left side and leave us with just the constant term, -12. On the right side, subtracting 6π from 7π gives us π.

So, our equation transforms into:

-12 = π - 9

Now, let's move the constant terms to the other side. To do this, we'll add 9 to both sides of the equation. Adding 9 to -12 gives us -3. On the right side, -9 and +9 cancel each other out, leaving us with just π.

This simplifies our equation to:

-3 = π

Wait a minute… This doesn't look right, does it? We've made a mistake somewhere in copying the question. This is a great reminder to always double-check your work and the original problem!

Let's assume the equation was intended to be:

$$\frac{3x-6}{3}=\frac{7x-9}{6}$$

Let’s redo the steps with this corrected equation.

Step 1 (Corrected Equation): Simplifying the Fractions

Multiplying both sides by the LCM, which is still 6, we get:

6 * ( (3x - 6) / 3 ) = 6 * ( (7x - 9) / 6 )

Simplifying, we have:

2 * (3x - 6) = 7x - 9

Step 2 (Corrected Equation): Distribute and Conquer

Distributing the 2 on the left side gives us:

6x - 12 = 7x - 9

Step 3 (Corrected Equation): Gathering Like Terms

Subtract 6x from both sides:

-12 = x - 9

Add 9 to both sides:

-3 = x

Step 4: The Solution – We Found It!

And there we have it! Our calculations reveal that x = -3. This matches option C in our list. We've successfully decoded the equation and found the correct value for x.

Therefore, the correct answer is C. -3

Key Takeaways – Wisdom for Future Equations

1. Simplify First: Always look for ways to simplify the equation before diving into complex calculations. This often involves clearing fractions or distributing terms.
2. Maintain Balance: Remember the golden rule – whatever you do to one side of the equation, do to the other.
3. Gather Like Terms: Group similar terms together to make the equation easier to solve.
4. Double-Check: Always double-check your work, including the original problem statement, to avoid errors.

Mastering Equations – A Journey, Not a Sprint

Solving equations is like learning a new language – it takes practice and patience. Don't get discouraged if you don't get it right away. The more you practice, the more fluent you'll become in the language of mathematics.

So, keep practicing, keep exploring, and keep unlocking those mathematical mysteries! You've got this, guys!