Unlocking The Equation: A Step-by-Step Guide To Solve For X

Hey Plastik Magazine readers! Ever stumbled upon an equation that looks like it's speaking a different language? Don't sweat it! Today, we're diving deep into the world of exponents and equations to solve for x in the equation: $\frac{3^{1-x}}{3^{x+2}}=9^{2-3 x}$. It might seem intimidating at first glance, but trust me, we'll break it down into bite-sized pieces, making it super easy to understand. Think of it like this: We are going to go on a math adventure, and by the end, you'll be the hero who tames this equation. Ready to jump in? Let's go!

Understanding the Basics: Exponents and Their Rules

Before we can solve for x, let's refresh our memory on some essential exponent rules. These rules are the secret weapons that will help us simplify the equation and get closer to finding the value of 'x'. Here are the key rules we'll be using:

• Rule 1: Division with the same base. When dividing exponents with the same base, you subtract the exponents. This is the cornerstone of simplifying the left side of our equation. Mathematically, it's expressed as: $a^m / a^n = a^{m-n}$.
• Rule 2: Power of a power. When raising a power to another power, you multiply the exponents. This will be super helpful when we deal with the $9^{2-3x}$ part of the equation. This rule is defined as: $(a^m)^n = a^{m*n}$.
• Rule 3: Expressing numbers with the same base. We'll want to get all the terms in our equation to have the same base. It will make the comparison and simplification process much easier. This is where we will recognize that 9 is actually $3^2$.

Alright, guys, these rules are the foundation we need. Don't worry if they seem a little abstract right now; we'll see them in action very soon. The best way to understand these rules is to practice, so keep them in mind as we work through the problem. With these rules in our math toolbox, we can simplify the equation and isolate 'x'. Think of it as preparing your ingredients before you start cooking—essential for a delicious result!

Step-by-Step Solution: Taming the Equation

Now, let's get our hands dirty and start solving the equation $\frac{3^{1-x}}{3^{x+2}}=9^{2-3 x}$. Here's our game plan:

Step 1: Simplify the Left Side.

Using Rule 1 (division with the same base), we can simplify the left side. Remember, when dividing exponents with the same base, we subtract the exponents. So, we have:

$\frac{3^{1-x}}{3^{x+2}} = 3^{(1-x)-(x+2)}$

Now, let's simplify the exponent:

$3^{(1-x)-(x+2)} = 3^{1-x-x-2} = 3^{-2x-1}$

So, our equation now looks like this: $3^{-2x-1} = 9^{2-3x}$. We've already made a big step by simplifying the left side.

Step 2: Express Both Sides with the Same Base.

To make our comparison easier, let's express both sides of the equation with the same base. Since 9 is $3^2$, we can rewrite the right side:

$9^{2-3x} = (3^2)^{2-3x}$

Now, using Rule 2 (power of a power), we multiply the exponents:

$(3^2)^{2-3x} = 3^{2*(2-3x)} = 3^{4-6x}$

Our equation now becomes: $3^{-2x-1} = 3^{4-6x}$. Things are starting to look really nice, right?

Step 3: Equate the Exponents.

When the bases are the same, for the equation to hold true, the exponents must be equal. This is the crucial step where we get rid of the exponents and solve for x. So, we set the exponents equal to each other:

$-2x - 1 = 4 - 6x$

Step 4: Solve for x.

Now we're left with a simple linear equation. Let's solve for x:

Add $6x$ to both sides:

$-2x + 6x - 1 = 4$

Simplify:

$4x - 1 = 4$

Add 1 to both sides:

$4x = 5$

Divide by 4:

$x = \frac{5}{4}$

And there you have it! We've found the value of x.

Verification: Ensuring Our Answer Is Correct

We've crunched the numbers and found our solution: $x = \frac{5}{4}$. But it's always smart to double-check our work. Let's plug this value back into the original equation to ensure it holds true. This is like a final test drive to make sure everything is working perfectly. Are you ready to verify?

Our original equation: $\frac{3^{1-x}}{3^{x+2}} = 9^{2-3x}$

Let's substitute $x = \frac{5}{4}$:

$\frac{3^{1-\frac{5}{4}}}{3^{\frac{5}{4}+2}} = 9^{2-3(\frac{5}{4})}$

Simplify the exponents:

$\frac{3^{-\frac{1}{4}}}{3^{\frac{13}{4}}} = 9^{2-\frac{15}{4}}$

$\frac{3^{-\frac{1}{4}}}{3^{\frac{13}{4}}} = 9^{-\frac{7}{4}}$

Simplify the left side using the rule of division with the same base:

$3^{-\frac{1}{4} - \frac{13}{4}} = 9^{-\frac{7}{4}}$

$3^{-\frac{14}{4}} = 9^{-\frac{7}{4}}$

Simplify the left side by reducing the fraction:

$3^{-\frac{7}{2}} = 9^{-\frac{7}{4}}$

Express 9 as $3^2$:

$3^{-\frac{7}{2}} = (3^2)^{-\frac{7}{4}}$

Simplify the right side:

$3^{-\frac{7}{2}} = 3^{-\frac{14}{4}}$

Simplify the right side by reducing the fraction:

$3^{-\frac{7}{2}} = 3^{-\frac{7}{2}}$

Both sides of the equation are equal! This confirms that our solution $x = \frac{5}{4}$ is correct. High five, guys! We did it. We took a complicated-looking equation and turned it into a solved problem. Verification is a crucial step; it is our safety net, making sure that the solution we found is the right one.

Key Takeaways: Mastering Exponential Equations

Awesome work, everyone! We've successfully solved for x in the equation $\frac{3^{1-x}}{3^{x+2}}=9^{2-3 x}$. But more importantly, we've learned some valuable lessons that we can apply to any exponential equation we encounter. Here's a quick recap of the key takeaways:

• Understand Exponent Rules: The foundation of solving exponential equations lies in understanding the rules of exponents. Mastering these rules is like having the keys to unlock a treasure chest.
• Simplify, Simplify, Simplify: Always aim to simplify the equation. This involves combining like terms, reducing fractions, and using exponent rules to make the equation easier to handle. Simpler equations are easier to solve.
• Get the Same Base: Strive to express both sides of the equation with the same base. This allows you to equate the exponents and solve for 'x'. It's the secret weapon in your arsenal.
• Check Your Work: Always verify your solution by plugging it back into the original equation. This crucial step ensures that your answer is correct and provides a sense of accomplishment. It's like a victory lap.

By following these steps and practicing consistently, you'll become a pro at solving exponential equations. Keep practicing, and don't be afraid to challenge yourself with more complex problems. You got this, guys! Remember, the more you practice, the easier it will become. And always remember to have fun with math—it's a journey of discovery and a celebration of problem-solving. Until next time, keep those equations in check!