Hey guys! Let's dive into solving a classic algebraic equation. We've got , and our mission, should we choose to accept it, is to find the value of x that makes this equation true. Don't worry, it's not as daunting as it looks! We'll break it down into simple, manageable steps. So, grab your pencils, and let's get started!
Step 1: Gather the x Terms
The golden rule of solving equations is to isolate the variable – in this case, x. To do that, we need to get all the terms containing x on one side of the equation. Currently, we have on the left and on the right. To consolidate these, we can add to both sides of the equation. This maintains the balance of the equation, which is super important. Think of it like a seesaw; whatever you do on one side, you have to do on the other to keep it level.
So, let's perform this operation:
Now, we simplify each side. On the left, combines to give us . On the right, cancels each other out, leaving us with just 17. Our equation now looks like this:
See? We're already making progress! We've managed to get all the x terms on one side. This is a crucial step because it simplifies the equation, making it easier to handle. Now we can move on to the next phase of our x-hunt. Remember, the key is to take it one step at a time, and each step brings us closer to the solution. By adding to both sides, we've effectively neutralized the x term on the right side and brought it over to the left, making our equation much cleaner and more manageable. This is a fundamental technique in algebra, and mastering it will help you tackle all sorts of equations with confidence.
Step 2: Isolate the x Term
Now that we have , our next goal is to isolate the term with x – that's the part. To do this, we need to get rid of the +2 that's hanging out on the left side. The way we do that is by performing the inverse operation. Since we're adding 2, we'll subtract 2 from both sides of the equation. Remember the seesaw analogy? What we do on one side, we must do on the other to keep things balanced.
So, let's subtract 2 from both sides:
On the left side, and cancel each other out, leaving us with just . On the right side, gives us 15. Our equation now looks like this:
Awesome! We're getting closer. We've successfully isolated the x term. Now, it's just a matter of getting x all by itself. This step is super important because it strips away the extra baggage around x, allowing us to see its true value. Think of it like clearing away the clutter on your desk so you can focus on the important stuff. By subtracting 2 from both sides, we've created a much simpler equation that's easier to solve. This technique of using inverse operations is a cornerstone of algebra, and you'll use it time and time again. So, let's keep going and get that x all by its lonesome!
Step 3: Solve for x
We've reached the final stretch! We're at , and our ultimate goal is to find out what x equals. Right now, x is being multiplied by 5. To undo this multiplication and get x by itself, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 5. You guessed it – we're still keeping that equation balanced, just like our trusty seesaw!
Let's divide both sides by 5:
rac{5x}{5} = rac{15}{5}
On the left side, the 5s cancel each other out, leaving us with just x. On the right side, 15 divided by 5 is 3. So, our equation simplifies to:
And there you have it! We've solved for x! The value of x that makes the original equation true is 3. x is equal to 3. This is the moment of triumph when all our hard work pays off. We've successfully navigated the algebraic maze and found our treasure. By dividing both sides by 5, we've effectively unwrapped x from its numerical cocoon and revealed its true identity. This step is the culmination of all our previous efforts, and it's a testament to the power of algebraic manipulation. So, let's bask in the glory of our solution and celebrate our algebraic prowess!
Step 4: Verify the Solution (Optional but Recommended)
To be absolutely sure we've nailed it, it's always a good idea to verify our solution. This is like double-checking your answer on a test – it gives you that extra peace of mind. To verify, we'll plug our solution, , back into the original equation and see if it holds true.
Our original equation was:
Now, let's substitute x with 3:
Let's simplify each side. On the left, is , so we have:
is , so the left side is .
On the right, is , so we have:
is , so the right side is also .
We end up with:
This is a true statement! Both sides of the equation are equal, which means our solution, , is correct. High five! Verifying the solution is like the final flourish on a masterpiece. It confirms that our efforts were successful and that we've arrived at the correct answer. By plugging our solution back into the original equation, we've put it to the ultimate test and it has passed with flying colors. This step not only gives us confidence in our answer but also reinforces our understanding of the equation and how the solution works. So, always remember to verify your solutions – it's the mark of a true algebraic champion!
Conclusion
We did it, guys! We successfully solved the equation and found that . We tackled this problem step by step, from gathering the x terms to isolating x and finally verifying our solution. This journey through the algebraic landscape has shown us the power of breaking down complex problems into simpler steps. Remember, solving equations is like building a puzzle – each step is a piece that fits together to reveal the final picture. And in this case, the picture is the value of x. So, keep practicing, keep exploring, and keep solving! You've got the skills, and you've got the determination. Now go out there and conquer those equations!