Solve For X: 9(x-3)=18 - Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into the exciting world of algebra to tackle a classic equation: 9(x-3)=18. Our mission? To crack the code and discover the value of our elusive variable, x. Don't worry if algebra sometimes feels like navigating a maze – we'll break it down step-by-step, making sure everyone's on board for the journey. We will equip you with the skills to solve similar problems with confidence. So, grab your thinking caps, and let's get started!
Decoding the Equation: A Step-by-Step Guide
Before we jump into the solution, let's take a moment to understand what this equation is telling us. We have 9 multiplied by the quantity (x-3), and this entire expression equals 18. Our goal is to isolate x on one side of the equation, effectively unraveling the operations that are currently obscuring its true value. Think of it like peeling back the layers of an onion, one step at a time, until we reach the core – our solution.
Step 1: The Distributive Property – Our First Key
The first hurdle we encounter is the parentheses surrounding (x-3). To overcome this, we'll employ a powerful tool called the distributive property. This property states that for any numbers a, b, and c:
a(b + c) = ab + ac
In simpler terms, we multiply the term outside the parentheses (in our case, 9) by each term inside the parentheses. Applying this to our equation, we get:
9 * x - 9 * 3 = 18
This simplifies to:
9x - 27 = 18
See how we've effectively removed the parentheses? We're one step closer to isolating x!
Step 2: Isolating the x Term – Moving the Numbers Around
Now that we have 9x - 27 = 18, our next task is to isolate the term containing x (which is 9x). To do this, we need to get rid of the -27 on the left side of the equation. How do we do that? By performing the inverse operation! Since we're subtracting 27, we'll add 27 to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance.
9x - 27 + 27 = 18 + 27
This simplifies to:
9x = 45
Excellent! We've successfully isolated the 9x term. x is almost within our grasp.
Step 3: Unveiling x – The Final Division
We're in the home stretch! We now have 9x = 45. To finally reveal the value of x, we need to undo the multiplication by 9. Again, we'll use the inverse operation – this time, division. We'll divide both sides of the equation by 9:
9x / 9 = 45 / 9
This simplifies to:
x = 5
Eureka! We've found it! The value of x that satisfies the equation 9(x-3)=18 is 5.
Confirming Our Victory: The Verification Step
But wait, how can we be absolutely sure our answer is correct? The beauty of algebra is that we can always verify our solution by plugging it back into the original equation. Let's substitute x = 5 into 9(x-3)=18:
9(5 - 3) = 18
9(2) = 18
18 = 18
It checks out! Both sides of the equation are equal, confirming that our solution, x = 5, is indeed correct. This verification step is a crucial habit to develop, as it provides peace of mind and helps catch any potential errors.
Mastering the Art of Equation Solving: Key Takeaways
We've successfully navigated the equation 9(x-3)=18 and emerged victorious. But the true value lies not just in solving this specific problem, but in the understanding of the underlying principles and techniques. Let's recap the key takeaways from our journey:
- The Distributive Property: This is a fundamental tool for simplifying expressions containing parentheses. Remember to multiply the term outside the parentheses by each term inside.
- Inverse Operations: To isolate a variable, we use inverse operations. Addition undoes subtraction, multiplication undoes division, and vice versa. The key is to perform the same operation on both sides of the equation to maintain balance.
- Verification: Always check your solution by substituting it back into the original equation. This ensures accuracy and builds confidence.
- Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This makes the process less daunting and reduces the chance of errors.
Level Up Your Algebra Skills: Practice Makes Perfect
Now that you've grasped the core concepts, the best way to solidify your understanding is through practice. Seek out similar equations and apply the techniques we've discussed. The more you practice, the more comfortable and confident you'll become in your ability to solve algebraic equations. Remember, even the most seasoned mathematicians were once beginners. It's all about consistent effort and a willingness to learn.
Beyond the Basics: Exploring More Complex Equations
The equation 9(x-3)=18 is a great starting point, but the world of algebra is vast and filled with more challenging puzzles. As you progress, you'll encounter equations with multiple variables, fractions, exponents, and more. But don't be intimidated! The fundamental principles we've learned here – the distributive property, inverse operations, and the importance of verification – will serve as your trusty compass as you navigate these complexities.
Consider equations like:
- 2x + 3y = 10
- x^2 - 4x + 3 = 0
- (x + 1) / (x - 2) = 5
Each of these presents its own unique challenges, but the core problem-solving strategies remain the same. Embrace the challenge, and remember that every equation you solve is a step forward on your mathematical journey.
Conclusion: The Power of Algebra
We've successfully solved for x in the equation 9(x-3)=18, and more importantly, we've gained valuable insights into the world of algebra. Algebra is more than just manipulating symbols and numbers; it's a powerful tool for problem-solving, critical thinking, and logical reasoning. The skills you develop in algebra will benefit you in countless areas of life, from everyday decision-making to advanced scientific pursuits.
So, keep exploring, keep practicing, and keep pushing your mathematical boundaries. The world of algebra awaits, filled with fascinating challenges and rewarding discoveries. You've got this!
