Solving 2(18x - 288) = 0: Find X & Verify Solution

by ADMIN 51 views

Hey guys! Today, we're diving into solving a linear equation, finding the value of x, verifying our solution, and even tackling the greatest common divisor (GCD). Buckle up, it's going to be a fun mathematical ride! Let's break down the equation 2(18x - 288) = 0 step by step. Our primary goal here is to isolate x and discover its true value. To do this effectively, we'll employ a series of algebraic manipulations, ensuring that each step preserves the equation's balance. Think of it like a delicate scale – whatever we do on one side, we must also do on the other. This principle of equality is the cornerstone of solving equations, allowing us to methodically strip away the layers surrounding x until it stands alone, revealing its numerical identity. Understanding this fundamental approach is crucial not only for this particular problem but also for tackling a wide array of algebraic challenges. So, let's embark on this mathematical journey together, unraveling the mystery of x and solidifying our understanding of equation solving.

Step-by-Step Solution

First, let's tackle the equation: 2(18x - 288) = 0. To kick things off, we need to simplify the equation by distributing the 2 across the terms inside the parenthesis. This means multiplying both 18x and -288 by 2. When we multiply 2 by 18x, we get 36x. And when we multiply 2 by -288, we get -576. So, the equation now looks like this: 36x - 576 = 0. This step is crucial as it clears the parenthesis, making the equation easier to work with. Distributive property, you are our friend! This transformation allows us to proceed with isolating x, our ultimate goal. Remember, the distributive property is a fundamental tool in algebra, allowing us to simplify expressions and equations by properly managing terms within parentheses. It's like unpacking a package – you need to deal with each item inside individually to understand the whole content. With the equation now in a more manageable form, we can move on to the next step in our quest to find the value of x. So, let's keep rolling and see what we can uncover!

Next up, we need to isolate the term with x on one side of the equation. Currently, we have 36x - 576 = 0. To get the x term by itself, we need to get rid of the -576. We can do this by adding 576 to both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. So, adding 576 to both sides, we get: 36x - 576 + 576 = 0 + 576. Simplifying this, the -576 and +576 on the left side cancel each other out, leaving us with 36x = 576. We're getting closer now! This step is a classic example of using inverse operations to isolate variables. Addition is the inverse operation of subtraction, and by adding 576 to both sides, we effectively neutralized the -576 on the left. The result is a much simpler equation that brings us one step closer to unveiling the value of x. Keep up the great work, guys! We're on the right track to solving this puzzle.

Now, we have 36x = 576. Our final step in isolating x is to get rid of the 36 that's multiplying it. To do this, we'll use the inverse operation of multiplication, which is division. We need to divide both sides of the equation by 36. So, we have: (36x) / 36 = 576 / 36. On the left side, the 36 in the numerator and the 36 in the denominator cancel each other out, leaving us with just x. On the right side, 576 divided by 36 equals 16. Therefore, we find that x = 16. Eureka! We've found the value of x. This final step of division is crucial in the process of solving for a variable. It undoes the multiplication that was previously applied to x, allowing us to reveal its true value. The beauty of algebra lies in this systematic approach of using inverse operations to isolate and solve for unknowns. We've successfully navigated the equation, and our reward is the solution: x equals 16. But our journey doesn't end here; we still need to verify this solution to ensure it holds true. So, let's move on to the verification step and confirm our mathematical triumph.

Verification

Okay, so we think x = 16, but let's make absolutely sure! To verify our solution, we're going to plug x = 16 back into the original equation: 2(18x - 288) = 0. This is like checking our work – we're going to see if our answer makes the equation true. So, wherever we see x in the original equation, we'll replace it with 16. This gives us: 2(18 * 16 - 288) = 0. Now, we need to follow the order of operations (PEMDAS/BODMAS) to simplify the expression inside the parenthesis first. This means doing the multiplication before the subtraction. Verification is a crucial step in problem-solving because it provides a safety net, ensuring that our solution is accurate. It's like a detective double-checking their evidence to make sure they've got the right suspect. By substituting our calculated value back into the original equation, we can confirm whether it satisfies the equation's conditions. This process not only validates our answer but also helps us catch any potential errors we might have made along the way. So, let's continue with our verification journey and see if x = 16 truly holds the key to this equation.

Let's continue simplifying. Inside the parentheses, we have 18 * 16, which equals 288. So now our equation looks like this: 2(288 - 288) = 0. Next, we perform the subtraction inside the parentheses: 288 - 288 = 0. Lookin' good so far! Our equation is now: 2(0) = 0. Finally, we multiply 2 by 0, which equals 0. So, we have 0 = 0. Victory! The equation holds true. This confirms that our solution, x = 16, is indeed correct. This step-by-step simplification within the verification process is essential. It allows us to meticulously track each operation and ensure that we're following the correct mathematical procedure. By breaking down the expression into smaller, manageable steps, we reduce the chance of making errors and gain a clearer understanding of how the solution satisfies the equation. The fact that we arrived at the true statement 0 = 0 is a testament to the accuracy of our calculations and the validity of our solution. So, we can confidently say that x = 16 is the answer. But wait, there's more! We still have one more task to tackle: finding the greatest common divisor (GCD).

Finding the Greatest Common Divisor (GCD)

Now, let's switch gears and find the greatest common divisor (GCD). What's that, you ask? The GCD of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In our original equation, 2(18x - 288) = 0, we have the numbers 18 and 288. We need to find the GCD of 18 and 288. There are a couple of ways to do this, but one common method is to list the factors of each number and find the largest one they have in common. Understanding the concept of GCD is crucial in various mathematical contexts, from simplifying fractions to solving number theory problems. It's a fundamental tool for identifying the common building blocks of numbers and finding the largest shared factor. This can be incredibly useful in real-world applications as well, such as dividing resources fairly or optimizing designs. So, let's dive into the process of finding the GCD of 18 and 288, exploring the factors of each number and uncovering their greatest common divisor. It's like a mathematical treasure hunt, where the prize is the largest shared factor.

Let's list the factors of 18. The factors of 18 are the numbers that divide evenly into 18. These are: 1, 2, 3, 6, 9, and 18. Got 'em all! Now, let's list the factors of 288. This one might take a bit longer, as 288 is a larger number. The factors of 288 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, and 288. Wow, that's a lot! Listing the factors of each number is a methodical way to identify all the possible divisors. It's like taking inventory of all the components that make up a number. This process can be time-consuming, especially for larger numbers, but it's essential for accurately determining the GCD. By carefully listing out the factors, we can visually compare the two sets of numbers and pinpoint the largest one they share. It's like comparing two toolboxes to see which tool is the biggest one present in both. So, with our lists of factors in hand, we're ready to move on to the next step: identifying the greatest common factor.

Now, let's compare the lists of factors and find the largest number that appears in both. Looking at the factors of 18 (1, 2, 3, 6, 9, 18) and the factors of 288 (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288), we can see that the largest number they have in common is 18. Aha! So, the GCD of 18 and 288 is 18. Identifying the greatest common factor from the lists is the final step in our GCD calculation. It's like the moment of discovery when you find the treasure you've been searching for. By comparing the lists, we visually pinpoint the largest number that both numbers share as a factor. This number represents the GCD, the greatest common divisor that evenly divides both numbers. The GCD, in this case, 18, is a significant value that reflects the shared divisibility of 18 and 288. With this final piece of the puzzle in place, we've completed our mathematical journey, from solving the equation to verifying the solution and finding the GCD. What a triumph!

Conclusion

So, guys, we successfully solved the equation 2(18x - 288) = 0, found that x = 16, verified our solution, and determined that the GCD of 18 and 288 is 18. High five! We tackled a multi-faceted problem and came out victorious. Remember, math can be fun and rewarding when we break it down step by step. Keep practicing, and you'll be solving equations and finding GCDs like a pro in no time! This comprehensive journey through solving the equation, verifying the solution, and finding the GCD highlights the interconnectedness of mathematical concepts. Each step builds upon the previous one, reinforcing our understanding and problem-solving skills. The ability to solve equations, verify solutions, and calculate GCDs are valuable tools in both mathematical and real-world contexts. So, let's celebrate our mathematical achievement and carry forward this knowledge to tackle future challenges with confidence. Keep exploring, keep learning, and keep the mathematical spirit alive!