Solving For X: Find The Values In 54/(3x) = 2x

Let's dive into the fascinating world of algebra and tackle a problem that involves finding the values of x in a given equation. Algebraic equations might seem daunting at first, but with a systematic approach and a sprinkle of mathematical curiosity, we can unravel their mysteries. In this article, we're going to break down the equation 54 / (3x) = 2x, step by step, to reveal the two possible values that x can hold. So, grab your mathematical toolkit, and let's embark on this algebraic adventure together!

Understanding the Equation

Before we jump into solving, let’s take a moment to understand the equation we’re dealing with: 54 / (3x) = 2x. This equation tells us that if we divide 54 by the product of 3 and x, the result is equal to 2 times x. Our mission is to find the values of x that make this statement true. This involves a combination of algebraic manipulation and a bit of clever thinking. Remember, in algebra, we treat both sides of the equation like a balancing scale. Whatever operation we perform on one side, we must also perform on the other to maintain the balance. This principle is our guiding star as we navigate through the solution.

Initial Assessment and Strategy

When faced with an algebraic equation, it's always a good idea to take a step back and assess the landscape before diving in headfirst. What kind of equation is it? What are the operations involved? Are there any immediate simplifications we can make? In our case, we have a rational equation, which means it involves fractions with variables in the denominator. Our primary strategy will be to eliminate the fraction by multiplying both sides of the equation by the denominator. This will transform the equation into a more manageable form, hopefully leading us to a solution. Additionally, we need to be mindful of potential values of x that would make the denominator zero, as these values would render the equation undefined. With our strategy in place, let’s roll up our sleeves and get to work!

Step-by-Step Solution

Now, let's embark on the journey of solving this equation. Each step is a stride closer to the solution, and it's crucial to understand the logic behind each move.

Step 1: Eliminate the Fraction

Our first goal is to get rid of the fraction, which is achieved by multiplying both sides of the equation by the denominator, 3x. This gives us:

54 / (3x) * (3x) = 2x * (3x)

On the left side, the (3x) in the numerator and denominator cancel out, leaving us with 54. On the right side, 2x multiplied by 3x results in 6x². So, our equation now looks like this:

54 = 6x²

This is a significant simplification! We've transformed the rational equation into a quadratic equation, which is a form we're more familiar with.

Step 2: Simplify the Equation

To make the equation even simpler, we can divide both sides by 6. This isolates the x² term:

54 / 6 = 6x² / 6

This simplifies to:

9 = x²

We're getting closer! Now, we have a clean and concise equation that directly relates x² to a constant.

Step 3: Solve for x

To find the values of x, we need to take the square root of both sides of the equation. Remember, when we take the square root, we must consider both positive and negative solutions:

√9 = ±√x²

This gives us:

±3 = x

So, the two possible values for x are 3 and -3.

Step 4: Verification

Before we declare victory, it's crucial to verify our solutions. This involves plugging each value of x back into the original equation to ensure it holds true. Let's start with x = 3:

54 / (3 * 3) = 2 * 3

54 / 9 = 6

6 = 6

This checks out! Now, let's try x = -3:

54 / (3 * -3) = 2 * -3

54 / -9 = -6

-6 = -6

This also checks out! Both values satisfy the original equation, confirming that our solutions are correct.

The Two Possible Values of x

After navigating through the algebraic terrain, we've successfully discovered the two possible values of x in the equation 54 / (3x) = 2x. These values are:

• x = 3
• x = -3

These solutions represent the points where the equation holds true, the values that perfectly balance the algebraic scale. It's a testament to the power of algebraic manipulation and the beauty of mathematical problem-solving. Remember, in mathematics, every problem is a puzzle waiting to be solved, and with the right tools and techniques, no challenge is insurmountable.

Importance of Verification

You might be wondering, why did we dedicate a whole step to verification? Well, in mathematics, accuracy is paramount. It's not enough to arrive at a solution; we must also ensure that the solution is correct. Verification serves as our safety net, catching any potential errors that might have crept in during the solving process. It's a crucial step in building confidence in our answers and ensuring the integrity of our mathematical work. In our case, by plugging the values of 3 and -3 back into the original equation, we confirmed that they indeed satisfy the equation, giving us the assurance that our solutions are valid.

Real-World Applications of Algebraic Equations

Algebraic equations aren't just abstract concepts confined to textbooks and classrooms. They're powerful tools that find applications in various real-world scenarios. From engineering and physics to economics and computer science, algebraic equations play a vital role in modeling and solving problems. For instance, engineers use equations to design structures and predict their behavior under stress, physicists use them to describe the motion of objects, economists use them to analyze market trends, and computer scientists use them to develop algorithms. The ability to solve algebraic equations is a fundamental skill that opens doors to numerous career paths and enables us to understand and shape the world around us. So, the next time you encounter an algebraic equation, remember that you're not just solving a mathematical problem; you're honing a skill that has far-reaching implications.

Conclusion

In this exploration of algebra, we've successfully navigated the equation 54 / (3x) = 2x and unearthed the two values of x that make it sing: 3 and -3. We've seen how algebraic manipulation, combined with a dash of careful verification, can lead us to the heart of a mathematical problem. This journey underscores the elegance and power of algebra as a tool for unraveling the mysteries of the mathematical world. Remember, every equation is a story waiting to be told, and with the right approach, you can become the storyteller, revealing the hidden values and relationships that lie within. Embrace the challenge, celebrate the solutions, and keep exploring the vast and fascinating landscape of mathematics!

For more insights into algebraic equations and problem-solving strategies, check out resources like Khan Academy's Algebra Section.