Solving For 12x + 10y Given (2/5)x + (1/3)y = 1
In the realm of linear equations, a fascinating problem presents itself: Given the equation (2/5)x + (1/3)y = 1, what is the value of 12x + 10y? This seemingly simple question opens a gateway to exploring the fundamental principles of algebraic manipulation and problem-solving strategies. Let's embark on a journey to unravel this mathematical puzzle, delving into the core concepts and techniques required to arrive at the solution. This article will not only provide the answer but also offer a detailed explanation of the steps involved, ensuring a thorough understanding of the underlying mathematical principles. We will explore various approaches to solving this problem, highlighting the most efficient and elegant methods. Whether you're a student honing your algebra skills or simply a math enthusiast, this guide will equip you with the knowledge and confidence to tackle similar challenges. Understanding the relationship between the given equation and the expression we need to evaluate is key to unlocking the solution. By carefully manipulating the equation, we can transform it into a form that directly reveals the value of 12x + 10y. This process involves strategic multiplication and simplification, emphasizing the importance of precision and attention to detail in algebraic manipulations. The article will also touch upon the broader implications of solving linear equations, demonstrating their relevance in various fields such as physics, engineering, and economics. By mastering these fundamental concepts, you'll gain a valuable toolset for tackling real-world problems that can be modeled using linear relationships. So, let's dive into the heart of the problem and discover the elegant solution that awaits us. This exploration will not only enhance your algebraic skills but also foster a deeper appreciation for the power and beauty of mathematics. As we progress, we'll break down the problem into manageable steps, making the solution process clear and accessible to everyone. Remember, the key to success in mathematics lies in understanding the underlying principles and applying them strategically. With that in mind, let's begin our journey to unlock the value of 12x + 10y.
H2: The Foundation Equation (2/5)x + (1/3)y = 1
Our journey begins with the bedrock equation: (2/5)x + (1/3)y = 1. This equation serves as the cornerstone of our problem, providing the fundamental relationship between the variables x and y. To effectively manipulate this equation and extract the desired value of 12x + 10y, we must first understand its structure and properties. This equation is a linear equation in two variables, which means it represents a straight line when plotted on a graph. The coefficients (2/5 and 1/3) determine the slope and intercepts of this line, while the variables x and y represent the coordinates of any point lying on the line. The constant term (1) defines the y-intercept of the line. To transform this equation into a more manageable form, we often seek to eliminate the fractions. This can be achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 15. Multiplying both sides by 15 yields a new equation that is equivalent to the original but without any fractional coefficients. This step is crucial for simplifying the subsequent algebraic manipulations and making the problem easier to solve. Furthermore, understanding the properties of equality is essential for working with equations. Any operation performed on one side of the equation must also be performed on the other side to maintain the balance and ensure that the equality holds true. This principle allows us to manipulate equations without changing their underlying meaning. In this case, multiplying both sides by 15 preserves the relationship between x and y, while simultaneously simplifying the equation. The resulting equation will be a linear combination of x and y, which will allow us to directly compare it with the expression 12x + 10y. By carefully analyzing the coefficients of x and y in the simplified equation, we can identify a multiplicative factor that will transform the left-hand side into 12x + 10y. This factor will then allow us to determine the value of the expression on the right-hand side, which will be the answer to our problem. This process highlights the importance of strategic algebraic manipulation in solving mathematical problems. By understanding the underlying principles and applying them thoughtfully, we can unlock the solutions to even the most challenging equations. The initial equation (2/5)x + (1/3)y = 1 is therefore not just a starting point but the key to unraveling the mystery of 12x + 10y.
H2: The Transformation Multiplying to Reveal the Solution
The core strategy to solve this equation hinges on a clever transformation. As highlighted in the previous section, our aim is to eliminate the fractions in the given equation (2/5)x + (1/3)y = 1. The most effective way to achieve this is by multiplying both sides of the equation by the least common multiple (LCM) of the denominators 5 and 3, which is 15. This seemingly simple step is a pivotal moment in our problem-solving journey, as it sets the stage for a more direct comparison with the target expression 12x + 10y. When we multiply both sides of the equation by 15, we're essentially scaling the entire equation without altering the fundamental relationship between x and y. This operation is justified by the properties of equality, which dictate that performing the same operation on both sides maintains the balance of the equation. The transformation yields the following: 15 * [(2/5)x + (1/3)y] = 15 * 1. Distributing the 15 on the left-hand side gives us (15 * 2/5)x + (15 * 1/3)y = 15. Simplifying the coefficients, we get 6x + 5y = 15. Now, we have a much cleaner and more manageable equation, free from fractions. This equation, 6x + 5y = 15, is equivalent to the original equation but is in a form that allows us to see a direct relationship with the expression 12x + 10y. This is where the brilliance of the transformation shines through. Notice that 12x + 10y is precisely twice the expression 6x + 5y. This crucial observation provides the key to unlocking the solution. If we multiply the entire equation 6x + 5y = 15 by 2, we'll obtain an equation where the left-hand side is exactly the expression we're looking for: 12x + 10y. This elegant manipulation highlights the power of strategic algebraic thinking. By recognizing the relationship between the transformed equation and the target expression, we can easily determine the value of 12x + 10y. The act of multiplying by 15 and then by 2 is a testament to the importance of simplification and strategic scaling in solving mathematical problems. It demonstrates how a series of carefully chosen operations can lead us to the desired solution in a straightforward and efficient manner. This transformation is not just a mathematical trick; it's a powerful technique that can be applied to a wide range of algebraic problems. By mastering this approach, you'll be well-equipped to tackle more complex equations and unlock their hidden solutions.
H2: The Grand Finale Calculating 12x + 10y
With the groundwork laid, we're now poised to calculate the value of 12x + 10y. As we established in the previous section, our transformed equation is 6x + 5y = 15. The crucial insight here is that the expression we want to find, 12x + 10y, is precisely twice the left-hand side of our transformed equation. To see this clearly, we can simply multiply the entire equation 6x + 5y = 15 by 2. This operation maintains the equality, thanks to the fundamental properties of algebraic equations. When we multiply both sides by 2, we get 2 * (6x + 5y) = 2 * 15. Distributing the 2 on the left-hand side gives us (2 * 6x) + (2 * 5y) = 30. Simplifying, we arrive at 12x + 10y = 30. And there it is! The value of 12x + 10y is revealed to be 30. This elegant solution is a testament to the power of strategic algebraic manipulation. By carefully transforming the original equation and recognizing the relationship between the transformed equation and the target expression, we were able to determine the answer with ease. The journey from the initial equation (2/5)x + (1/3)y = 1 to the final answer of 12x + 10y = 30 underscores the importance of methodical problem-solving. Each step, from eliminating the fractions to recognizing the multiplicative factor, played a crucial role in unlocking the solution. This problem serves as a valuable example of how algebraic techniques can be used to solve seemingly complex problems. By breaking down the problem into manageable steps and applying the appropriate operations, we can arrive at the answer with confidence. The solution 12x + 10y = 30 is not just a number; it's the culmination of a logical and strategic problem-solving process. It demonstrates the beauty and elegance of mathematics, where seemingly disparate pieces can be brought together to form a cohesive and satisfying solution. This understanding not only answers the specific question at hand but also equips you with a powerful set of tools for tackling future mathematical challenges. The final answer, 30, is the result of a carefully orchestrated sequence of algebraic manipulations. It stands as a testament to the power of mathematical reasoning and the elegance of well-chosen transformations.
H2: Conclusion and Key Takeaways
In conclusion, we've successfully navigated the problem of finding the value of 12x + 10y given the equation (2/5)x + (1/3)y = 1. The journey began with understanding the fundamental equation, then strategically transforming it by eliminating fractions and recognizing the relationship between the transformed equation and the target expression. The key to unlocking the solution was the realization that 12x + 10y is precisely twice the expression 6x + 5y, which we obtained by multiplying the original equation (after clearing fractions) by a factor of 15. Multiplying the transformed equation 6x + 5y = 15 by 2 directly revealed the answer: 12x + 10y = 30. This problem highlights several important takeaways for mathematical problem-solving. Firstly, strategic algebraic manipulation is crucial. By carefully choosing the operations we perform on an equation, we can transform it into a more manageable form and reveal hidden relationships. Secondly, recognizing patterns and relationships is essential. In this case, recognizing that 12x + 10y is twice 6x + 5y was the key to the solution. Thirdly, understanding the properties of equality is fundamental. We relied on the fact that performing the same operation on both sides of an equation maintains the equality, allowing us to manipulate the equation without changing its underlying meaning. This problem also underscores the importance of breaking down complex problems into smaller, more manageable steps. By systematically addressing each step, we were able to arrive at the solution in a clear and logical manner. Furthermore, this exercise demonstrates the power of linear equations and their ability to model relationships between variables. Linear equations are ubiquitous in mathematics and science, and mastering the techniques for solving them is a valuable skill. The solution we found, 12x + 10y = 30, is not just a numerical answer; it's the result of a methodical and insightful problem-solving process. It showcases the beauty and elegance of mathematics, where careful reasoning and strategic manipulation can lead to clear and satisfying solutions. By understanding the techniques used in this problem, you'll be better equipped to tackle a wide range of algebraic challenges and appreciate the power of mathematical thinking. The journey from the initial equation to the final answer has provided valuable insights into the art of problem-solving and the elegance of mathematical reasoning. The value of 12x + 10y, which is 30, is not just a number but a testament to the power of strategic manipulation and insightful observation in the realm of mathematics.
