Solving Proportions A Step By Step Guide To 5/(2x-3) = 2/x
At the heart of mathematics lies the concept of proportions, which express the equality of two ratios. Mastering the art of solving proportions is crucial for various mathematical applications, ranging from simple scaling problems to complex scientific calculations. This article delves into the intricacies of solving the proportion $\frac{5}{2x-3}=\frac{2}{x}\, providing a step-by-step guide that will empower you to confidently tackle similar problems.
Understanding Proportions
Before we embark on solving the given proportion, let's first establish a solid understanding of what proportions are and why they are so important. A proportion is essentially a statement that two ratios are equal. A ratio, in turn, is a comparison of two quantities, often expressed as a fraction. For instance, the ratio of 5 apples to 3 oranges can be written as $\frac{5}{3}\. When we say that two ratios are in proportion, we mean that they represent the same relationship between quantities. This concept is fundamental to many real-world scenarios, such as scaling recipes, converting units, and determining the relative sizes of objects.
Proportions are a cornerstone of mathematical problem-solving. They allow us to establish relationships between quantities and make predictions based on known information. In fields like engineering, architecture, and finance, proportions are indispensable tools for ensuring accuracy and maintaining scale. For example, architects use proportions to create blueprints that accurately represent the dimensions of a building, while engineers rely on them to calculate the stress on structures and ensure their stability. The ability to solve proportions efficiently is therefore a valuable skill across numerous disciplines.
Setting up the Proportion
The given proportion, $\frac{5}{2x-3}=\frac{2}{x}\, presents us with an equation where two ratios are set equal to each other. Our goal is to find the value of the unknown variable, x, that satisfies this equation. To do this, we'll employ a technique known as cross-multiplication, which leverages a fundamental property of proportions. This property states that if two ratios are equal, then the product of the means equals the product of the extremes. In other words, if $\frac{a}{b} = \frac{c}{d}\, then ad = bc. This seemingly simple rule forms the basis for solving a wide range of proportional equations.
Before we apply cross-multiplication, it's crucial to ensure that our proportion is properly set up. We must clearly identify the numerators and denominators of each ratio. In our case, the first ratio is $\frac{5}{2x-3}\, where 5 is the numerator and (2x - 3) is the denominator. The second ratio is $\frac{2}{x}\, with 2 as the numerator and x as the denominator. Once we have correctly identified these components, we can confidently proceed with cross-multiplication. This methodical approach is essential for avoiding errors and ensuring that we arrive at the correct solution. Remember, a clear understanding of the problem setup is half the battle in solving any mathematical equation.
Cross-Multiplication: The Key to Solving
Now that we have our proportion set up correctly, we can apply the principle of cross-multiplication. This technique involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. In our equation, $\frac{5}{2x-3}=\frac{2}{x}\, this translates to multiplying 5 by x and 2 by (2x - 3). This gives us the equation 5x = 2(2x - 3). Cross-multiplication effectively eliminates the fractions, transforming the proportion into a more manageable linear equation. This step is pivotal in the solving process, as it allows us to isolate the variable x and determine its value.
The resulting equation, 5x = 2(2x - 3), is a crucial stepping stone in our solution. It represents a simplified form of the original proportion, where the equality of the ratios is now expressed in terms of a linear relationship. However, we're not quite at the finish line yet. We still need to further simplify the equation and isolate x to find its value. The next step involves applying the distributive property to expand the right side of the equation, which will pave the way for collecting like terms and ultimately solving for x. Cross-multiplication is a powerful tool, but it's just one piece of the puzzle in our journey to solve this proportion.
Simplifying the Equation
With the equation 5x = 2(2x - 3) in hand, our next task is to simplify it. This involves applying the distributive property, which states that a*(b + c) = ab + ac. In our case, we need to distribute the 2 on the right side of the equation across the terms inside the parentheses. This means multiplying 2 by 2x and 2 by -3. Performing these multiplications yields the equation 5x = 4x - 6. By applying the distributive property, we have successfully removed the parentheses and further simplified the equation.
This simplified equation, 5x = 4x - 6, brings us closer to isolating the variable x. Our next goal is to collect like terms, which means gathering all the x terms on one side of the equation and the constant terms on the other side. This will allow us to isolate x and determine its value. The process of simplifying equations is a fundamental skill in algebra, and mastering it is essential for solving a wide range of mathematical problems. Each step we take, from applying the distributive property to collecting like terms, brings us closer to the solution.
Isolating the Variable
Now, let's isolate the variable x in the equation 5x = 4x - 6. To do this, we need to get all the x terms on one side of the equation and the constant terms on the other. A common strategy is to subtract 4x from both sides of the equation. This will eliminate the x term on the right side and move it to the left side. Performing this subtraction, we get 5x - 4x = 4x - 6 - 4x. Simplifying both sides, we have x = -6. This step is crucial because it brings us to the solution by isolating x.
With x now isolated, we have arrived at the solution: x = -6. This means that the value of x that satisfies the original proportion, $\frac{5}{2x-3}=\frac{2}{x}\, is -6. However, before we declare victory, it's important to verify our solution. This involves plugging the value of x back into the original equation to ensure that both sides are equal. This step is a critical safeguard against errors and confirms that our solution is indeed correct. The isolation of the variable is a key milestone, but verification is the final stamp of approval.
Verifying the Solution
Before we definitively declare x = -6 as the solution, it's essential to verify it. This involves substituting -6 for x in the original proportion, $\frac{5}{2x-3}=\frac{2}{x}\, and checking if both sides of the equation are equal. Substituting x = -6, we get $\frac{5}{2(-6)-3}=\frac{2}{-6}\. Simplifying the left side, we have $\frac{5}{-12-3}=\frac{5}{-15} = -\frac{1}{3}\. On the right side, we have $\frac{2}{-6} = -\frac{1}{3}\. Since both sides are equal to -\, our solution x = -6 is verified.
This verification step is not merely a formality; it's a crucial safeguard against errors. By plugging our solution back into the original equation, we confirm that it satisfies the proportional relationship. In cases where the equation involves more complex expressions or potential extraneous solutions, verification becomes even more critical. It ensures that we have not only solved the equation correctly but also that our solution is valid within the context of the problem. Therefore, always make verification a standard part of your problem-solving routine.
Final Answer
After diligently working through the steps of cross-multiplication, simplification, isolation, and verification, we have arrived at the final answer. The solution to the proportion $\frac{5}{2x-3}=\frac{2}{x}\ is x = -6. This integer value represents the point of equilibrium between the two ratios, where the proportional relationship holds true. Our journey through this problem has not only yielded a solution but also reinforced the importance of methodical problem-solving in mathematics.
The process we've followed – setting up the proportion, cross-multiplying, simplifying, isolating the variable, and verifying the solution – is a valuable template for tackling similar problems. By mastering these steps, you'll be well-equipped to solve a wide range of proportions and apply this knowledge to various mathematical and real-world scenarios. The final answer, x = -6, is a testament to the power of these techniques and the satisfaction of arriving at a correct solution through careful and systematic effort.
Therefore, the solution to the proportion is x = -6.
