Solving The Equation 5x - 16/x = -16 A Step-by-Step Guide

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In this article, we will delve into the process of solving the equation 5x - 16/x = -16. This equation falls under the category of rational equations, which are equations containing fractions with variables in the denominator. Solving such equations requires a systematic approach to eliminate the fractions and transform the equation into a more manageable form, typically a quadratic equation. We will explore each step in detail, providing clear explanations and justifications to ensure a comprehensive understanding of the solution. Mastering the techniques for solving rational equations is crucial for various mathematical and scientific applications, making this a valuable skill for students and professionals alike. This guide aims to provide not just the solution, but also the underlying concepts and strategies that can be applied to a wider range of similar problems. Whether you're a student grappling with algebra or someone looking to refresh your mathematical skills, this article will provide a clear and structured approach to solving this specific equation and rational equations in general.

Before diving into the solution, let's first understand the nature of the equation 5x - 16/x = -16. This is a rational equation because it involves a term where the variable x appears in the denominator. The presence of the fraction 16/x is what makes it a rational equation. To solve this, our primary goal is to eliminate the fraction. This is typically achieved by multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is simply x, as it is the only denominator present. Multiplying by x will clear the fraction and transform the equation into a quadratic form. Recognizing the structure of the equation is crucial because it dictates the appropriate method for solving it. If we were dealing with a linear equation, we would use a different approach. If it were a system of equations, yet another method would be required. Therefore, identifying the equation type – in this case, a rational equation – is the critical first step in finding the solution. The subsequent steps will involve algebraic manipulation to isolate the variable and find its possible values.

To solve the equation 5x - 16/x = -16, we will follow a series of algebraic steps to isolate the variable x. Here's a detailed breakdown of the process:

  1. Eliminate the fraction: To get rid of the fraction, we multiply both sides of the equation by x. This gives us: x(5x - 16/x) = -16x This simplifies to: 5x² - 16 = -16x

  2. Rearrange into a quadratic equation: To solve for x, we need to rearrange the equation into the standard quadratic form, which is ax² + bx + c = 0. Adding 16x to both sides, we get: 5x² + 16x - 16 = 0

  3. Solve the quadratic equation: Now we have a quadratic equation in the form ax² + bx + c = 0, where a = 5, b = 16, and c = -16. We can solve this using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) Plugging in the values, we get: x = (-16 ± √(16² - 4 * 5 * (-16))) / (2 * 5) x = (-16 ± √(256 + 320)) / 10 x = (-16 ± √576) / 10 x = (-16 ± 24) / 10

  4. Find the two possible solutions: The ± sign indicates that there are two possible solutions for x. Let's calculate them:

    • Solution 1: x = (-16 + 24) / 10 = 8 / 10 = 4/5
    • Solution 2: x = (-16 - 24) / 10 = -40 / 10 = -4

After obtaining potential solutions, it's essential to verify them by substituting them back into the original equation. This step ensures that our solutions are valid and do not lead to any undefined expressions, such as division by zero. Let's verify each solution:

  1. Verifying x = 4/5: Substitute x = 4/5 into the original equation 5x - 16/x = -16: 5*(4/5) - 16/(4/5) = -16 4 - 16*(5/4) = -16 4 - 20 = -16 -16 = -16 The equation holds true, so x = 4/5 is a valid solution.

  2. Verifying x = -4: Substitute x = -4 into the original equation 5x - 16/x = -16: 5*(-4) - 16/(-4) = -16 -20 - (-4) = -16 -20 + 4 = -16 -16 = -16 The equation also holds true for x = -4, confirming it as a valid solution.

Both solutions, x = 4/5 and x = -4, satisfy the original equation. This verification step is critical because it confirms the accuracy of our calculations and the validity of our solutions. It also helps to identify any extraneous solutions that might arise due to the algebraic manipulations involved in solving rational equations. In this case, both solutions are valid, demonstrating the thoroughness of our solution process.

In conclusion, we have successfully solved the equation 5x - 16/x = -16. By systematically eliminating the fraction, rearranging the equation into a quadratic form, and applying the quadratic formula, we found two solutions: x = 4/5 and x = -4. We then verified both solutions by substituting them back into the original equation, confirming their validity.

The process of solving this equation highlights the importance of understanding the underlying principles of algebra, particularly when dealing with rational equations. The key steps involved – eliminating fractions, rearranging into standard forms, and verifying solutions – are applicable to a wide range of algebraic problems. Mastering these techniques is essential for success in mathematics and related fields.

This detailed guide provides not only the solution to the specific equation but also a comprehensive understanding of the methodology involved. By following this step-by-step approach, you can confidently tackle similar problems and enhance your problem-solving skills in mathematics. Remember, practice is key to mastering these concepts, so try applying these techniques to other rational equations to solidify your understanding.