Problem 19 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{x}{5}-\frac{5}{x}=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5\) and \(x = -5\).

Step 1: Equal terms to zero

To start, it would be beneficial to get both terms on one side of the equation and equal to 0: \[\frac{x}{5}-\frac{5}{x}=0 \Rightarrow \frac{x}{5}-\frac{5}{x} = 0.\]

Step 2: Eliminate the denominators

Next step is to multiply by 5x to eliminate the denominators:\[5x \left(\frac{x}{5}-\frac{5}{x}\right)=5x*0 \Rightarrow x^2-25=0.\]

Step 3: Solve the equation

Now, solve for \(x\) by using the square root:\[x^2 = 25 \Rightarrow x = \pm \sqrt{25}= \pm 5.\]

Step 4: Check for extraneous solutions

Remember that x cannot be zero, as this will make the second term undefined. Since neither of the solutions, \( 5\) or \(-5\), make \(x\) zero in the original equation, both solutions are valid.

Key Concepts

Algebraic Techniques

Algebraic techniques form the cornerstone of solving equations, as they provide systematic methods to isolate variables and find solutions. In the exercise given, the initial technique used is to set the rational equation equal to zero. This fundamental step creates the basis for further manipulation. After establishing equality with zero, the next technique employed is clearing the denominators. This involves multiplying each term by the least common denominator, which in this case is the product of the individual denominators, 5x. By doing so, we convert the rational equation into a polynomial one, specifically a quadratic equation. This simplification is pivotal as it allows us to use additional techniques, such as factoring or applying the quadratic formula, to solve for the unknown variable x.

Key to mastering these algebraic steps is practice, along with a thorough understanding of the properties of equality and operations with fractions. To improve student's grasp of these techniques, it's essential to present examples that involve varying levels of complexity and to encourage the formulation of alternative strategies for solving similar problems.

Equation Solving

Equation solving is the process of finding the values of variables that satisfy a given mathematical statement. The presented exercise involves solving a quadratic equation, which is a specific type of polynomial equation. The equation is transformed into a standard form, where the powers of x are in descending order and set equal to zero. This methodical approach paves the way for identifying potential solutions through factoring, using the quadratic formula, or taking square roots, as appropriate.

To enhance one's ability to solve equations, understanding the fundamental principles, such as balancing equations and performing inverse operations, is crucial. It is also beneficial to explore the reasons why certain methods work to strengthen the conceptual understanding and provide a framework for tackling more complex equations. Reinforcement can come from working out multiple problems and varying the types of equations students encounter, including multi-step and real-world application equations.

Rational Expressions

Rational expressions represent ratios of polynomial expressions, similar to how a fraction is a ratio of numbers. They can often lead to equations where the variable appears in the denominator. Solving equations with rational expressions requires a clear understanding of how to manipulate these ratios and maintain equivalencies. A common approach is to find a common denominator to combine terms, as seen in the step-by-step solution where multiplying by 5x eliminated the fractions altogether.

An essential reminder when working with rational expressions is to watch for restrictions on the variables that prevent division by zero. In the educational content, emphasizing the importance of checking for extraneous solutions, which are apparent solutions that do not actually satisfy the original equation, is vital. To enhance this lesson, examples should highlight scenarios where extraneous solutions arise, and students should be coached on how to verify their answers by substituting back into the initial equation.