Problem 31 - Intermediate Algebra

Question

Solve each equation. $$ \frac{5}{n-3}-\frac{3}{n+3}=1 $$

Step-by-Step Solution

Answer

The solutions are \( n = 6 \) and \( n = -5 \).

Step 1: Identify the Denominators

First, observe the denominators in the equation: \( n-3 \) and \( n+3 \). These need to be considered when finding a common denominator as we will eventually clear the fractions.

Step 2: Find the Common Denominator

The common denominator of the fractions is \((n-3)(n+3)\). This is the product of both denominators, which allows us to eliminate the fractions by multiplying through by this common denominator.

Step 3: Clear the Fractions

Multiply every term in the equation by the common denominator \((n-3)(n+3)\): \[ (n-3)(n+3) \left(\frac{5}{n-3} - \frac{3}{n+3}\right) = (n-3)(n+3) \times 1 \]This simplifies to: \[ 5(n+3) - 3(n-3) = (n-3)(n+3) \]

Step 4: Expand and Simplify

Expand both sides of the equation:\[ 5n + 15 - 3n + 9 = n^2 - 9 \]Combine like terms:\[ 2n + 24 = n^2 - 9 \]

Step 5: Rearrange into a Quadratic Equation

Move all terms to one side of the equation to form a quadratic equation:\[ 0 = n^2 - 2n - 33 \]

Step 6: Solve the Quadratic Equation

Factor the quadratic equation:\[ n^2 - 2n - 33 = (n - 6)(n + 5) \]Set each factor equal to zero and solve:\[ n - 6 = 0 \] or \[ n + 5 = 0 \]So, \( n = 6 \) or \( n = -5 \).

Step 7: Verify the Solutions

Check each potential solution in the original equation. For \( n = 6 \), \[ \frac{5}{6-3} - \frac{3}{6+3} = \frac{5}{3} - \frac{3}{9} = \frac{5}{3} - \frac{1}{3} = 1 \]For \( n = -5 \), \[ \frac{5}{-5-3} - \frac{3}{-5+3} = \frac{5}{-8} - \frac{3}{-2} = \frac{5}{-8} + \frac{3}{2} = \frac{-5}{8} + \frac{12}{8} = 1 \]Both solutions satisfy the original equation.

Key Concepts

Solving Rational Equations

Rational equations are equations containing fractions with polynomials in their numerators and/or denominators. Understanding how to solve these requires a few important steps. The key idea is to eliminate fractions to simplify the equation. This involves using the concept of the common denominator.

Here’s a simple approach to tackling rational equations like the one in our exercise:

• Identify the denominators in the equation. In our example, they are \(n-3\) and \(n+3\).
• Find a common denominator, which allows you to multiply through and clear the fractions.
• Once the fractions are cleared, you can work with a simplified equation, often transforming it into a form you are more familiar with, like a quadratic equation.

This step-by-step approach simplifies what could initially seem complex. Practice these steps with different examples until they become second nature!

Quadratic Equations

Quadratic equations appear frequently in algebra, often taking the form \( ax^2 + bx + c = 0 \). They arise during the process of solving rational equations after clearing fractions. Simplifying and rearranging the equation can result in a standard quadratic form.

To solve a quadratic equation, you typically use one of these methods:

• Factoring: This involves expressing the quadratic equation as a product of two binomials. For example, the equation \(n^2 - 2n - 33 = 0\) is factored into \((n-6)(n+5)\).
• Quadratic Formula: This formula \(n = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) can be used to find solutions for any quadratic equation.
• Completing the square or graphing may also be used in more complex scenarios.

This aspect of solving equations requires careful attention to identify and correctly manipulate terms. Each method has its advantages, and practicing them helps in becoming proficient in solving quadratics.

Common Denominator

A common denominator is essential when working with rational equations. It is a common multiple of the denominators of all fractions involved in the equation. By using a common denominator, you can transform the equation to eliminate fractions.

Here's how you find and use a common denominator:

• Identify each distinct denominator in the equation. In our example, these are \(n-3\) and \(n+3\).
• The common denominator is the product of these denominators, \((n-3)(n+3)\).
• Multiply every term in the equation by the common denominator, effectively eliminating the fractions.

The result is a simplified equation without fractions, making it easier to solve. Finding a common denominator is a powerful tool in algebra and is often the first step in solving rational equations. Understanding it can make manipulating and solving equations much more manageable.