Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$\frac{1}{x-3}+\frac{1}{x+3}=\frac{10}{x^{2}-9}$$

Step-by-Step Solution

Answer

The solution to the given equation is \(x = 5\).

Step 1: Check for undefined values

Given that the denominator of each fraction cannot equal zero, x cannot be 3 or -3.

Step 2: Multiply each term by the common denominator

Multiplying each term by \(x^{2}-9\), the equation becomes \(x+3 + x-3 = 10\).

Step 3: Simplify the equation

Simplifying gives \(2x = 10\). This simplifies into \(x = 5\).

Key Concepts

Common Denominator

When dealing with rational equations, one useful strategy is to find a common denominator. This involves identifying a common base for the fractions that make up the equation. In our case, we see three fractions with denominators: \( x-3 \), \( x+3 \), and \( x^{2}-9 \).

• The denominator \( x^{2}-9 \) is actually a special kind of expression called a difference of squares, which can be factored into \( (x-3)(x+3) \).
• Thus, the common denominator for these fractions is simply \( x^{2}-9 \) because \( x^{2}-9 \) equals \( (x-3)(x+3) \).
• By multiplying the entire equation by this common denominator, we eliminate the fractions allowing us to work with a much simpler linear equation, which is often easier to solve.

Finding a common denominator helps eliminate fractions and makes complex-looking equations simpler to solve, turning what might seem an intimidating problem into a more straightforward one.

Undefined Values

Before solving any rational equation, it’s crucial to determine which values of \( x \) would cause the equation to become undefined. This happens when any denominator equals zero, as division by zero in mathematics is undefined.

• Look at the denominators: the potential values for \( x \) that could lead to a zero denominator are when \( x-3=0 \) or \( x+3=0 \).
• Solving these equations, we get \( x=3 \) and \( x=-3 \) as values that make the denominator zero.
• These are points where the original equation becomes undefined, and as such, they are not part of the solution set.

Identifying undefined values early is integral as it avoids mistakes where such solutions might accidentally be included, ensuring a correct and complete solution to the problem.

Equation Simplification

In solving rational equations, simplifying the expressions after eliminating fractions is key to finding the solution. With the fractions removed by multiplying through by the common denominator, we now face a simpler equation:

• In this scenario, after clearing the fractions, the equation simplifies to \( x+3 + x-3 = 10 \).
• Combine like terms, which results in \( 2x = 10 \).
• This linear equation in one variable can be easily solved by isolating \( x \) to find \( x = 5 \).

The simplification turns the problem-solving process into just organizing like terms and dealing with a straightforward operation, making it more accessible and manageable. It's this step-by-step dismantling of complexity that leads to efficient and accurate solutions.