Problem 44 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{1}{x+3}+\frac{5}{x^{2}-9}=\frac{2}{x-3} $$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Recognize Common Denominator

Identify the denominators in the given equation, \(x+3\), \((x-3)(x+3)\) which simplifies to \(x^2 - 9\), and \(x-3\). The common denominator is \((x+3)(x-3)\), since it encompasses the factors of all other denominators.

Step 2: Rewrite Fractions with Common Denominator

Rewrite each term with the common denominator: \(\frac{(x-3)}{(x+3)(x-3)} + \frac{5}{(x+3)(x-3)} = \frac{2(x+3)}{(x+3)(x-3)}\).

Step 3: Combine and Simplify

Combine the numerators over the common denominator: \(\frac{x-3+5}{(x+3)(x-3)} = \frac{2(x+3)}{(x+3)(x-3)}\). Simplifying the numerator gives: \(\frac{x+2}{(x+3)(x-3)}\).

Step 4: Eliminate Denominator

Since the denominators are the same, set the numerators equal to each other: \(x+2 = 2(x+3)\).

Step 5: Solve the Linear Equation

Expand and simplify the equation: \(x + 2 = 2x + 6\). Rearrange terms to find \(x = -4\).

Step 6: Verify Solution

Substitute \(x = -4\) back into the original denominators to ensure it doesn't make any undefined (denominator cannot be zero). For \(x+3 = -1 eq 0\) and \(x-3 = -7 eq 0\). Therefore, \(x = -4\) is a valid solution.

Key Concepts

Common Denominator

In algebra, working with fractions often requires a common denominator. This is essential for adding or subtracting fractions. Having a common denominator makes combining them possible. Think of a common denominator as a shared base for comparison or calculation. In this exercise, we identified the denominators: \(x+3\), \((x-3)(x+3)\), and \(x-3\).

• The goal is to find a denominator that encompasses all denominators. Therefore, the common denominator we chose is \((x+3)(x-3)\).
• This choice combines both factors present in all denominators, including simplifying \(x^2 - 9\) to \((x+3)(x-3)\).

This is important so that each term can be rewritten with this common denominator, allowing for straightforward combination.

Numerator Simplification

After establishing a common denominator, we turn our attention to simplifying the numerators. This step involves rewriting the fractions to have the shared denominator. Next, we combine the numerators over the common denominator.

• In the step-by-step solution, the numerators to be combined are \(x-3\) and \(5\) over our common denominator \((x+3)(x-3)\).
• Adding these numerators results in \(x+2\). This simplifies the fraction to \(\frac{x+2}{(x+3)(x-3)}\).

Simplifying the numerators makes it easier to eliminate the denominator later and solve the equation.

Verify Solution

Verification is a crucial last step in solving equations. After finding a solution, like \(x = -4\), it must be checked against the original equation to ensure its validity. This involves substituting the solution back into the original equations and ensuring no fraction becomes undefined.

• When substituting \(x = -4\), check that none of the denominators become zero. For \(x+3\) and \(x-3\), neither becomes zero.
• If a denominator were zero, it would indicate division by zero, which is undefined. Thankfully, \(x = -4\) does not cause this problem here.

Once verified, we can confidently say that the solution is correct and valid.

Solving Algebraic Equations

Solving linear (or equivalent) algebraic equations involves several critical steps. It's about finding the value of \(x\) that satisfies the equation. This process typically includes:

• Firstly, eliminate denominators by equating numerators since they are over a common denominator.
• Then, simplify and rearrange terms to isolate \(x\). In our case, this meant solving \(x + 2 = 2(x+3)\).
• Simplifying this, we rearranged to \(x + 2 = 2x + 6\), and solving gives \(x = -4\).

Understanding these steps can improve your problem-solving skills in algebra, enabling you to handle similar exercises with confidence.