Problem 48 - Algebra and Trigonometry

Question

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 to the equation is \(x = -4\).

Step 1: Identify Terms and Simplify

First, notice the denominators of the fractions: \(x+3\), \(x^2-9\), and \(x-3\). Recognize that \(x^2-9\) is a difference of squares, which can be factored as \((x-3)(x+3)\). Rewrite the equation as: \[\frac{1}{x+3} + \frac{5}{(x-3)(x+3)} = \frac{2}{x-3}.\]

Step 2: Find a Common Denominator

The common denominator for all fractions is \((x-3)(x+3)\). We'll multiply each term by this common denominator to eliminate the fractions. This gives:\[\frac{1 \cdot (x-3)}{x+3} + \frac{5}{(x-3)(x+3)}(x-3)(x+3) = \frac{2 \cdot (x+3)}{x-3}.\]

Step 3: Simplify and Expand

After multiplying, the equation becomes:\[(x-3) + 5 = 2(x+3).\]This simplifies and expands to:\[x - 3 + 5 = 2x + 6.\]

Step 4: Combine Like Terms

Combine the constants on the left side:\[x + 2 = 2x + 6.\]

Step 5: Solve for x

Reorder and rearrange the equation to isolate \(x\). Subtract \(x\) from both sides to get:\[2 = x + 6.\]Subtract 6 from both sides to find:\[2 - 6 = x.\]This simplifies to:\[x = -4.\]

Step 6: Verify the Solution

Substitute \(x = -4\) back into the original equation to ensure there are no extraneous solutions due to restrictions on the denominators. Since none of the original denominators become zero (i.e., \(-4 eq -3\) and \(-4 eq 3\)), \(x = -4\) is a valid solution.

Key Concepts

Factoring

Factoring is a crucial skill in solving algebraic equations, especially those involving polynomials. It involves breaking down a complex expression into simpler, multiplyable factors. Consider the polynomial expression given by \(x^2 - 9\) in our problem. This is a perfect example of a difference of squares—an expression that can be rewritten as the product of two binomials. Specifically, the difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Applying this formula, \(x^2 - 9\) becomes \((x-3)(x+3)\).
Factoring helps simplify equations, making them easier to solve. By expressing complex terms as products of simpler ones, we set the stage for effective manipulation, like finding common denominators.

Common Denominators

Finding a common denominator is key when dealing with equations involving fractions. This method simplifies the process of adding or subtracting fractions, making them easier to manage. In our equation, three different denominators are present: \(x+3\), \((x-3)\), and \((x^2-9)\) which equals \((x-3)(x+3)\) after factoring.

• Finding the least common denominator (LCD) involves taking the most complex denominator, which, in this case, is \((x-3)(x+3)\).
• Multiply every term in the equation by this LCD to eliminate the denominators. This turns the fractions into simpler expressions, allowing for straightforward algebraic manipulations.

By finding a common denominator, we transform the problem from one of fractions into more familiar linear expressions.

Simplifying Expressions

Simplifying expressions refers to reducing them into their simplest form by performing operations like combining like terms, distributing factors, or eliminating fractions through common denominators. In our example, after factoring and finding a common denominator, we apply these operations.
For instance, multiplying each fraction by the common denominator \((x-3)(x+3)\) clears the fractions. The resulting equation \((x-3) + 5 = 2(x+3)\) becomes free of fractions, making it easier to manage.
Simplifying involves:

• Expanding expressions by using the distributive property, which helps eliminate parentheses.
• Combining like terms, which simplifies further by grouping constants and like variables together.

This process makes the solution clear and extracts the essential form of the equation, paving the way for solving.

Solving Equations

Solving equations, especially linear ones, involves finding values for variables that satisfy the equation. Once you have simplified the equation, as in the previous steps, you're left with a straightforward task. We ended with the equation \(x + 2 = 2x + 6\).
Solving involves:

• Rearranging the equation: Subtract \(x\) from both sides to get \(2 = x + 6\).
• Isolating the variable \(x\): Subtracting \(6\) from both sides then gives \(x = -4\).

This reflects the solution in its simplest form where \(x\) is isolated on one side of the equation.
After solving, always verify your result by substituting it back into the original equation. This ensures accuracy and confirms that no restrictions in the problem were violated, such as denominators turning zero.