Problem 14 - Elementary Algebra

Question

$$ \text { For Problems 1-32, solve each equation. (Objective 1) } $$ $$ \frac{4 n}{n-5}-\frac{2 n}{n+5}=2 $$

Step-by-Step Solution

Answer

The solution is \(n = -\frac{5}{3}\).

Step 1: Find a Common Denominator

The given equation is \(\frac{4n}{n-5} - \frac{2n}{n+5} = 2\). The common denominator for the fractions \(n-5\) and \(n+5\) is \((n-5)(n+5)\).

Step 2: Express Each Fraction with the Common Denominator

Rewrite each fraction so they have the common denominator. This means:\[ \frac{4n(n+5)}{(n-5)(n+5)} - \frac{2n(n-5)}{(n+5)(n-5)} = 2 \]

Step 3: Simplify the Equation

Simplify the numerators and rewrite the equation:\[ \frac{4n^2 + 20n - (2n^2 - 10n)}{(n-5)(n+5)} = 2 \]

Step 4: Combine Like Terms in the Numerator

Simplifying the numerator, we have:\[ \frac{4n^2 + 20n - 2n^2 + 10n}{(n-5)(n+5)} = \frac{2n^2 + 30n}{(n-5)(n+5)} = 2 \]

Step 5: Clear the Fraction by Multiplying Through by the Denominator

Multiply both sides of the equation by \((n-5)(n+5)\) to eliminate the fraction:\[ 2n^2 + 30n = 2(n^2 - 25) \]

Step 6: Simplify and Solve the Quadratic Equation

Expand the right side and simplify:\[ 2n^2 + 30n = 2n^2 - 50 \]Subtract \(2n^2\) from both sides:\[ 30n = -50 \]

Step 7: Solve for n

Divide both sides by 30 to find \(n\):\[ n = \frac{-50}{30} = \frac{-5}{3} \]

Step 8: Check Your Solution

Substitute \(n = -\frac{5}{3}\) back into the original equation to verify that it satisfies the equation. Upon substitution and simplification, both sides of the equation equal, confirming our solution is correct.

Key Concepts

Common Denominator

Finding a common denominator is a crucial step when solving rational equations, especially when dealing with algebraic expressions. For instance, in the equation \( \frac{4n}{n-5} - \frac{2n}{n+5} = 2 \), each term of the fractions has different denominators, \( n-5 \) and \( n+5 \). We find the common denominator by multiplying these distinct denominators together, giving us \((n-5)(n+5)\).
This allows us to combine the fractions into a single rational expression since they share the same base. Common denominators help us:

• Simplify equations to make them easier to solve.
• Combine multiple fractions into a single fraction.
• Avoid errors when adding or subtracting fractions.

Once fractions share the same denominator, you can focus on simplifying their numerators. This foundation is key to moving forward in solving algebraic equations.

Quadratic Equation

A quadratic equation is a polynomial equation where the highest power of the variable is two. In this exercise, after clearing the fraction by multiplying the entire equation by the common denominator \((n-5)(n+5)\), we are led to a new equation: \[ 2n^2 + 30n = 2(n^2 - 25) \]This equation simplifies to a basic quadratic form: \[ 30n = -50 \]The quadratic nature involves several operations, depending on the context:

• Completing the square
• Factoring
• Using the quadratic formula

However, in this case, solving was straightforward as the terms \(2n^2\) on both sides cancel out. Quadratic equations appear frequently in algebra and can represent various physical phenomena, such as projectile motion.

Fraction Operations

When dealing with rational equations, understanding fraction operations is fundamental. The primary operations include:

• Addition and Subtraction: Fractions must share a common denominator to be directly added or subtracted. Otherwise, you'll need to find an equivalent form with a common base.
• Multiplication: Multiply the numerators together and the denominators together.
• Division: Flip the second fraction and multiply.

In the equation \(\frac{4n(n+5)}{(n-5)(n+5)} - \frac{2n(n-5)}{(n+5)(n-5)} = 2\), we utilize these operations:

• Combine terms by focusing on the numerators once the denominators are the same.
• Simplify the equation by clearing fractions, utilizing multiplication across the entire equation with the common denominator.

Mastering these operations empowers you to tackle a wide range of algebra problems.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations that collectively form terms and equations. In solving rational equations like this example, you encounter these expressions in both the numerators and denominators:

• \(4n(n+5)\) and \(2n(n-5)\): These expressions involve multiplication and can be expanded and simplified.
• Polynomials: Represent multiple terms, where the degree indicates the highest power of the variable.

Understanding how to navigate and manipulate algebraic expressions is key:

• Expanding terms: Helps in simplifying the entire equation.
• Combining like terms: Eases reduction into simpler forms for further operations.

This proficiency not only aids in rational equations but is fundamental across all algebraic problem-solving tasks.