Problem 23 - Algebra for College Students

Question

Solve each equation. $$ \frac{2 n}{6 n^{2}+7 n-3}-\frac{n-3}{3 n^{2}+11 n-4}=\frac{5}{2 n^{2}+11 n+12} $$

Step-by-Step Solution

Answer

The solution is \(n = \frac{7}{2}\).

Step 1: Factor Denominators

First, factor each of the denominators. For the first denominator, \(6n^2 + 7n - 3\), factor it to get \[(3n - 1)(2n + 3)\]For the second denominator, \(3n^2 + 11n - 4\), factor it to get \[(3n - 1)(n + 4)\]For the third denominator, \(2n^2 + 11n +12\), factor it to get \[(2n + 3)(n + 4)\].

Step 2: Identify the Least Common Denominator (LCD)

The least common denominator is the product of all unique factors:\((3n - 1)(2n + 3)(n + 4)\).

Step 3: Rewriting Each Fraction with the LCD

Rewrite each fraction so that they all have the LCD as their denominator.For \(\frac{2n}{(3n-1)(2n+3)}\), multiply both numerator and denominator by \((n+4)\) to get:\[\frac{2n(n+4)}{(3n-1)(2n+3)(n+4)}\]For \(\frac{n-3}{(3n-1)(n+4)}\), multiply both numerator and denominator by \((2n+3)\) to get:\[\frac{(n-3)(2n+3)}{(3n-1)(2n+3)(n+4)}\]For \(\frac{5}{(2n+3)(n+4)}\), multiply both numerator and denominator by \((3n-1)\) to get:\[\frac{5(3n-1)}{(3n-1)(2n+3)(n+4)}\].

Step 4: Combine Terms and Eliminate the Denominator

Combine the fractions into a single equation using the common denominator:\[\frac{2n(n+4) - (n-3)(2n+3)}{(3n-1)(2n+3)(n+4)} = \frac{5(3n-1)}{(3n-1)(2n+3)(n+4)}\]Since the denominators are the same, we can eliminate them and equate the numerators:\[2n(n+4) - (n-3)(2n+3) = 5(3n-1)\]

Step 5: Simplify and Solve for \(n\)

Expand and simplify the left side:\[2n^2 + 8n - (2n^2 + 3n - 6n - 9) = 5(3n - 1)\]This simplifies to:\[2n^2 + 8n - 2n^2 - 3n + 6n + 9 = 15n - 5\]Combine like terms:\[11n + 9 = 15n - 5\]Rearrange to solve for \(n\):\[4n = 14\]\[n = \frac{14}{4}\]Simplify the fraction to get:\[n = \frac{7}{2}\]

Step 6: Verify the Solution

Substitute \(n = \frac{7}{2}\) back into the original equation to verify that both sides are equal. After substitution and simplification, both sides should match, confirming \(n = \frac{7}{2}\) is the correct solution.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in solving rational equations. It involves breaking down complex polynomial expressions into simpler factors that can be easily managed. Polynomials are expressions like \(6n^2 + 7n - 3\) or \(3n^2 + 11n - 4\).

• Factoring helps identify common factors across different terms, which simplifies the equation.
• In our example, you need to know how to factor quadratics. For instance, the polynomial \(6n^2 + 7n - 3\) can be factored into \((3n - 1)(2n + 3)\).

To factor a quadratic polynomial, look for two numbers that multiply to give you the constant term (the last number) and add up to the linear coefficient (the middle number). Factoring makes finding the least common denominator more efficient because it simplifies the expression's components.

Least Common Denominator

The least common denominator (LCD) is a shared multiple of all the denominators in a rational equation. Finding the LCD is vital because it allows you to combine fractions by giving them a common denominator.

• To determine the LCD, list all the unique factors from each denominator.
• In the provided exercise, the LCD of the denominators \((3n - 1)(2n + 3)\), \((3n - 1)(n + 4)\), and \((2n + 3)(n + 4)\) is \((3n - 1)(2n + 3)(n + 4)\).

The LCD serves as a universal denominator, ensuring all terms within the equation align, which is essential for correctly performing algebraic operations. Ensuring each fraction shares the LCD simplifies the combination of fractions into a single expression.

Combining Fractions

Combining fractions means rewriting each fraction so they have a common denominator. This concept allows you to subtract, add, or compare fractions effectively.

• Use the LCD to adjust each fraction's numerator and denominator. Multiply the numerator and the denominator by factors that achieve this common denominator.
• For example, for \(\frac{2n}{(3n - 1)(2n + 3)}\), multiply both by \((n + 4)\) to adjust the fraction. Similarly, modify the other fractions using appropriate factors.

Once all fractions share the LCD, you can combine their numerators into one fraction. After combining, you can solve for the unknown variable by focusing on the numerators alone since the denominators are identical.

Polynomial Simplification

Polynomial simplification in rational equations involves reducing expressions to their simplest form by eliminating like terms and performing algebraic operations.

• Start by expanding all expressions, distributing terms appropriately.
• In this exercise, the expression \(2n(n+4) - (n-3)(2n+3)\) becomes simplified by expanding and combining like terms.

Simplification helps isolate the variable for solving. By equating numerators when the fractions have the same denominator, you simplify the expressions on both sides of the equation. This enables easier identification of solutions, like solving \(11n + 9 = 15n - 5\) to find \(n\). Proper simplification ensures accuracy and clarity, making it easier to solve the entire rational equation.