Problem 15 - Elementary Algebra

Question

$$ \text { For Problems 1-32, solve each equation. (Objective 1) } $$ $$ \frac{3}{t^{2}-4}+\frac{5}{t+2}=\frac{2}{t-2} $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{11}{3} \).

Step 1: Identify the Denominators

Examine the given equation: \( \frac{3}{t^2-4} + \frac{5}{t+2} = \frac{2}{t-2} \). The denominators are \( t^2-4 \), \( t+2 \), and \( t-2 \). Notice that \( t^2-4 = (t+2)(t-2) \). This means the common denominator is \( (t+2)(t-2) \).

Step 2: Clear the Fractions

Multiply every term in the equation by the common denominator \( (t+2)(t-2) \) to eliminate the fractions: \( (t+2)(t-2) \cdot \frac{3}{t^2-4} + (t+2)(t-2) \cdot \frac{5}{t+2} = (t+2)(t-2) \cdot \frac{2}{t-2} \) This simplifies to: \( 3 + 5(t-2) = 2(t+2) \)

Step 3: Simplify the Equation

Expand and simplify both sides: \( 3 + 5t - 10 = 2t + 4 \) This results in: \( 5t - 7 = 2t + 4 \)

Step 4: Solve for t

Isolate \( t \) by first subtracting \( 2t \) from both sides: \( 5t - 2t - 7 = 4 \) Simplify further: \( 3t - 7 = 4 \) Add 7 to both sides: \( 3t = 11 \) Finally, divide by 3: \( t = \frac{11}{3} \)

Step 5: Check for Extraneous Solutions

Substitute \( t = \frac{11}{3} \) back into the original denominators to ensure no division by zero occurs. The denominators are \( t^2-4 \), \( t+2 \), and \( t-2 \). With \( t = \frac{11}{3} \), none of these expressions equals zero, confirming the solution is valid.

Key Concepts

Rational Equations

Rational equations are equations that involve at least one rational expression. A rational expression is essentially a fraction in which the numerator and/or the denominator are polynomials. These types of equations require specific approaches to solve because of their fractional components.

Working with rational equations involves the following steps:

• Identify and isolate the rational expressions in the equation.
• Aim to eliminate the fractions to simplify the equation, often through finding a common denominator.
• Ensure that the solution you find doesn't result in division by zero, which would render the equation undefined.

Rational equations are commonly encountered in algebra problems, and mastering them is crucial as they form the basis for more advanced mathematical topics.

Common Denominators

In solving rational equations, a key step is finding a common denominator. This allows us to eliminate the fractions so that we can solve the equation more straightforwardly.

A common denominator is essentially a shared multiple of the denominators in your equation. In the provided exercise, the denominators are \( t^2-4 \), \( t+2 \), and \( t-2 \). We recognize that \( t^2-4 \) is the same as \((t+2)(t-2)\), so that product becomes our common denominator.

Once the common denominator is identified:

• Multiply every term in the equation by this common denominator to clear the fractions.
• This transforms the rational equation into a simpler form, often a linear or quadratic equation, that can be solved with basic algebraic techniques.

Utilizing a common denominator is a powerful method for turning complex fractional equations into manageable standard equations.

Solving Equations Steps

Once the fractions in a rational equation are cleared, the task is to solve the resulting equation. Following consistent steps can make this process efficient and accurate.

Here’s a typical approach:

• Simplify Both Sides: Expand and simplify each side of the equation to make it easier to work with.
• Isolate the Variable: Use algebraic operations to gather all terms involving the variable on one side and constant terms on the other.
• Solve for the Variable: Complete the algebraic manipulations to find the value of the variable.
• Check for Extraneous Solutions: Substitute your solution back into the original fraught equation fractions to ensure it results in non-zero denominators.

In the sample problem, these steps led to finding \( t = \frac{11}{3} \), and verifying that this solution does not make any denominator zero confirms the solution's validity.