Problem 16 - Elementary Algebra

Question

$$ \text { For Problems 1-32, solve each equation. (Objective 1) } $$ $$ \frac{t}{2 t-8}+\frac{16}{t^{2}-16}=\frac{1}{2} $$

Step-by-Step Solution

Answer

The solution is \( t = -12 \).

Step 1: Identify the Problem Type

We have a rational equation to solve, involving fractions with variable expressions in the denominators.

Step 2: Factor Denominators

Factor the quadratic denominator: \(t^2 - 16 = (t+4)(t-4)\). Notice that \(2t - 8\) can be factored as \(2(t-4)\).

Step 3: Find the Common Denominator

The least common denominator (LCD) for all fractions is \(2(t+4)(t-4)\) since it includes all factors from each denominator.

Step 4: Clear Fractions by Multiplying Through by LCD

Multiply every term by \(2(t+4)(t-4)\) to eliminate the denominators: \[ t \cdot (t+4) + 16 \cdot 2 = (1/2) \cdot 2(t+4)(t-4) \] simplifying to\[ t(t+4) + 32 = (t+4)(t-4) \]

Step 5: Expand and Simplify the Equation

Expand each side:\[ t^2 + 4t + 32 = t^2 - 16 \]Then simplify and combine like terms.

Step 6: Eliminate Identical Terms

Since \(t^2\) appears on both sides, subtract \(t^2\) from both sides to get:\[ 4t + 32 = -16 \]

Step 7: Solve for the Variable

Solve for \(t\) by first subtracting 32 from both sides to obtain:\[ 4t = -48 \]Then divide by 4:\[ t = -12 \]

Step 8: Verify the Solution

Substitute \(t = -12\) back into the original equation to ensure no zero division and that the equation balances. Check that the denominators do not become 0 and both sides of the equation equal when simplified.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is an essential step in simplifying many algebraic equations, including rational equations. To factor a quadratic expression like \( t^2 - 16 \), we need to recognize it as a difference of squares. A difference of squares has the form \( a^2 - b^2 \), which factors into \( (a + b)(a - b) \). Applying this to \( t^2 - 16 \), we identify \( t^2 \) as \( a^2 \) and \( 16 \) as \( 4^2 \), giving us:

• \( t^2 - 16 = (t + 4)(t - 4) \)

Sometimes, expressions aren't immediately in a factorable form and require simplification first. As seen with \( 2t - 8 \), we factor out the greatest common factor (GCF). The GCF of \( 2t - 8 \) is 2:

• \( 2t - 8 = 2(t - 4) \)

This simplification makes it easier to find common denominators later on.

Least Common Denominator

Finding the least common denominator (LCD) is crucial for solving rational equations. The LCD is the smallest expression that each denominator can divide without leaving a remainder. For the equation\[\frac{t}{2t-8} + \frac{16}{t^2-16} = \frac{1}{2},\]we start by factoring the denominators:

• \( 2t - 8 = 2(t - 4) \)
• \( t^2 - 16 = (t + 4)(t - 4) \)

The LCD must include all unique factors from each denominator, which here are \( 2 \), \( (t - 4) \), and \( (t + 4) \). Thus, the LCD is:

• \( 2(t + 4)(t - 4) \)

By expressing each fraction with this common denominator, we can more easily eliminate the fractions and solve the equation.

Solving Equations

Solving equations, particularly those involving fractions, often involves clearing fractions first. In this exercise, we achieve this by multiplying through by the least common denominator (LCD):

• Multiply every term by \( 2(t + 4)(t - 4) \)

This step effectively removes the fraction and simplifies the process of solving the equation:\[ t \cdot (t+4) + 16 \cdot 2 = \frac{1}{2} \cdot 2(t+4)(t-4) \]Simplifying gives:\[ t(t+4) + 32 = (t+4)(t-4) \]Next, expand and simplify each side:

• Left: \( t^2 + 4t + 32 \)
• Right: \( t^2 - 16 \)

Eliminate identical terms by subtracting \( t^2 \) from both sides:\[ 4t + 32 = -16 \]From there, continue solving for \( t \). This step-by-step process allows for finding the variable reliably and accurately.

Variable Expressions in Denominators

Variable expressions in denominators pose unique challenges in algebra because they introduce restrictions and the potential for undefined expressions. In this problem, the expressions \( 2t-8 \) and \( t^2-16 \) appear in the denominators. These expressions must never lead to division by zero, which defines the constraints of the equation. In mathematical terms, a denominator becomes zero when:

• \( 2t - 8 = 0 \) implies \( t = 4 \)
• \( t^2 - 16 = 0 \) implies \( t = 4 \) or \( t = -4 \)

Therefore, \( t \) must not equal 4 or -4, as these would make the original fractions undefined.
Understanding these restrictions is crucial when verifying solutions. Always recheck potential solutions to ensure they don't violate these constraints, thus confirming the solution's validity.