Questions: Solve the following equation. 1/(y-5)+1/20=-5/(4y-20) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a fraction. Use a comma to separate answers as needed.) B. The solution set is y y is a real number, y ≠ . (Type an integer or a fraction. Use a comma to separate answers as needed.) C. The solution set is ∅.

Solution Steps

To solve the given equation, we need to find the value of \( y \) that satisfies the equation. Here are the high-level steps:

1. Identify a common denominator for the fractions on both sides of the equation.
2. Combine the fractions on the left-hand side.
3. Simplify the equation to isolate \( y \).
4. Solve for \( y \) and check for any restrictions or extraneous solutions.

We start with the equation: \[ \frac{1}{y-5} + \frac{1}{20} = \frac{-5}{4y - 20} \]

To solve the equation, we first find a common denominator for the left-hand side. The common denominator is \(20(y-5)\). Rewriting the left-hand side gives: \[ \frac{20 + (y-5)}{20(y-5)} = \frac{-5}{4y - 20} \]

This simplifies to: \[ \frac{y + 15}{20(y-5)} = \frac{-5}{4(y-5)} \] Cross-multiplying leads to: \[ 4(y + 15) = -100 \]

Expanding and simplifying gives: \[ 4y + 60 = -100 \implies 4y = -160 \implies y = -40 \]

Final Answer