What is the solution to the equation $\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}$?
The objective is to find the solution(s) for the rational equation \[\frac{y}{y-4} - \frac{4}{y+4} = \frac{32}{y^2-16}.\] First, we analyze the constraints on the variable \(y\). The denominators are \(y-4\), \(y+4\), and \(y^2-16\). Since the division by zero is undefined, we must have \(y-4 eq 0\) and \(y+4 eq 0\). This implies that \(y eq 4\) and \(y eq -4\). We also observe that \(y^2-16 = (y-4)(y+4)\), which is consistent with our domain constraints. Thus, the solution set must exclude \(\pm 4\).
To solve the equation, we find a common denominator for the fractions on the left side, which is \((y-4)(y+4) = y^2-16\). Multiplying the entire equation by this least common denominator, we get: \[(y-4)(y+4) \left( \frac{y}{y-4} - \frac{4}{y+4} \right) = (y-4)(y+4) \left( \frac{32}{y^2-16} \right).\] This simplifies to: \[y(y+4) - 4(y-4) = 32.\] Expanding the terms, we obtain: \[y^2 + 4y - 4y + 16 = 32,\] which further simplifies to: \[y^2 + 16 = 32.\] Subtracting 16 from both sides yields \(y^2 = 16\), so the potential solutions are \(y = 4\) and \(y = -4\).
Finally, we check the potential solutions against our initial domain constraints. We determined that \(y eq 4\) and \(y eq -4\) to ensure the expressions in the original equation are defined. Since our potential solutions are exactly the values that make the denominators zero, they are extraneous. Therefore, there are no valid solutions that satisfy the original equation. Looking at the provided options, the correct conclusion is that there is "no solution".
Rational equations often appear in physics, such as calculating the effective resistance of circuits or the time taken for two objects to meet when moving at different speeds. For example, if you are modeling the time \(t\) it takes to fill a pool using two pipes, you might encounter an equation like \(\frac{1}{t-2} + \frac{1}{t+3} = \frac{1}{5}\). Solving such equations helps ensure that the calculated time or physical quantity is mathematically consistent and within the real-world limits defined by the system's setup.