Step-by-step explanation:
To solve the equation
\[
\frac{y}{y - 4} - \frac{4}{y + 4} = \frac{32}{y^2 - 16},
\]
we first recognize that \( y^2 - 16 \) can be factored using the difference of squares:
\[
y^2 - 16 = (y - 4)(y + 4).
\]
Thus, the equation becomes
\[
\frac{y}{y - 4} - \frac{4}{y + 4} = \frac{32}{(y - 4)(y + 4)}.
\]
To combine the fractions on the left-hand side, we need a common denominator. The common denominator is \((y - 4)(y + 4)\). We rewrite each term with this common denominator:
\[
\frac{y(y + 4)}{(y - 4)(y + 4)} - \frac{4(y - 4)}{(y - 4)(y + 4)} = \frac{32}{(y - 4)(y + 4)}.
\]
Simplify the numerators:
\[
\frac{y^2 + 4y}{(y - 4)(y + 4)} - \frac{4y - 16}{(y - 4)(y + 4)} = \frac{32}{(y - 4)(y + 4)}.
\]
Combine the fractions:
\[
\frac{y^2 + 4y - 4y + 16}{(y - 4)(y + 4)} = \frac{32}{(y - 4)(y + 4)}.
\]
This simplifies to:
\[
\frac{y^2 + 16}{(y - 4)(y + 4)} = \frac{32}{(y - 4)(y + 4)}.
\]
Since the denominators are the same, we equate the numerators:
\[
y^2 + 16 = 32.
\]
Solve for \(y\):
\[
y^2 + 16 = 32,
\]
\[
y^2 = 16,
\]
\[
y = \pm 4.
\]
However, we need to check for any restrictions. The original denominators \(y - 4\) and \(y + 4\) cannot be zero:
- \(y - 4 \neq 0 \implies y \neq 4\)
- \(y + 4 \neq 0 \implies y \neq -4\)
Thus, \(y = 4\) and \(y = -4\) are both invalid solutions. Therefore, the equation has no valid solutions.
\[
\boxed{\text{No solution}}
\]
