To solve the equation [tex]\(\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}\)[/tex], we can follow these steps:
### Step 1: Factor the denominator
First, recognize that [tex]\(y^2 - 16\)[/tex] can be factored into [tex]\((y - 4)(y + 4)\)[/tex]. Rewrite the equation using this factorization:
[tex]\[
\frac{y}{y-4} - \frac{4}{y+4} = \frac{32}{(y-4)(y+4)}
\][/tex]
### Step 2: Find a common denominator
The common denominator of the fractions on the left side of the equation is [tex]\((y-4)(y+4)\)[/tex]. Express each fraction with this common denominator:
[tex]\[
\frac{y(y+4) - 4(y-4)}{(y-4)(y+4)} = \frac{32}{(y-4)(y+4)}
\][/tex]
### Step 3: Simplify the numerator
Simplify the numerator of the left-hand side:
[tex]\[
y(y+4) - 4(y-4) = y^2 + 4y - 4y + 16 = y^2 + 16
\][/tex]
Now, the equation becomes:
[tex]\[
\frac{y^2 + 16}{(y-4)(y+4)} = \frac{32}{(y-4)(y+4)}
\][/tex]
### Step 4: Set the numerators equal to each other
Since the denominators are the same, set the numerators equal to each other:
[tex]\[
y^2 + 16 = 32
\][/tex]
### Step 5: Solve for [tex]\(y\)[/tex]
Solve the equation [tex]\(y^2 + 16 = 32\)[/tex]:
[tex]\[
y^2 + 16 = 32 \implies y^2 = 32 - 16 \implies y^2 = 16 \implies y = \pm 4
\][/tex]
### Step 6: Consider the domain
Recall that the original equation has denominators [tex]\(y - 4\)[/tex] and [tex]\(y + 4\)[/tex], which cannot be zero. Therefore, [tex]\(y = 4\)[/tex] and [tex]\(y = -4\)[/tex] would make the denominators zero, which are not valid solutions in this context.
Therefore, there is no valid value of [tex]\(y\)[/tex] that satisfies the equation.
### Conclusion
The solution to the equation [tex]\(\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}\)[/tex] is no solution.
