what is the solution of $\frac{-8}{2y - 8}=\frac{5}{y + 4}-\frac{7y + 8}{y^{2}-16}$?\n$y=-4$\n$y=-2$\n$y =…

Explanation:

Step1: Factor the denominators

First, factor $2y - 8=2(y - 4)$ and $y^{2}-16=(y + 4)(y - 4)$. The common - denominator of the right - hand side is $2(y + 4)(y - 4)$.

Step2: Rewrite the equation with a common denominator

Rewrite the equation $\frac{-8}{2(y - 4)}=\frac{5}{y + 4}-\frac{7y + 8}{(y + 4)(y - 4)}$. Multiply each term by $2(y + 4)(y - 4)$: [ \begin{align*} \frac{-8}{2(y - 4)}\times2(y + 4)(y - 4)&=\frac{5}{y + 4}\times2(y + 4)(y - 4)-\frac{7y + 8}{(y + 4)(y - 4)}\times2(y + 4)(y - 4)\ -8(y + 4)&=5\times2(y - 4)-2(7y + 8) \end{align*} ]

Step3: Expand the expressions

Expand the left - hand side: $-8(y + 4)=-8y-32$. Expand the right - hand side: $5\times2(y - 4)=10y-40$ and $2(7y + 8)=14y + 16$. So the equation becomes $-8y-32=10y-40-(14y + 16)$.

Step4: Simplify the right - hand side

$-8y-32=10y-40 - 14y-16$. Combine like terms on the right - hand side: $10y-14y-40-16=-4y-56$. So the equation is $-8y-32=-4y-56$.

Step5: Solve for y

Add $8y$ to both sides: $-32 = 4y-56$. Add 56 to both sides: $4y=24$. Divide both sides by 4: $y = 6$.

Answer:

$y = 6$