ANSWER
No solution
EXPLANATION
[tex] \frac{y}{y - 4} + \frac{y}{y - 4} = \frac{32}{ {y}^{2} - 16} [/tex]
The least common multiple is
[tex] {y}^{2} - 16 = (y - 4)(y + 4)[/tex]
We multiply through by the LCM to obtain,
[tex](y - 4)(y + 4) \times \frac{y}{y - 4} + (y - 4)(y + 4) \times \frac{y}{y - 4} = \frac{32}{ {y}^{2} - 16} \times (y - 4)(y + 4)[/tex]
We cancel out to get,
[tex]y(y + 4) + y(y - 4) = 32[/tex]
We expand to get,
[tex] {y}^{2} + 4y + {y}^{2} - 4y = 32[/tex]
We now group like terms to obtain,
[tex] {y}^{2} + {y}^{2} + 4y - 4y = 32[/tex]
We simplify to get,
[tex]2 {y}^{2} + 0 = 32[/tex]
This gives,
[tex]2 {y}^{2} = 32[/tex]
We divide through by 2.
[tex]y ^{2} = 16[/tex]
Taking square root of both sides gives,
[tex]y = \pm \sqrt{16} [/tex]
[tex]y = \pm4[/tex]
This implies that,
[tex]y = 4 \: or \: y = - 4[/tex]
But the above solutions are not within the domain of the function, which is
[tex]y \ne4 \: or \: y \ne - 4[/tex]
Therefore the equation has no solution.
