If x + (5)/(y) = 4 and y + (5)/(x) = -4 , then what is (x + y)...

Given equations are $x + \dfrac{5}{y} = 4$ and $y + \dfrac{5}{x} = -4$.

Adding the two equations, we get:

$x + y + \dfrac{5}{x} + \dfrac{5}{y} = 0$

$(x + y) + 5\left(\dfrac{x + y}{xy}\right) = 0$

$(x + y)\left(1 + \dfrac{5}{xy}\right) = 0$

This implies either $x + y = 0$ or $1 + \dfrac{5}{xy} = 0$.

Subtracting the second equation from the first, we get:

$x - y - \dfrac{5}{x} + \dfrac{5}{y} = 8$

$(x - y) + 5\left(\dfrac{x - y}{xy}\right) = 8$

$(x - y)\left(1 + \dfrac{5}{xy}\right) = 8$

If $1 + \dfrac{5}{xy} = 0$, the second factor becomes zero, which gives $0 = 8$, a contradiction. Thus, $1 + \dfrac{5}{xy} \neq 0$.

Therefore, we must have $x + y = 0$.

Answer: $0$