Easier without algebra

As $x,y \ne 0$ , divide the first equation through by $xy$ to find that $\dfrac{1}{y} - \dfrac{1}{x} = 1$ . Add this to the second equation to find that

$\left(\dfrac{1}{y} - \dfrac{1}{x}\right) + \left(\dfrac{1}{x} + \dfrac{1}{y}\right) = 1 + 5 \Longrightarrow \dfrac{2}{y} = 6 \Longrightarrow \dfrac{1}{y} = \boxed{3}$ .

Given

1) $x - y = xy$

2) $1/x + 1/y = 5$

From 2),

$(x + y)/xy = 5$

with 1)

$x + y = 5xy = 5(x - y) = 5x - 5y$

$6y = 4x$

$y = 2x/3$

Substituting into 1)

$x - (2x/3) = x(2x/3)$

Since x $\neq$ 0,

$1 - 2/3 = 2x/3$

$3 - 2 = 2x$

$x = 1/2$

$y = 2(1/2)/3 = 1/3$

$1/y = 3$