For this one, we solve the first equation alone without the second equation.

Since log x y and log y x are reciprocals,
Let log x y = u

2u + 2/u = 5
2u2 + 2 = 5u
2u2 - 5u + 2 = 0
(2u - 1) (u - 2) = 0
u = 1/2 or u = 2

This means that

log x y = 1/2 or log x y = 2

Converting to index,

x^1/2 = y or x^2 = y
x = y^2 or x^2 = y

For each case, we solve the equation together with the other equation xy = 8.

For x = y^2,
y^2 * y = 8
y^3 = 8
y = 2
x = 2^2 = 4

For x^2 = y,
x * x^2 = 8
x^3 = 8
x = 2
y = 2^2 = 4

The solutions sets are x = 4, y = 2 or x = 2, y = 4 (which are expected to be symmetrical since the equations are symmetrical)