It is known that both sides of the equation are divided by xy, and x/y+4*y/x-z/xy=3. Because x/y+4*y/x≥2√4=4, z/xy≥ 1, xy/z≤ 1.

When xy/z takes the maximum value, xy/z= 1, x 2-4xy+4y 2 = 0, and x=2y, so 2/x+1/y+2/z = 2/y+1/y.

There are many methods, for example, let t = 2/y+1/y 2, then t * y 2-2y+1= 0, and get the discriminant △=4-4t≥0, t≤ 1. In addition, when t= 1, we can find x = z = 2 and y = 1.