To solve this system of equations, we can start by solving the first equation for one of the variables. Let's solve for y in terms of x:

Xy = 10
y = 10/x

Now, we can substitute y = 10/x into the second equation:

(lg x)(lg(10/x)) = -2
(lg x)(lg 10 - lg x) = -2
(lg x)(1 - lg x) = -2
lg x - (lg x)^2 = -2
(lg x)^2 - lg x - 2 = 0

Let's make a substitution to make the equation simpler. Let p = lg x:

p^2 - p - 2 = 0
(p - 2)(p + 1) = 0

From this, we get two possible solutions for p:

p = 2 or p = -1

Now, we need to find the corresponding values for x using the property of logarithms:

lg x = 2 or lg x = -1

x = 10^2 or x = 10^-1
x = 100 or x = 0.1

Now that we have found possible values for x, we can find the corresponding values for y using y = 10/x:

When x = 100: y = 10/100 = 0.1
When x = 0.1: y = 10/0.1 = 100

Therefore, the solutions to the system of equations are x = 100, y = 0.1 and x = 0.1, y = 100.