To solve this system of equations, we can first solve for one variable in terms of the other in the first equation and substitute it into the second equation.

From the first equation:
x + y = π/4
=> y = π/4 - x

Substitute y in terms of x into the second equation:
tan(x) - tan(π/4 - x) = 1/6

tan(x) - (tan(π/4)tan(x) + tan(x)tan(π/4))/(1 + tan(π/4)tan(x)) = 1/6

tan(x) - (1tan(x) + tan(x)1)/(1 + 1*tan(x)) = 1/6

tan(x) - (2tan(x))/(1 + tan(x)) = 1/6

Multiply both sides by (1 + tan(x)):
(1 + tan(x))(tan(x)) - 2tan(x) = (1/6)(1 + tan(x))

Expand:
tan(x) + tan^2(x) - 2tan(x) = 1/6 + 1/6tan(x)

Combine like terms:
tan^2(x) - tan(x) - 1/6 = 1/6tan(x) + 1/6

Rearrange the equation and simplify:
tan^2(x) - 7/6tan(x) - 1/3 = 0

This is a quadratic equation in terms of tan(x). To solve for tan(x), we can use the quadratic formula:

tan(x) = [7/6 ± sqrt((7/6)^2 + 4*(1/3))]/2

tan(x) = [7/6 ± sqrt(49/36 + 4/3)]/2
tan(x) = [7/6 ± sqrt(49/36 + 48/36)]/2
tan(x) = [7/6 ± sqrt(97/36)]/2
tan(x) = (7 ± sqrt(97))/12

Therefore, the solutions for tan(x) in this system of equations is (7 + sqrt(97))/12 and (7 - sqrt(97))/12.