This confusion is due to the fact that the system of two equations I wrote in my comment above was originally crammed onto one line, rather than being separated onto two lines. Sorry! This formatting error made the comment hard to read, and has since been corrected. The correct system of equations to solve is:

p(sick1) x 0.42 + p(sick2) x 0.58 = 1 x 0.10

p(sick1) = p(sick2) x 0.60

Instead of:

.42* p(sick1) + .58*p(sick2) = .238

.1 p(sick1) + 1.38p(sick2) = .238.