It turns out, not surprisingly, that I needed to make eqn1 and eqn2 symbolic objects, which I did by wrapping sym() around each one. Now everything works fine. Yippee!
eqn1 = sym((120*Lt*pi*Rt)/(14400*Lt^2*pi^2 + Rt^2)^(1/2) - 338920901889975/17592186044416)
eqn2 = sym((42000*Lt*pi*Rt)/(1764000000*Lt^2*pi^2 + Rt^2)^(1/2) - 8473022547249375/17592186044416)
S = solve([eqn1, eqn2])