This may be what you want:
Simplify@Solve[{s == t s + t a + t b + 1, a == t c + t e,
b == t d + t f, c == t s + t b, d == t s + t a, f == t c,
e == t d}][[1]]
Strangely enough, this gives no answer:
Solve[{s == t s + t a + t b + 1, a == t c + t e, b == t d + t f,
c == t s + t b, d == t s + t a, f == t c, e == t d}, {a, b, c, d, e,
f}]