In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let Λ = b 2 log α 2 - b 1 log α 1 - b 3 log α 3 ≠ 0 with b 1 , b 2 , b 3 positive integers and α 1 , α 2 , α 3 positive multiplicatively independent rational numbers greater than 1 . Let α j 1 = α j 1 / α j 2 with α j 1 , α j 2 coprime positive integers ( j = 1 , 2 , 3 ) . Let α j ≥ max { α j 1 , e } and assume that gcd ( b 1 , b 2 , b 3 ) = 1 . Let b ' = b 2 log α 1 + b 1 log α 2 b 2 log α 3 + b 3 log α 2 and assume that B ≥ max { 10 , log b ' } . We prove that either { b 1 , b 2 , b 3 } is c 4 , B -linearly dependent over ℤ (with respect to a 1 , a 2 , a 3 ) or Λ > exp - C B 2 ∏ j = 1 3 log a j , where c 4 and C = c 1 c 2 log ρ + δ are given in the tables of Section 6. Here b 1 , b 2 , b 3 are said to be ( c , B ) -linearly dependent over ℤ if d 1 b 1 + d 2 b 2 + d 3 b 3 = 0 for some d 1 , d 2 , d 3 ∈ ℤ not all 0 with either (i) 0 < | d 2 | ≤ c B log a 2 min { log a 1 , log a 3 } , | d 1 | , | d 3 | ≤ c B log a 1 , log a 3 , or (ii) d 2 = 0 and | d 1 | ≤ c B log a 1 log a 2 and | d 3 | ≤ c B log a 2 log a 3 . In particular, we obtain c 4 < 9146 and C < 422 , 321 for all values of B ≥ 10 , and for B ≥ 100 we have c 4 ≤ 5572 and C ≤ 260 , 690 . . More complete information is given in the tables in Section 6. We prove this theorem by modifying the methods of P. Philippon, M. Waldschmidt, G. Wüstholz, et al. In particular, using a combinatorial argument, we prove that either a certain algebraic variety has dimension 0 or b 1 , b 2 , b 3 are linearly dependent over ℤ where the dependence has small coefficients. This allows us to improve Philippon’s zero estimate, leading to the interpolation determinant being non-zero under weaker conditions.