Summary: In this paper we establish several equivalent approximate conditions for the implication involving vector-valued functions: \[ x\in C, \quad G(x)\in -S\Longrightarrow F(x)+y\notin -\overset{\circ}{K}, \tag{\(\alpha_y\)} \] where \(X\), \(Y\), \(Z\) are real locally convex Hausdorff topological vector spaces, \(S\) is nonempty convex cone in \(Z\), \(F:X\to Y^\bullet\), \(G:X\to Z^\bullet\) are proper mappings, \(\emptyset\ne C\subset X\), and \(y\) is a given element in \(Y\). Such a pair of \((\alpha_y)\) with its equivalent condition is called an approximate Farkas lemma corresponding to the system given in \((\alpha_y)\). These versions of approximate Farkas lemmas are then used to study optimality conditions, zero duality gap, and strong duality for the vector problem. In the case where \(Y=\mathbb{R}\), the results are even still new for scalar convex optimization without closedness and lower semi-continuity imposed on the data.