In this work, a kind of distributed constraintcoupled optimization problem is studied. We first propose an augmented primal-dual dynamics and study its convergence, but the need for knowing the violation of the coupled constraint prevents it from being implemented distributedly. By comprehending a classical continuous-time distributed optimization algorithm from a new perspective, the novel implicit tracking approach is proposed to track the violation distributedly, which leads to the implicit tracking-based distributed augmented primaldual dynamics (IDEA). To deal with the cases where local constrained sets exist, we further develop a projected variant of IDEA, i.e., Proj-IDEA. Under the undirected and connected communication topology, the convergence of IDEA can be guaranteed when local objective functions are only convex, and if local objective functions are strongly convex and smooth, IDEA can converge exponentially with a quite weak condition about the constraint matrix. In line with IDEA, Proj-IDEA can also guarantee convergence when local objective functions are only convex. Furthermore, we also explore the convergences of IDEA and Proj-IDEA under the directed, strongly connected, and weight-balanced communication topology. Finally, numerical experiments are taken to corroborate our theoretical results.