Consider the following linear programming (hereafter, problem [1]) $$ \max_{y\in \mathbb{R}^J}c'y\\ \text{s.t. } b_t' y \leq a_t \text{ }\forall t\in \{1,...,T\} $$ where $c$ is a $J\times 1$ vector of reals, $y$ is a $J\times 1$ vector of unknowns, $b_t$ is a $J\times 1$ vector of reals, $a_t$ is a real scalar.
It can be shown that this problem is equivalent to problem [2] $$ \min_{x\in \mathbb{R}^T} x'a\\ \text{s.t. } x\geq 0_T\\ \sum_{t=1}^T x_t b_t=c $$ where $a\equiv (a_1,...,a_T)$, $x$ is a $T\times 1$ vector of unknowns, $0_T$ is the $T\times 1$ vector of zeros, $x_t$ is the $t$-th element of $x$.
I'm trying to understand under which conditions on the elements of [1] we can ensure uniqueness of the solution, $x$, of [2]. The hint of the exercise in the book is "constraint qualification". I'm a beginner and I can't find any simple explanation online on what it means and how I can apply it here. Could you help?