A system of linear equations Consider a system of linear equations $Ax=b$. Can it be that for a generic choice of $A$ and $b$, elements of $x$ are distinct? 
 A: If I understand correctly, you're asking if the set $G$ of pairs $(A,b) \subset \mathbb{R}^{n\times n} \times \mathbb{R}^{n}$ such that the coordinates of a solution $x = (x_1, \ldots, x_{n})$ of $Ax = b$ are necessarily pairwise distinct can be called generic.
If that is indeed the question, the answer is yes and I hope I haven't goofed below.
In fact, $G$ is the open and dense subset of $\mathbb{R}^{n\times n} \times \mathbb{R}^{n}$ consisting of pairs $(A,b)$ such that $\det{A} \neq 0$ and $b \notin A(\Delta)$, where $\Delta = \{x \in \mathbb{R}^{n}\,:\,\exists i\neq j \;\text{such that}\; x_i = x_j\}$ is the large diagonal of $\mathbb{R}^{n}$. If $\det{A} = 0$ there is no solution if $b \neq 0$ and otherwise $x = 0$ is a solution.
To see that $G$ is open, observe that it is the preimage of the open set $\mathbb{R}^{n} \smallsetminus \Delta$ under the continuous map $(A,b) \mapsto A^{-1}b$ defined on the open subset $\operatorname{GL}(n,\mathbb{R}) \times \mathbb{R}^{n}$ of $\mathbb{R}^{n \times n} \times \mathbb{R}^n$. To see that $G$ is dense, let $(A,b) \in \mathbb{R}^{n \times n} \times \mathbb{R}^{n}$ be arbitrary. Choose $A_{k} \in \operatorname{GL}(n,\mathbb{R})$ such that $A_{k} \to A$. Since $\mathbb{R}^{n} \smallsetminus A_{k}(\Delta)$ is open and dense, we know that $B = \bigcap_{k} (\mathbb{R}^{n} \smallsetminus A_{k}(\Delta))$ is dense in $\mathbb{R}^{n}$ by the Baire category theorem, so we may choose $b_{k} \in B$ such that $b_{k} \to b$. Obviously, $(A_k, b_k) \in G$ and $(A_k, b_k) \to (A,b)$.
In other words, "the solution $(x_1, \cdots, x_n)$ of a generic system of $n$ linear equations $Ax = b$ in $n$ variables over $\mathbb{R}$ has pairwise distinct coordinates".
