Suppose we have the following linear system $$\begin{bmatrix} a_1^1 & \ldots & a_1^{N+1}\\ \vdots & \ddots & \vdots \\ a_{N-1}^1 & \ldots & a_{N-1}^{N+1} \\ 1 & \ldots & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ \vdots \\ \vdots \\ x_{N+1} \end{bmatrix}=\begin{bmatrix} b_1\\ \vdots \\ b_{N-1} \\ 1 \end{bmatrix},$$ where $a_n^i$ and $b_n$ are all known constants that are strictly positive for $n=1,\ldots,N-1$ and $i=1,\ldots,N+1$.
Is there a general form of the solution to the above system such that $$\begin{bmatrix} x_1 \\ \vdots \\ \vdots \\ x_{N+1} \end{bmatrix}=v_1+\alpha v_2,$$ where $v_1$ and $v_2$ are fixed vectors in $\mathbb{R^{N+1}}$ and $\alpha$ is a free variable in $\mathbb{R}$?
Thanks in advance for any help!
Edit From Gerry Myerson's reply, I realized that we need to add the following constraints to make sure that the system has a solution $$0 < a_n^{N+1} < a_n^N < \ldots < a_n^1$$ for all $n=1,\ldots,N-1$. Also, $x_{i}$ is strictly positive for $i=1,\ldots,N+1$.
The system might not have any solutions, e.g., $$\pmatrix{1&1&1\cr1&1&1\cr}\pmatrix{x_1\cr x_2\cr x_3\cr}=\pmatrix{2\cr1\cr}$$
EDIT: Despite the edit to the question, the system still might have no solutions, e.g., $$\pmatrix{4&3&2&1\cr4&3&2&1\cr1&1&1&1\cr}\pmatrix{x_1\cr x_2\cr x_3\cr x_4\cr}=\pmatrix{2\cr1\cr1\cr}$$