I have the following system of simultaneous dot products in $\mathbb{R}^3$ which I am trying to solve for $x$:
\begin{eqnarray} x \cdot t & = & p \cdot t \\ x \cdot n & = & \frac{1}{k} + p \cdot n \\ x \cdot (k'n - k^2t - k\tau b) & = & p \cdot (k'n - k^2t - k\tau b) \end{eqnarray}
in which "$\cdot$" is the dot product, $(t, n, b)$ is an orthonormal basis of vectors, $p$ is a constant vector and $k, \tau$ are scalars (if it helps, this is a differential geometry problem taken from do Carmo, Exercise 3.3/10c with $(t, n, b)$ the moving trihedron and $k, \tau$ the curvature and torsion of a curve).
The solution is known to be
x = p + \frac{1}{k}n + \frac{k'}{k^2\tau}b
which I can verify but not derive... I tried a lot to coerce the above into the form $Ax = b$, but to no avail.
My question: how can the solution be derived from the given equations?