= Solution
In the nonsingular case the unique solution is
$$
\boxed{\mathbf x=\mathbf a+\frac{\mathbf b\cdot\mathbf a}{1-\mathbf b\cdot\mathbf c}\,\mathbf c,
\qquad \mathbf b\cdot\mathbf c\ne1}.
$$
Geometrically this is a single selected point of the <straight line> through $\mathbf a$ parallel to $\mathbf c$.
When $\mathbf b\cdot\mathbf c=1$ and $\mathbf b\cdot\mathbf a=0$, the solution set is that entire <straight line>, $\mathbf a+\lambda\mathbf c$. The remaining case has empty solution set. In <matrix> terms the equation is $(I-\mathbf c\mathbf b^T)\mathbf x=\mathbf a$, a <rank-one matrix update> of the <identity matrix>.
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