<s>[INST] Is it possible to solve for $3$ or more unknowns with only $1$ equation Example: 
$$x+3y+7z=221$$
or
$$2x+5y+z+9i=383$$
 [/INST] If the only thing you have is the constraint $$x + 3y+ 7z = 221$$ then you know that $x = 221-3y-7z$ and so all solutions to your problem can be characterized as
$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix}
= \begin{pmatrix} 221-3y-7z \\ y \\ z \end{pmatrix}
= \begin{pmatrix} 221 \\ 0 \\ 0 \end{pmatrix}
+ y\begin{pmatrix} -3 \\ 1 \\ 0 \end{pmatrix}
+ z\begin{pmatrix} -7 \\ 0 \\ 1 \end{pmatrix}
$$
</s>