Examining parametric equations

Question

By examining parametric equation of a line, how is it possible to determine by inspection whether the line is parallel to either the X-Axis or Y-Axis

Answer 1

if the line has the equation $$ax+by+c=0$$ we get $$x=-c/a$$ as a line which is parallel to the $y$ axes and $$y=-c/b$$ which is parallel to the $x$ axes

Answer 2

Yes. Write the parametric equation vectorially: if the straight line passes through the point $A$ and has directing vector $\vec u$, a parametric equation is $$M=A+t\,\vec u,\quad t\in\mathbf R,$$

and it is parallel to the $x$ axis (resp. the $y$ axis) if $\;\vec u\;$ has second coordinate (resp. first coordinate) equal to $0$.

Explicitly, this means $y$ is independent of $t$ (resp. $x$ is).