Write the plane $y + 2z = 1$ in a parametric form

Question

I'm asked to write the plane $$y + 2z = 1$$ in a parametric form.

I know how to do this with variables like $$ax + by + cz = d$$ but the lack of an $x$-variable stumps me. How should I go about doing it?

Answer 1

It means $x$ can take any value you want.

One possible parametrisation is $(t,2u,\frac{1}{2}-u)$

Answer 2

Since $x$ does not appear in the equation $y+2z=1$, $x$ is simply independent of the values of $y$ and the corresponding $z$. We can thus use any other variables to parametrize $x$. For instance, $t:=x$.

Then we first parametrize $y$ or $z$. Here, we let $u=y$. Since $y$ and $z$ are dependent of each other, $z$ can be parametrize in terms of $u$, which is $\displaystyle z=\frac{1-u}2$.

Thus the possible parametrization is $$(t,u,\frac{1-u}2), t,u\in\Bbb{R}$$