Finding a vector function for the curve of intersection of two surfaces

Question

I wanted to find a vector function $\mathbb{r}(t)$ for the curve of intersection between $z=\sqrt{x^2+y^2}$ and $z = y+5$. I understand that one way would be to let $x=t$ and then we can set $ \sqrt{x^2+y^2} = y + 5$ to get a function $y = f(x)$. Is letting $x=t$ my only choice of parameterization? Is letting $y = t$ valid? What about $z = t$? Why or why not? My concern with letting $y =t$ is that when we equate the two surfaces, solving for $x$ does not yield a function of $y$ since a square root is involved.