Parametrising $x+y+z=1$ subject to $x,y,z \ge 0$ with one variable $t$

Question

Title says it all. I am to find the contour integral

$$\oint_C \vec F(\vec r)\ \cdot d\vec r$$

For the surface ascribed in the title. To do this, I am to parametrise $x$, $y$, and $z$ as a function of $t$. Given the equation,

$$x=1-y-z$$ $$y=1-x-z$$ $$z=1-x-y$$

I don't see any way this can be parametrised with one variable.