In one of the example in my calculus textbook. It asks to find a parametric representation of the curve of intersection of the two surfaces $x^2+y+z=2$ and $xy+z=1$. And the solution goes like this.
By subtraction, we obtain $x^2+y-xy=1$.
let $x=t$, we have $y=(1-t^2)/(1-t)=1+t$
By substitution, we can obtain a parametrization of the curve $\vec r(t) = t\hat i + (1+t) \hat j + (1-t-t^2)\hat k$
However, I wonder why we do not have to handle the case $t=1$ where a division by $0$ would occur when expressing $y$ in terms of $t$.
It doesn't matter since in the original equation and the final one, that case is included. In other words note that when $x=t=0$ in both equations, we obtain that $y=1.$