Basic stuff but I'm not sure where I'm going wrong. Much appreciated if someone could check through my working to see where I misstep!
$$x(t)=t+\frac{1}{t}, y(t)=1-\frac{1}{t}$$
So: $xt-1=t^2\,\,\,\,(1)$ $\,\,\,\,\,1-yt=t \,\,\,(2)$
$\implies xt-1=1-2yt+(yt)^2$
$t(x+2y)-2=(yt)^2 \implies x+2y=yt+\frac{2}{t}$ Note: $\frac{1}{t}=x-t$ so
$x+2y=yt+2(x-t)=t(y-2)+2x$
$$\therefore t=\frac{2y-x}{y-2}$$
Sub into $(2)$ to get: $1-y(\frac{2y-x}{y-2})=\frac{2y-x}{y-2}$.
simplifying the expression is easy from there, but plotting the parametric curve at this point and comparing it to the plot of the above cartesian expression, I'm obviously wrong at this point. My question: where's the mistake(s)? Apologies for messy working.
$$y = 1 - \frac{1}{t} \implies yt= t - 1 \implies t-yt=1$$ so what you labelled as (2) is incorrect.