Answer
$y=\displaystyle \frac{2-x}{2x-1},\ \ x\displaystyle \lt \frac{1}{2}.$
Work Step by Step
From the parametric equation for x, x varies from $\displaystyle \frac{1}{2}$ (as t nears -1) to $-\infty $ (as t nears 1). So, $x\displaystyle \lt \frac{1}{2}.$
$x=\displaystyle \frac{t}{t-1}=\frac{t-1+1}{t-1}=1+\frac{1}{t-1}$
$x-1=\displaystyle \frac{1}{t-1}$
$t-1=\displaystyle \frac{1}{x-1}$
$t=\displaystyle \frac{1}{x-1}+1=\frac{1+x-1}{1-x}$
$t=\displaystyle \frac{x}{x-1}$
Substituting into the parametric equation for y,
$y=\displaystyle \frac{\frac{x}{x-1}-2}{\frac{x}{x-1}+1}=\frac{\frac{x-2(x-1)}{x-1}}{\frac{x+x-1}{x-1}}=\frac{2-x}{2x-1}$
$ \fbox{$y=\displaystyle \frac{2-x}{2x-1},\ \ x\lt \frac{1}{2}$}.$
To graph, create a table using several values for t, and then calculate $(x(t), y(t)),$ plotting the points as you go. Join with a smooth curve, noting the direction in which $t$ increases.
