Parametric equations, eliminating the parameter: $x = t/(t-1)$, $y = (t-2)/(t+1)$

Question

I need to convert this parametric equation into Cartesian

$$x = \frac{t}{t-1}$$ $$y = \frac{t-2}{t+1}$$

I'm trying to solve for $x$. I get to this step: $x(t-1) = t$

I've looked up the steps and solution to this problem and they go from $x(t-1) = t$ to $t = x/(x-1)$. I don't know how they did that. The full solution is $y = (2-x)/(2x-1)$.

Answer 1

$$x(t-1) = t \\ xt-x=t \\ xt-t=x \\ (x-1)t = x \\ t = \frac{x}{x-1}$$

Answer 2

Hint:

The second is

$y=1-\frac{3}{t+1}$

thus $t=\frac{3}{1-y}-2$ which we replace in the expression of $x$.