Is the system represented by the equation $y(t) = x(2t)$ time invariant?
From your solution: 

I followed the following algorithm:

$$ y(t) =x(2t) $$
$$ y_1(t) = x_1(2t)$$

Let  $$x_2(t) = x_1(t-t_0)  ~~~\text{and}~~~ y_2(t) = x_2(2t) $$

On this following step (time sampling of the shifted argument) you make the usual mistake:

$$\implies y_2(t) = x_1(2(t - t_0))$$ which should instead be :
$$\implies y_2(t) = x_2(2t) = x_1(t - t_0)|_{t=2t} = x_1(2t - t_0)$$

The remaining parts follow as usual to show that the time scaler is a time-varying system...