I have to determine if the system is linear, time-invariant, and/or causal
The input to this system is denoted by \$x(t)\$ and its output by \$y(t)\$.


To prove that the system is homogeneous, what you need to show is
that


  if input \$x(t)\$ produces output \$y(t)\$,
  then the output produced by input \$\alpha\cdot x(t)\$ is \$\alpha\cdot y(t)\$.


Here, \$\alpha\$ is an arbitrarily chosen real number, that is, the highlighted
statement must hold regardless of the value of \$\alpha\$.  Similarly,
the requirement must be satisfied regardless of the choice of input \$x(t)\$.
In other words, finding one pair of input and output signals \$x(t)\$ and \$y(t)\$ 
and one real number \$\alpha\$ for which the highlighted statement is true
is not sufficient; it has to hold for all such choices.
To prove that the system is additive, what you need to show is that


  if inputs \$x_1(t)\$ and \$x_2(t)\$ produce outputs \$y_1(t)\$
  and \$y_2(t)\$ respectively, 
  then the output produced by input \$x_1(t)+x_2(t)\$ is 
  \$y_1(t)+y_2(t)\$.


Once again, cherry-picking is not allowed; the statement has to hold
for all choices of input signals and corresponding output signals.


It is not too hard to verify that your system is both homogeneous
and additive.