I want to numerize each equation in my system by my own numbering , also for the system

documentclass[12pt,a4paper]{article}

usepackage{amsmath}
usepackage{cases}

begin{document}

Half-hearted recommendation
begin{subnumcases} {label{system}}
  x-y = 1label{x}
  2,x-y=3 label{y}
end{subnumcases}
Provide the solution to equation (ref{system}) subtracting
(ref{x}) from (ref{y}).

But if you must have three separate numbers, you could put them in by
hand where cases expects a qualification
begin{equation}
label{system2}
  begin{cases} 
  x-y = 1 &qquad text{(a)}
  2,x-y=3&qquad text{(b)}
  end{cases}
end{equation}
But those braces don't seem appropriate at all.
begin{equation}
label{system3}
begin{aligned}
   x-y &= 1 &qquad& text{(a)}
 2,x-y&= 3 &qquad& text{(b)}
end{aligned}
end{equation}
but probably best is this
begin{subequations}
label{system4}
  begin{align}
   x-y &= 1 label{xx}
 2,x-y&= 3 label{yy}
end{align}
end{subequations}
combined eq.~(ref{system4}).
What it sounds like you are asking for is this atrocity
begin{align}
  x-y &= 1 tag{a}
      &           
 2,x-y&= 3 tag{b}
end{align}
which looks awful.

end{document}