hmm...I'm not quite sure what you are asking here:
I have an example in my workbook that might help you:
Solve the linear system in the variables x, y ,z which has the row echelon form:
[1 0 1 -2
0 2 2 4
0 0 0 0]
Last row reads 0 = 0. z is not specified, so let z =t.
Second equation reads
2y + 2z = 4 ---> y + z =2
But z=t, so y = 2-t
First equation reads x + z = -2
but z=t, so x = -2-t
Hence solution is {(x,y,z): x=2-t, y =2-t,z=t, t for all real numbers}
or
{(x,y,z) = (2, 2, 0) + t(-1,-1,1), t for all real numbers}
Thats how I've taught to show the solution of a linear system.
ANd yes, in general you should say t for all real numbers...or if you have more than one parameter, eg: s and t, then you should write, "s,t for all real numbers"
Hope I've been some help