x-y+3z=19
-2x+2y-6z=9
3x+5y+2z=3
D =
|1 -1 3|
|-2 2 -6| = 0
|3 5 2|
so, no solution exists
Note that dividing #2 by -2,
x-y+3z=19
-2x+2y-6z=9
can be written as
x-y+3z=19
x-y+3z= -9/2
Those two lines are parallel, so they cannot intersect in a solution.