Question

X+y+z=1
2x+y-z=3
3x+y-3z=5

Answer 1

notice the y term has a coefficient of 1 in each equation, so it is easy to
eliminate the y term:
subtract the 1st from the 2nd: x - 2z = 2
subtract the 2nd from the 3rs: x - 2z = 2
ahh, the same equation for both actions.

I don't know how detailed you are into the study of these systems
but each of your equations can be represented by a plane.
In this case the 3 planes all intersect in a straight line, visualize the
spine of a text book with 3 pages fanned out.