2x + y = 6; 6x + 3y = 12; The graph of the above system of equations is overlapping, parallel or intesecting?

For a system of lines a₁x + b₁y + c₁ = 0; a₂x + b₂y + c₂ = 0

If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are overlapping (or coincident or same lines)
If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are parallel
If a₁/a₂ = b₁/b₂, the lines are intersecting
 
Here we have the system 2x + y - 6 = 0; 6x + 3y - 12 = 0.
We can see that
2/6 = 1/3 = -6/-12
 
So, the lines are parallel lines. Their graphs will be parallel to each other and they'll never touch or intersect each other.