Decide the relative position of the lines 3x+3y-12=0 and 6x-3y+6=0

To decide the relative position of the given lines, we'll
solve the system formed of the given equations of the
lines.

If the system has solutions, that means that the
lines are intercepting each other in a certain point. If the system has no solutions,
that means that the lines are not intercepting each
other.

We'll divide the 1st equation by
3:

x + y - 4 = 0

We'll
re-write the 1st equation:

x+y =
4

x = 4 - y (1)

We'll divide
the 2nd equation by 3:

2x - y + 2 = 0
(2)

We'll substitute (1) in (2):

2(4 - y) - y =
-2

We'll remove the
brackets:

8 - 2y - y =
-2

We'll combine like terms and we'll subtract 8 both
sides:

-3y = -2 - 8

-3y =
-10

We'll divide by -3:

y =
10/3

We'll substitute y in
(3):

x = 4 - 10/3

x =
(12-10)/3

x =
2/3

The solution of the system represents the
coordinates of the intercepting point of the given lines: (2/3 ;
10/3).