add 6 to both sides of the first eqution and leave the second equation as is to get:
2x + 3y = 6
4x + 6y = 0
multiply both sides of the first equation by 2 and leave the second equation as is to get:
4x + 6y = 12
4x + 6y = 0
subtract the second equation from the first to get:
0 = 12.
all the variable disappeared and the equation is untrue, so there is no solution which means the lines never intersect which means the lines are parallel.
alternatively, convert both equations to slope intercept form to get:
4x + 6y = 12 becomes 6y = -4x + 12 which becomes y = -4/6 * x + 2.
4x + 6y = 0 becomes 6y = -4x + 0 which becomes y = -4/6 * x.
the equations have the same slope and a different y-intercept which means they are parallel.
from the following graph, you can see that both forms of each equation are identical because they draw the same line.
these two lines will always be 2 vertical units from each other, no matter what the value of x is.
this can be seen on the graph at x = -6, x = 0, and x = 6.
