By the graphical method, decide consistency and solve:
(i). \(3x + y + 4 = 0\) and \(6x - 2y + 4 = 0\)
(ii). \(x - 2y = 6\) and \(3x - 6y = 0\)
(iii). \(x + y = 3\) and \(3x + 3y = 9\)
(i). Consistent with a unique solution: \(x = -1\), \(y = -1\).
(ii). Inconsistent (parallel). No solution.
(iii). Consistent and dependent: infinitely many solutions (the line \(x + y = 3\)).
First equation: \(3x + y + 4 = 0\).
Move terms: \(y = -3x - 4\).
This is a straight line with slope \(-3\).
Second equation: \(6x - 2y + 4 = 0\).
Rearrange: \(-2y = -6x - 4\).
Divide by \(-2\): \(y = 3x + 2\).
This line has slope \(3\).
Since slopes are different, the two lines will intersect at one point.
To find the point, set the two expressions for \(y\) equal:
\(-3x - 4 = 3x + 2\).
Bring terms together: \(-3x - 3x = 2 + 4\).
\(-6x = 6\).
So, \(x = -1\).
Put \(x = -1\) in \(y = -3x - 4\):
\(y = -3(-1) - 4 = 3 - 4 = -1\).
Solution is \((x, y) = (-1, -1)\).
Since they intersect, the system is consistent and has a unique solution.
First equation: \(x - 2y = 6\).
Second equation: \(3x - 6y = 0\).
Notice: if we multiply the first equation by 3, we get:
\(3x - 6y = 18\).
But the second equation is \(3x - 6y = 0\).
Left-hand sides are the same, but right-hand sides are different (18 vs 0).
This means the two lines are parallel and never meet.
So, the system is inconsistent (no solution).
First equation: \(x + y = 3\).
Second equation: \(3x + 3y = 9\).
Divide the second equation by 3:
\(x + y = 3\).
This is exactly the same as the first equation.
So both equations represent the same line.
That means there are infinitely many solutions (all points on the line \(x + y = 3\)).
The system is consistent and dependent.