1) The system below is consistent and dependent
x+3y=4
3x+9y=12
If we multiply the first equation by 3:
3(x+3y=4)→
3x+9y=12
We get exactly the second equation, then the system is consistent and dependent
2) The system below is inconsistent
3x-4y=12
6x-8y=21
If we multiply the first equation by 2:
2(3x-4y=12)→
6x-8y=24
We don't get exactly the second equation, it's different the right side of the equation (24 versus 21), then the system is inconsistent
3) The system below is consistent and independent
2x-3y=8
-3x+2y=8
If we multiply the first equation by 3:
3(2x-3y=8)→
6x-9y=24
If we multiply the second equation by -2:
-2(-3x+2y=8)→
6x-4y=-16
We get two different equations, then the system is consistent and independent