x + y = 4 and x - y = 6
Adding the two equations, we get:
2x = 10
x = 5
Substituting x = 5 into either of the original equations, we get:
5 + y = 4
y = -1
Therefore, the system is inconsistent.
x + y = 8 and 2x + 2y = 16
Dividing the second equation by 2, we get:
x + y = 8
This is the same as the first equation, so the system is dependent.
x - y = 4 and x - y = 8
Subtracting the second equation from the first, we get:
0 = -4
This is not a true statement, so the system is inconsistent.
2x + 2y = 9 and 3x + 2y = 12
Subtracting the first equation from the second, we get:
x = 3
Substituting x = 3 into either of the original equations, we get:
2(3) + 2y = 9
4 + 2y = 9
2y = 5
y = 5/2
Therefore, the system is consistent and independent, with solution (x, y) = (3, 5/2).