Solutions:

5x + 3y = 0 --> 5(3) + 3(7) 0             no

6x + 7y = 4 --> 6(3) + 7(7) 4

5x + 3y = 0 --> 5(4) + 3(5) 0             no

6x + 7y = 4 --> 6(4) + 7(5) 4

1) 2x - y = 4 --> y = 2x - 4

    -6x + 3y = -18 --> 3y = 6x - 18, y = 2x - 6

no solution

2) 6x - 2y = -2 --> -2y = -6x - 2, y = 3x + 1

    -3x - 7y = 17 --> -7y = 3x + 17, y = -3/7x - 17/7

one solution

3) 21x - 7y = 7 --> -7y = -21x + 7, y = 3x - 1

    -3x + y = -1 --> y = 3x - 1

infinitely many solutions

3x + 2y = 2 --> 2y = -3x + 2, y = -3/2 x + 1

4x + 8y = 3 --> 8y = -4x + 3, y = -1/2x + 3/8