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In the first case, x is fixed ans y is the unknown. In the second case, y is fixed ans x is the unknown. In the third case, both x and y are unknown, and the set of solutions is of dimension 1, hence the need for a parameter r1.

The fact that the parameter r1 is in direct bijection with y is only a consequence of the fact that your equation is very simple (affine).

In the first case, x is fixed ans and y is the unknown. In the second case, y is fixed ans and x is the unknown. In the third case, both x and y are unknown, and the set of solutions is of dimension 1, hence the need for a parameter r1.

The fact that the parameter r1 is in direct bijection with y is only a consequence of the fact that your equation is very simple (affine).