A simple diagram with original axes in blue.

The coordinates of point E are (1,1)

A translation defined by x -> x + 2, y -> y + 1 moves point E to point D, with coordinates (3,2)

If the x axis is moved 2 steps left and the y axis is moved one step down then the coordinates of the original point E in the moved axes are (3,2)

This will be the case for any original point – the coordinates of each one of them will be the same as the coordinates of their new positions under the translation (in the original coordinate system).

This can be seen to be true for the other rigid motions, for example

rotation about the origin through an angle theta is equivalent to a rotation about the origin of the axes through an angle minus theta. So there is a one to one correspondence between rigid motions and change of axes (scales preserved).

On a lighter note, it does seem easier to rotate a pair of axes than rotating the whole plane ! ! !