Eliminating the arbitrary constant

Hi,

I am having trouble understanding the steps needed to eliminate the arbitrary constants from the equation y = Ax + 3. The answer in the book is y = x (dy/dx) + 3. Here are my steps along with my reasoning for each step (please correct my reasoning so I can finally understand how to do this):

y = Ax + 3

dy/dx = A (I am still left with A, so I need to find another way to get rid of A)

y = Ax + 3 (starting over)

(y - 3)/x = A (isolate A)

(y - 3)/x = dy/dx (replace A with dy/dx - is this correct? if so, why?)

y - 3 = (dy/dx)x

y = (dy/dx)x + 3 (isolate y. Now we are left with an equation that does not have A in it )