4x-3y=15, 3y=4x-15, y=4x/3-5.
y is the vertical axis. If values of y are to be reduced by 3 units ("3 units down") then the constant -5 needs to be reduced by 3:
y=4x/3-8 which is the same as 3y=4x-24 or 4x-3y=24.
4x-3y=15 translates to 4x-3y=24 when shifted down three units.