A quick question... when do you add the lamba times constraint and when do you subtract? Or does it matter?
In fact, I have always thought of it as "gradient of object function equals $\displaystyle \lambda$ times gradient of constraint function" rather than adding or subtracting.
$\displaystyle \nabla F= \lambda\nabla G$ is, of course, the same as $\displaystyle \nabla F- \lambda \nable G= \nabla(F- \lambda G)= 0$ but $\displaystyle \nabla(F+ \lambda G)= 0$ involves only changing the sign on $\displaystyle \lambda$ which is unknown, anyway.