Yes, but since it is equated to zero, they have simplified by multiplying through by negative 1, I am guessing so that there is only 1 negative term, rather than two.
Yes, algebraically there would be no problem in doing that if you wish. I was taught the equivalent notation:
If given subject to the constraint , then we wish to solve the following system:
Then we draw an implied relationship between the two independent variables from the first to equations which we then use in the third to get the critical point(s).