No. There is no such requirement when solving MLE problems. I would suggest that you misunderstand MLE (or basic algebra) if you state that as fact.
If your goal is to solve the problem
f(x,y) == 0
for x, SUBJECT to the constraint that y is fixed, then this is just a call to solve, solving for the variable x. You will then get a relation for x, AS A FUNCTION OF y.
You could then substitute that expression for x into the second equation, then solving for y, and eventually recovering x once y is known. All of this is basic high school algebra, how you might solve the problem using pencil and paper. It is also similar to what you will get when you just call solve on the two equations.
So, lets see what solve does, if you just throw both equations into solve.
syms x y
E1 = x + y == 3;
E2 = x^2 + y^2 == 25;
xy = solve(E1,E2);
xy.x
ans =
41^(1/2)/2 + 3/2
3/2 - 41^(1/2)/2
xy.y
ans =
3/2 - 41^(1/2)/2
41^(1/2)/2 + 3/2
Now, as you MIGHT do it using pencil and paper, solving for x first, then eliminating x from the second equation.
x_y = solve(E1,x)
x_y =
3 - y
subs(E2,x,x_y)
ans =
(y - 3)^2 + y^2 == 25
y_final = solve(subs(E2,x,x_y),y)
y_final =
3/2 - 41^(1/2)/2
41^(1/2)/2 + 3/2
Now recover x.
x_final = subs(x_y,y,y_final)
x_final =
41^(1/2)/2 + 3/2
3/2 - 41^(1/2)/2
Same result.
