Determine the elevator's acceleration

[y90x]

Up on the man and down on the elevator is correct. Your equation in post #22 is also correct. Where does your equation in post #24 come from?

My teacher explained to use two different equations when two variables are unknown, I was trying to find another equation to substitute T with

 

[SammyS]

My teacher explained to use two different equations when two variables are unknown, I was trying to find another equation to substitute T with

You have the correct information for the man in post #5.

Fn=209N (the scale reading)
Fg=mg
=(82kg)(9.8) = 803.6 N

And the force exerted on the rope is T
So the formula would be,

(Fn +T)-Fg=ma

Am I wrong or missing something ?

Use 209 Newtons for F_n and mg for F_g, of course with 82 kg for the man's mass.

Then as @kuruman wrote, your equation in post #22 is correct.

Oh, I see
So then it’ll be
T-209N-40g=40a ?

Use those two equations to eliminate T (or eliminate a if you'd rather find the tension).

 

[haruspex]

Wouldn’t it be the other way around ? Up on the man and down on the dumbwaiter ?

Right, but in post #16 you had an equation for the dumb waiter with T...+Fn = ...
If the tension is acting upwards on the dumb waiter and the normal force is acting downwards then they cannot both be positive on the same side of tne equation.

 

[kuruman]

I have a suggestion for OP, @y90x. Why don't you write two equations for Newton's 2nd Law, F_net = ma, one for which the system is the man and one for which the system is the dumbwaiter and in which only symbols appear, not numbers, and post them. One (or more) of us will review these equations and point out what is wrong (if anything) with them. At this point, it seems to me there is some tail-chasing going on here so a fresh start would be beneficial.