The landlord can accept a partial payment of rent, then serve a 3 day notice for the balance.
As a practical matter, you may be able to negotiate an agreement to leave voluntarily by a date certain in exchange for a waiver of the outstanding rent. This approach would help you avoid an eviction on your record, which will make it more difficult to rent in the future.
Mr. Kane is correct, the landlord may accept a partial payment and then serve a 3-day notice to pay rent or quit for the unpaid balance. This is a procedure I recommend to my landlord clients as a matter of course.
You might have a defense based upon the representation by the landlady that she would accept partial payment and hold off on the eviction, but I sure would not place much hope in that. Assume that if the landlady files an eviction action she will obtain a judgment of possession against you.
In reality you have to work something out with your landlady. Keep in mind that just because she served the notice does not mean necessarily that she will proceed with the eviction. Again, I often advice my clients to serve the 3-day notice to pay rent or quit immediately but further advise them that they do not have to file the eviction right away if they want to work with the tenant. This may be the case with you. Under any circumstance, it appears that you cannot afford this apartment. Try to find something you can afford and work out a schedule with your landlady before she has to resort to an eviction. Believe me, she would much prefer to work with you than to have to go through the eviction process.