On December 1, 2005, Cutting Edge Traded An Old Piece Of Equipment
For A Newer Model. The Following Information Pertains To This
Transaction:

Old Machine:
Original Cost 18,000
Accumulated Depreciation As Of 12/1/05 13,500

New Machine:
List Price 22,500
Cash Price Without Trade-in 21,000
Cash Paid With Trade-in 19,050

Can You Help Me Make The Journal Entry To Record The Purchase Of The New Machine And Also The Journal Entry If The Cash Paid Was 16,050?

Dr. New Machine 21,000 (value w/o trade in)
Dr. Acc. Amort - Old Machine 13,500 (given)
Cr. Old Machine 18,000 (given)
Cr. Cash 19,050 (given)
Dr. Loss 2,550

The cash paid is 19,050 - as stated in your question:
Cash Paid With Trade-in 19,050