On January 4, 2013, Illinois Company purchased new equipment for $550,000 that had a useful life of four years and a salvage value of $50,000.

To determine the journal entries for the purchase of new equipment by Illinois Company on January 4, 2013, we need to consider the initial cost, useful life, and salvage value.

1. Determine the initial cost: The equipment was purchased for $550,000.

2. Determine the useful life: The equipment has a useful life of four years.

3. Determine the salvage value: The equipment will have a salvage value of $50,000 at the end of its useful life.

Now, let's calculate the annual depreciation using the straight-line method:

Depreciation expense per year = (Initial cost - Salvage value) / Useful life

Depreciation expense per year = ($550,000 - $50,000) / 4
Depreciation expense per year = $500,000 / 4
Depreciation expense per year = $125,000

Now, we can calculate the journal entries:

On January 4, 2013:
Equipment (Asset) Dr - $550,000
Cash (Asset) or Accounts Payable (Liability) Cr - $550,000

This entry records the purchase of the equipment with the corresponding decrease in cash or increase in accounts payable.

To record annual depreciation expense:

At the end of each year (for four years):
Depreciation Expense (Income Statement - Operating Expense) Dr - $125,000
Accumulated Depreciation (Asset - Contra-Asset) Cr - $125,000

This entry records the depreciation expense for each year and reduces the carrying value of the equipment.

At the end of the useful life:

Accumulated Depreciation (Asset - Contra-Asset) Dr - $500,000
Equipment (Asset) Cr - $500,000

This entry removes the accumulated depreciation from the books and writes off the remaining book value of the equipment.

Please note that the above journal entries are simplified, and you should consult an accountant for your specific situation.