Accounting entry question?

If you:

1) credit cash(deposit) and debit an equity account (capital investment).

2)debit cash (pay bill) and credit the expense account

What would be your offsetting entry to reduce your equity account? or I should I do the entry another way?

You've got everything backwards.

1. To record a cash deposit as an investment to your company:
debit Cash
credit Equity Account

2. To pay a bill
debit Expense Account
credit Cash

Now for your question, I assume that you wanted to know how the payment of the bill in cash would offset the equity account. There is no need to do an offsetting entry to reflect the bill payment's effect on the capital account.

All your profits and losses as computed in the Income Statement goes straight to your Capital Accounts in the Balance Sheet as an addition (Net Income) or deduction (Net Loss). Net Income or Net Loss is computed by deducting expenses (like your paid bill) from Revenues.

The effect of your bill payment or more specifically the Expense the company incurred, will reduce the capital account when the closing entries are prepared at the end of your accounting period or end of the month closing entries in preparation for your Income Statement (also called a Profit and Loss [P&L] Statement)

The accounts for Cash and Equity goes to the Balance Sheet w/c shows your company's Resources and Liabilities ( Assets = Liabilities + Capital) while the Expense account goes to the income statement which shows your profits or losses (Revenues - Cost and Expenses = Net Profit or Net Loss).

Your entry #2 is your entry that reduces your equity account, but to see it that way you have to understand that all of your income and expense accounts are really a subset of your equity account, specifically, the details of your retained earnings account and you will only close that at the end of the year.

You can see it a little more clearly if you jump ahead to the end of your fiscal year and think about closing all of your income and expense accounts into Retained Earnings, which is an equity account.

If that expense were your only transaction for the year you would close your books with a credit to the expense account and a debit to retained earnings. That would be your reduction to your equity account that you are looking for, but it doesn't show up until the end of the year.

The whole reason for income and expense accounts is so that you can see where your money came from and where it went to and manage your business or personal finances more effectively.

If you didn't care where it came from or where it went (the Income Statement) and were only concerned with how much you have left (the Balance Sheet) you could ignore the income and expense accounts and just credit cash and debit equity but then there wouldn't be any need for accounting courses or accountants and we'd all have to get real jobs.

Oops, Dimos is right, you have your entries backwards - I should have read more carefully. After you fix your entries (a deposit is a debit to cash, a bill payment is a credit to cash) then my explanation works.