"On the balance sheet, after the first year it shows $80,000 of assets. However, the equity shows $100,000 of additional paid in capital. It's out of balance."

Did you forget that the $20,000 is part of Expenses?

Example (how I teach this):

Assets = Liab + Equity; the data tells a story, so let's turn that around...

Subtract Liab from each side (the formula stays equivalent).

Assets (what you Own) minus Liab (what you owe) = Equity or Net Assets or Ownership position

You own a $500,000 house; you have $300,000 mortgage; that means you have $200,000 Equity. Or, as I explain that in class: "If I get hit by a bus, you can sell my assets, pay off my debts, and party with the difference."

Income and expense ("temporary" accounts) always affect the balance sheet, because the changes here become Net Income(Loss) reflected in Retained Earnings (Equity). It doesn't matter how you slice-and-dice equity (RE, Capital Contributions, etc), it's all just "the Net position for this entity."

So, an Asset that is reducing in value means that reduction is taken as Expense. Let's do that math:

$100,000 Asset

=

$0 Liab (for example)

$100,000 Equity

Next year:

$100,000 Asset -$20,000 asset reduction (depreciation) = $80,000 total assets

$0 Liab

$100,000 Equity minus the New Expense of that $20,000 from the P&L (Income statement) = $80,000

The Balance Sheet balances.

Second year:

$100,000 Asset - $40,000 asset reduction (depreciation = $60,000 total assets

$0 Liab

$80,000 Equity minus the New Expense of that $20,000 from the P&L (Income statement) = $60,000

The Balance Sheet balances.

Rinse and repeat.