After the bank said no, a new plan across two states.

Equity is a property's value less the debt backed by it. It is the owner's share of the value, rather than cash in an account. Some may help you borrow more. How much you can use depends on the lender's view of the value, its loan limits and what you can afford to repay.
Updated
Equity changes when debt or property value changes. Paying off some debt grows your share if the value stays the same. A higher value can grow the gap too. A lower value shrinks it, even if the debt has not changed.
The lender sets the value it uses to decide on a loan. An app's estimate, an agent's opinion or a sale next door does not set that figure. The bank need not accept any of them. Your equity figure is an estimate until the relevant value has been checked.
Lenders tend to leave a margin between the loan and property value. Borrowing above 80% may mean paying LMI, which insures the lender. It can also mean different approval rules. This is why a first estimate often takes 80% of the value, less the loan you still owe.
This estimate does not give you a right to borrow that amount. A lender may allow less. Some loans allow more, with extra conditions. Other loans backed by the property, fees and checks of what you can repay also affect the money a lender can advance.
You borrow against the property, using it to back a loan. You are not spending cash you already own. Cash paid out from this new loan adds to your debt. The lender still needs proof that you can afford the extra payments.
Which home backs each loan is a separate choice. An equity loan on one home and a loan to buy another can each have their own security. Security is the property backing a debt. Cross-collateralisation instead links homes to back borrowing together. This affects later sales and loan changes.
For investors
A deposit borrowed against equity can leave debt on the old home as well as the new one. Counting only the new home's loan misses some of the borrowing cost. What the extra money pays for decides whether its interest may be claimed for tax. The linked article explains ways to use equity. This page defines your share of the value and the margin a lender keeps.
Common questions
Keep learning
Equity is one piece of it. Lending rules decide how much you can borrow, and how soon you can buy again. Next, read about loan-to-value ratio, lenders mortgage insurance and cross-collateralisation.
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