Every buyout starts with one table: where the money comes from (sources) and where it goes (uses). The two totals must be equal.
Uses are what the deal costs: the purchase price of the business (enterprise value, usually EBITDA x an entry multiple), transaction fees for advisers, financing fees for the lenders, and any cash left on the balance sheet to run the business. In a take-private you buy the shares at a premium to today's price and repay the target's existing debt.
Sources are how it's paid for. Debt is sized as a multiple of EBITDA, often in layers (tranches) such as a term loan and senior notes. The private equity sponsor's equity fills the gap:
sponsor equity = total uses - total debt
That plug is the number the sponsor cares about most, because returns are measured on it. Less equity means a bigger multiple of money if things go well, and more risk if they don't.