Home Equity Loan – How accessible is the amount determined?

I am trying to get a home equity loan to consolidate debts. I just read that banks use a price below the purchase or appraised value to determine how much to give. Is this correct, or whether they will really use the difference between the appraised value and what you owe on your mortgage? Does anyone know what the interest rates are looking like now, if someone has a poor credit rating?
Payments by credit card or mortgage are all made in the same way. What is the algebraic expression using the principles, P, remaining loan after a payment is made? Keep in mind that interest accrues at a rate of i, p, and you can make payments on the original amount is due L.

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