Too "much" equity, unable to refinance

[David Hernandez]

Hello,

I started renting my house last year (as I moved back with my family) and I'm ready to purchase a second house. I saw a good deal, but the price is just a little bit too high for me to qualify (debt-to-income ratio).

Debts:     1) Car payment ($400)
              2) Mortgage ($807)

Income:   1) W-2 job (60k/yr)
              2) Rental ($1100)

I figured I could refinance my home to get a lower mortgage payment while lowering my interest rate (current rate=5%). The home is worth 200k and I only owe 35k, but banks tell me that is too low of an amount and I don't qualify for refinancing.

Can I pull out enough money from my equity on the mortgage and then refinance the whole debt? That would lower my payments and allow me to give a bigger down payment. I know that refinancing is a new mortgage under new terms, and that the lender pays the first mortgage off before giving me the new one. I'm confused as to where the money from my equity comes from. Is it a different loan than my mortgage? If it is can it be combined with my mortgage since I'd be borrowing from the same lender?