The cheap house isn’t always a possibility.
First, some lenders have a minimum amount they will lend. You might have great credit and a down payment and a beautiful smile, but if they have a $45,000.00 minimum, they are not going to give you a mortgage to buy that $30,000.00 house.
Second, CONDITION counts! Lenders are not going to lend you money to buy a “fixer upper” unless you are using a (somewhat complicated!) rehab loan like an FHA 203k. If you are using a regular mortgage, the house must be worth at the time the loan closes, what is being loaned on it.