OK, you put down $40,000 on a house that costs more. How does the seller get the rest of their money? You make payments to them? THAT is a mortgage. It's a private mortgage, not a mortgage through a bank, but it's still a mortgage.
The only way to buy a house without borrowing the money from someone is to pay 100% cash. Are you going to save until you get $350,000? And save that money while paying rent? It can be done, but by the time you have saved that much, the same house will cost far, far more than what it does today.
Mortgages are smart. Yes, you have to pay interest, but what you are doing is leveraging your investment. You are getting to control a large asset for a small investment. If your credit and income are good enough, you might be able to buy a house for $350k by putting down 10%, $35,000.
You own it for ten years, make the payments on it that, if your income and credit were good enough for the lender to give you the loan, you will have no trouble paying for. Now, let's say values keep going up and in ten years, the house is worth half a million. You made $150k in ten years on an investment of $35,000. To turn that same $35k into double in ten years, or $70k, in the stock market, you would need a consistent rate of return of 7.2%. To turn $35k into $150k in ten years in the stock market, you would need a consistent rate of return of about 15%. And we all know that the stock market has ups and downs.
So now, subtract the 4% interest you're paying on the mortgage over those ten years from the 15% you got, and you have owned an investment that gave you 11% a year. That covers the interest. The principal you paid is still yours, it's the equity in the home after paying down the loan. And you owned it, you didn't have a pile of receipts from renting.
If you have managed to save that much money, you are WAY ahead of most first time buyers. Go sit down with someone at your bank and discuss what is involved in getting a mortgage. They can tell you the income you need to buy a house at any price, approximately how much the payments will be. There's no obligation. Once you have the roadmap and are ready, then you can start shopping, for a house and for the best mortgage rates out there.