Selling a house

Today we are as close as we have ever come to selling our house and buying a new one. I've done a lot of math regarding this, and today I'll write about one aspect. We are asking $180,000 for our house, and we owe $155,000. If there were no commissions, fees or other costs to selling a house then we could keep $180,000 - $155,000 = $25,000. But the truth is that there are such things. It's complicated to explain all the costs of selling a house. Some are calculated as percentages of the sell price, others are flat fees. But it is probably a safe assumption to say that the cost of selling the house will be 10% of the price at which it is sold. Now, I can from this number calculate our profit from selling the house at various prices, but it would have more explanatory power to have a single equation whose features explain the general relationship between the sell price and our profit.

So, here goes. Let x be the sell price and y be the profit. The equation would be y = x - 155,000 - 0.1x, which would simplify to y = 0.9x - 155,000. This equation says a number of things about how the sell price determines our profit.

(1) The sell price and profit are linearly related. One thing this means is that the graph is a line. Now, it is a fact that the graph is a line, but it does not enlighten the situation at all. Something better is this: being linearly related means that if the sell price increases by a certain amount, the increase in profit will be some constant multiple of that amount. Mathematician or not, everyone would believe that if the sell price were to increase, then so would the profit, but for a linear equation the point is that it doesn't matter what the sell price increases from. An increase in the sell price from $1 to $2 has exactly the same effect as an increase from $100 to $101. The key to understanding precisely what this means is by examining the slope of the line.

(2) The slope of the line is 0.9/1, and here's what that means: if the sell price increases by $1, then the profit will increase by $0.9. But any other proportion is also true, so if the sell price changes by $1,000, then the profit changes by $900. So, a $1,000 increase in the selling price will mean an additional $900 in profit.

(3) The x-intercept of the line represents the sell price at which the profit is zero. For our equation y = 0.9x - 155,000, the x-intercept is 155,000/0.9 = 172,223. So, for every $1,000 that we sell above that, we profit $900. But if we sell any less than $172,000, we would have to pay to sell the house.