Algebra help needed. I am needing to find price for max revenue

Question

I can sell 500 units at $40 each. If I raise the price by a $1, I can sell 10 units less. What is my break even price? I can find the answer by manually calculating each iteration, but need an algebraic way/formula to calculate.

Answer 1

500 sell at 40 each, 490 at 41

slope of line through (40,500), (41,490) is (500-490)/(40-41)=-10

line = y = -10x + b, plug in either point to find b

500 = -400+b

b = 900

y = -10x +900 = units sold at price = x

revenue = XY = x(-10x+900) = -10x^2 + 900x

set = 0 to find break even points

-10x^2 +900x = 0

x(-10x +900) = 0 set the factors = 0

x= 0 and x = 90, $90 price will break even for revenue