A city commission has proposed two tax bills. The first bill requires that a homeowner pay $1800 plus 3% of the assessed home value in taxes. The second bill requires
taxes of $200 plus 8% of the assessed home value. What price range of home assessment would make the first bill a better deal for the homeowner?
The way that I would attack this problem is by first setting up two equations. Let the first equation be modeled by y=0.03x+1800 (where y is equal to the taxes and x is equal to the home value). Keeping these same variable definitions, I make an equation for the second bill : y=0.08x+200. Now I want to see, where do these two taxes bills cost the same? To do this I set the two equations equal to one another to get that 0.03x+1800=0.08x+200. Now I simply solve the equation. Subtract 200 from each side and 0.03x from each side to get 1600=0.05x. Divide each side by 0.05 to get that x=32000. Now that’s where they’re equal. Since you know from equations that Bill 1 has a more gradual slope, you know that from $32,000 on, it will always be a better deal. That is why the range x>$32,000 is the best deal for homeowners with the first bill!