# Seems like a simple systems of equations problem...

• May 3rd 2013, 07:30 AM
kingdom
Seems like a simple systems of equations problem...
HELP!
Two cars... One costs \$299 per month and gets 20mi/gal. Other costs \$199 per month and gets 40mi/gal. Assuming gas costs \$3/gal, at how many miles will cost be equal?
Keep getting negative miles as my answer????
• May 3rd 2013, 07:35 AM
emakarov
Re: Seems like a simple systems of equations problem...
The second car costs less per month and gets better mileage. So its total cost will be less for any (positive) number of miles.
• May 3rd 2013, 07:41 AM
kingdom
Re: Seems like a simple systems of equations problem...
What would the equations be? Since they have different gas mileage, they must have different slope, so the lines must cross somewhere??
• May 3rd 2013, 07:49 AM
kingdom
Re: Seems like a simple systems of equations problem...
Am I missing something? Again, if the rates are not the same (taking into account gas mileage of each car), the two lines must cross somewhere...
• May 3rd 2013, 08:09 AM
emakarov
Re: Seems like a simple systems of equations problem...
Quote:

Originally Posted by kingdom
What would the equations be?

Why don't you post your version for checking?

Quote:

Originally Posted by kingdom
Again, if the rates are not the same (taking into account gas mileage of each car), the two lines must cross somewhere...

They do, but the crossing point corresponds to a negative number of miles, as you have discovered. If you drive zero miles, the first car costs \$299 and the second one costs \$199, i.e., the first car costs more. With each additional mile, the first car spends more fuel, so the difference in total cost increases instead of decreasing. Therefore, the difference will never become zero for a positive number of miles.

What about negative miles? Driving a negative mile subtracts money from the total cost, i.e., it adds money to your pocket. The first car adds more than the second one for each negative mile, so the difference in total cost decreases and eventually becomes zero. Of course, this only makes mathematical sense; this does not make sense financially.

The reasoning above is supposed to give an intuition about what is going on, but ultimately we can perform the calculations and compute the crossing point precisely.

I think it is most likely that the problem was copied incorrectly (e.g., the cost of the second car is in fact \$399 per month because more efficient cars usually cost more). Otherwise, the problem's author made a mistake.
• May 3rd 2013, 04:14 PM
Wilmer
Re: Seems like a simple systems of equations problem...
Quote:

Originally Posted by kingdom
Two cars... One costs \$299 per month and gets 20mi/gal. Other costs \$199 per month and gets 40mi/gal. Assuming gas costs \$3/gal, at how many miles will cost be equal?

199 + 33 1/3 * 3 = 299; 33 1/3 * 40 = 1333 1/3 miles.
The \$299 car stays parked: they're even (Sleepy)