# The Restaurant Bill

RAY: Here’s the answer. Well, it doesn't take too much. The only two numbers I know that make 63 when multiplied together are nine and seven.

TOM: So, that must be the answer, so you can work backwards and write the equation.

RAY: If x equals the number of people and y equals the amount paid. So, x times y equals 63, the original number of people.

TOM: Ah! So, if we hadn't sneaked out the window, then --

RAY: x times y --

TOM: x times y would have been 63.

RAY: -- would have been enough and you wouldn't have been able to figure it out.

TOM: x times y is 63.

RAY: Right. So you also know that if you add two to the number of people, so x plus two --

times, in parentheses, y minus two, which is the number of people that we're short --

has got to equal 63 also.

So, you multiply it all out, and what you get is - you get a long expression, and what you end up doing is you substitute. You get rid of the y, and you substitute the 63 over x for y. And when you do that, you wind up with something called a quadratic equation. Is that right?

TOM: Of course! Any kid will figure this out in five minutes. Just give them those two equations. (x + 2)(y - 2) = 63, and xy = 63. Solve that. You can't do that; you're out! That's it!

RAY: Well, it finally ends up with x^2 - 2x - 63 = 0. And when you factor that, it comes out to (x + 7)(x - 9) = 0. Right?

TOM: Yeah. I see it.

RAY: And so, one answer is x = 9. And the other one is x = -7.

TOM: That's how many people there were originally, minus seven. No one showed up.

RAY: And we left and we stiffed them, so they had to come up with two more bucks apiece. If you think you have any questions about that, write to my brother. Who won?

TOM: So, each of the seven people pays nine bucks.

RAY: Right. Whereas each of the nine people would have paid seven bucks. Who’s our winner?

TOM: The winner is Tucker Matsik from Lombard, IL. Congratulations, Tucker!

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