Changing Word Problems to Algebra

Date: 4/4/96 at 14:58:19
From: Anonymous
Subject: Applications: Maximum and Minimum

The difference of twice one number and three times another number 
is 24. Find the numbers if their product is a minimum.

I've just finished a chapter on completing the square and word
problems dealing with numbers, perimeters, and merchandising.  So
my problem is that I'm not sure if I started out right. I got
2x - 3y = 24.  Is this right, and if so what next so that I can 
continue?

Date: 4/5/96 at 11:14:7
From: Doctor John
Subject: Re: Applications: Maximum and Minimum

You are on the right track.  The next thing you need to do is to 
find an expression for the product of these two numbers.  

Solve the equation you have for y.  You get y = (2/3)x - 8.  
Then it is clear that xy is can also be written as x[(2/3)x - 8].  
Simplifying a little, you get (2/3)x^2 - 8x.

This is a quadratic function.  All you have to do now is find the 
minimum of the parabola that it describes.

-Doctor John,  The Math Forum