SOLUTION: According to data from a Texas agricultural report, the amounts of nitrogen (lb/acre), phosphate (lb/acre) and labor (hr/acre) needed to grow honeydews, yellow onions, and lettuce

Question 489762: According to data from a Texas agricultural report, the amounts of nitrogen (lb/acre), phosphate (lb/acre) and labor (hr/acre) needed to grow honeydews, yellow onions, and lettuce are given by the following table:

YELLOW
HONEYDEW ONIONS LETTUCE

NITROGEN 120 150 180

PHOSPHATE 180 80 80

LABOR 4.97 4.45 4.65

If a farmer has 220 acres, 29,100 pounds of nitrogen, 32,600 pounds of phosphate, and 1061 hours of labor available, can he use all of his resources completely? If so, how many acres should he allot for each crop?

We set up the initial equations:

120H + 150Y + 180L = 29,100

180H + 80Y + 80L = 32,600

4.97H + 4.45Y + 4.65L = 1061 OR 497H + 445Y + 465L = 106,100 (easier than decimals)

The instructor is working with systems of linear equations and matrices. We do not have a graphing calculator.
Thank you for your help!

Answer by chessace(471) (Show Source): You can put this solution on YOUR website!
A graphing calculator won't make much difference.
Linear equations and matrices should work, espically Gaussian elimination.
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One issue you all may be having is the definition of H, Y, and L. As stated above these are all fractions per acre, then when you are done, multiply all by 220.
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I would have started with the equation
H + Y + L = 220 which will result in the answer requested without further step and is easier to work with than those biggies.
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Of course, it's silly to try to use up all your resources, This situation cries out for Linear Programming (needed: Prices per crop), but that is another subject.

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