Linear Equation

Question

A farmer has 1,000 acres of land on which he grows corn, wheat, and soybeans. It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow soybeans. Because of market demand the farmer will grow twice as many acres of wheat as of corn. He has allocated $53,000 for the cost of growing his crops. How many acres of each crop should he plant?

Answer 1

X = acres of corn

Y = acres of wheat

Z = acres of soybeans

 

X+Y+Z=1000

45x+60Y+50Z=53000

2X=Y

2x-y=0

solve using substitution

 

X+2X+Z = 1000

3X+Z=1000

z=1000-3x

 

45X+60Y+50Z=53000

45X +60(2X) + 50(1000-3X) =53000

45X +120X+50000-150X = 53000

15X=3000

x=200

 

Once you solve for X you can use 2X=Y to find Y and X+Y+Z=1000 to find Z.

 

You can also solve this with a matrix:

 

[ 1   1   1  1000 

 45  60 50 53000

 2    -1   0  0]

 

row reduce to find x y and z (or use a graphing calculator).