An animal feed is to be made from corn soybeans and cottonse

An animal feed is to be made from corn, soybeans, and cottonseed. Determine how many units of each ingredient are needed to make a fee that supplies 1800 units of fiber, 2800 units of fat, and 2200 units of protein, given that 1 unit of each ingredient provides the number of units shown in the table below. The table states, for example, that a unit of corn provides 10 units of fiber, 30 units of fat, and 20 units of protein.

Solution

Solution:

Let x, y, z = amt of corn, soybeans, cottonseed respectively :
Three equations:
pro: 20x + 40y + 25z = 2200   ____________________(1)
fat: 30x + 20y + 40z = 2800   ____________________(2)
fib: 10x + 20y + 30z = 1800   ____________________(3)
:
Multiply the 1st equation by -1 and add the 3 equations.
- 20x - 40y - 25z = - 2200
   30x + 20y + 40z = 2800
   10x + 20y + 30z = 1800
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20x + 0y + 45z = 2400

Multiply the 3rd equation by -1 and add to the 2nd equation

30x + 20y + 40z = 2800
-10x - 20y - 30z = -1800
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20x + 0y + 10z = 1000

Two equation where we are dealing only with x and z

20x + 45z = 2400
20x + 10z = 1000

subtracting eliminates x, find z

35z = 1400 => z = 40 units of cottonseed

Use equation: 20x + 10z = 1000; substitute z = 40

=> 20x + 10(40) = 1000 => 20x = 600 => x = 30 units of corn

Use equation: 10x + 20y + 30z = 1800; substitute x = 30, z = 40

=> 10(30) + 20y + 30(40) = 1800 => 20y = 300 = > y = 15 units of Soybeans