Formulate the situation as a system of linear equations Be s

Formulate the situation as a system of linear equations. Be sure to state clearly the meaning of each variable. Solve using the Gauss-Jordan method. State your final answer in terms of the original question. (If the system is inconsistent, answer INCONSISTENT. If the system is dependent, parametrize the solutions in terms of the parameters u and v.) A large commercial farm needs 31,500 pounds of potash, 60,000 pounds of nitrogen, and 22,500 pounds of phosphoric acid. Three brands of fertilizer, GrowRite, MiracleMix, and GreatGreen, are available and contain the amounts of potash, nitrogen, and phosphoric acid per truckload listed in the table. How many truckloads of each brand should be used to provide the required potash, nitrogen, and phosphoric acid?

x1 =_______________ truckloads of GrowRite

x2 =_______________ truckloads of MiracleMix

x3 =_______________ truckloads of GreatGreen

(Pounds per truckload)

GrowRite MiracleMix GreatGreen

Potash 400 600 700

Nitrogen 500 1000 1600

Phosphoric acid 300 400 500

Solution

Let the farm purchase x₁ , x₂ and x₃ truckloads of GrowRite, MiracleMix and Great Green Fertilizers. All these fertilizers contain potash, nitrogen, and phosphoric acid. As per the given information, the quantity ( in pounds)of potash, nitrogen, and phosphoric acid nutrients purchased by the farm are 400x₁+600x₂+ 700x₃(potash), 500x₁+1000x₂+ 1600x₃(Nitrogen) and 300x₁+400x₂+500x₃(Phosphoric acid) respectively. Further, since the farm’s requirements of nutrients are 31,500 pounds of potash, 60,000 pounds of nitrogen, and 22,500 pounds of phosphoric acid respectively, therefore, we have

400x₁+600x₂+ 700x₃ = 31500 or, on dividing both the sides by 100, 4x₁+6x₂+ 7x₃ = 315…(1)

500x₁+1000x₂+ 1600x₃=60000 or, on dividing both the sides by 100, 5x₁+10x₂+ 16x₃=600…(2)

300x₁+400x₂+500x₃ = 22500 or, on dividing both the sides by 100, 3x₁+4x₂+5x₃ = 225 …(3)

The augmented matrix of this system of linear equations is A=

4

6

7

315

5

10

16

600

3

4

5

225

To solve the above system of linear equations, we will reduce the matrix A to its RREF as under:

Multiply the 1st row by ¼

Add -5 times the 1st row to the 2nd row

Add -3 times the 1st row to the 3rd row

Multiply the 2nd row by 2/5

Add 1/2 times the 2nd row to the 3rd row

Multiply the 3rd row by 5/6

Add -29/10 times the 3rd row to the 2nd row

Add -7/4 times the 3rd row to the 1st row

Add -3/2 times the 2nd row to the 1st row

Then the RREF of A is

1

0

0

20

0

1

0

10

0

0

1

25

Thus, the solution to the above system of linear equations is x₁ = 20, x₂ = 10 and x₃ = 25. Therefore, the commercial farm will use x₁ =20 truckloads of GrowRite, x₂ = 10 truckloads of MiracleMix, and x₃ = 25 truckloads of GreatGreen fertilizers.