Problem 12. In order to grow a certain crop, it is recommended that each square

Question

Problem 12. In order to grow a certain crop, it is recommended that each square foot of ground be treated with 10 units of phosphorous, 9 units of potassium, and 19 units of nitrogen. Suppose there are three brands of fertilizer on the market, say A, B, C. One pound of brand A contains 2 units of phosphorous, 3 units of potassium, and 5 units of nitrogen. One pound of brand B contains 1 unit of phosphorous, 3 units of potassium, and 4 units of nitrogen. One pound of brand C contains only 1 unit of phosphorous and 1 unit of nitrogen. Determine whether or not it is possible to meet exactly the recommendation by applying some combination of the three brands of fertilizer. If it is possible to meet the recommendation exactly, give specific numbers of pounds of each brand to use. If it is not possible, clearly state why.

Explanation / Answer

Let x pound of A, y pound of B and z pound of C
Then:
unit of phophorous = 10
2x+y+z= 10 ..eqn 1

unit of potassium = 9
3x+3y = 9 ...eqn 2

unit of nitrogen = 19
5x+4y+z = 19 ...eqn 3

we need tosolve these 3 equations:
3x+3y=9
x+y=3
x=3-y

put this in eqn 1 and eqn 3
2x+y+z= 10
2(3-y)+y+z=10
6-2y+y+z=10
-y+z = 4

5x+4y+z = 19
5(3-y) + 4y+z=19
15-y+z=19
z-y = 4

so eqn 1and eqn 3 are same. We needed 3 equation for solving x,y,z but here we are getting only 2 since eqn 1 and 3 are turning out be same

Answer: Not Possible

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