Formulate a system of equations

Question

Lawnco produces three grades of commercial  fertilizer. A 100 pound bag of grade A fertilizer contains 18 lbs of nitrogen, 4 lbs of phosphate, and 5 lbs of potassium. A 100 bag of grade B fertilizer contains 20 lbs of nitrogen, and 4 lbs of both phosphate and potassium. A 100 lb bag of grade C fertilizer contains 24 lbs of nitrogen, 3 lbs of phosphate, and 6 lbs of potassium. How many 100 lb bags of each of the three grades of fertilizers should Lawnco produce if:

A.) 26,400 lb of nitrogen, 4900 lb of phosphate and 6200 lb of potassium are available and all the nutrients are used.

B.) 21,800 lb of nitrogen, 4200 lb of phosphate, and 5300 lb of potassium are available and all the nutrients are used?

Write the answer using a system of equations.

Answer 1

Let X, Y, and Z be the number of 100 lb bags of grade A, B, C

 

Set the equations up for the components as

 

18 X + 20 Y + 24 Z = 26,400 Nitrogen balance

  4 X +   4 Y +   3 Z =   4,900 Phosphate balance

  5 X +   4 Y +   6 Z =   6,200 Potassium balance

 

Solve this system of equations

 

X = 400, Y = 600, Z= 300 for the first one

 

In the second case replace the right hand side with values given.

 

X = 500, Y = 400, Z= 200 for the second case

Answer 2

Bag A contains 18ni + 4ph + 5po.

Bag B contains 20ni + 4ph + 5po.

Bag C contains 24ni + 3ph + 6po.

 

Let's say I make n bags of A, m bags of B and r bags of C.

 

then 18n + 20m + 24r = 26,400

         4n +  4m  +  3r =  4,900

 

and    5n  + 4m  +  6r = 6,200

 

We have 3 equations with 3 unknowns. Solving this system gives n = 400, m = 600, and r = 300.

 

For the second question do the same thing, the solution is n = 500, m = 400, and r = 200