   Chapter 2.2, Problem 73E

Chapter
Section
Textbook Problem

The problems in Exercises 61-82 correspond to those in Exercises 23-44, Section 2.1. Use the results of your previous work to help you solve this problems.LAWN FERTILIZERS Lawnco produces three grades of commercial fertilizers. A   100 -lb bag of grade   A fertilizer contains 18 lb of nitrogen, 4 lb of phosphate, and 5 lb of potassium. A 100 -lb bag of grade B fertilizer contains 20 lb of nitrogen and 4 lb each of phosphate and potassium. A 100 -lb bag of grade C fertilizer contains 24 lb of nitro-gen, 3 lb of phosphate, and 6 lb of potassium. How many 100 -lb bags of each of the three grades of fertilizers should Lawnco produce if 26 , 400 lb of nitrogen, 4900 lb of phosphate, and 6200 lb of potassium are available and all the nutrients are used?

To determine

The number of 100-lb bags will be produced in each grade.

Explanation

Given:

18 lb of nitrogen, 4 lb of phosphate, and 5 lb of potassium is contained in a 100-lb bag of grade A fertilizer.

20 lb of nitrogen and 4 lb each of phosphate and potassium is contained in a 100-lb bag of grade B fertilizer.

24 lb of nitrogen, 3 lb of phosphate, and 6 lb of potassium is contained in a 100-lb bag of grade C fertilizer.

Calculation:

Consider that the number of grade A fertilizer is represented as x, the number of grade B fertilizer is represented as y and the number of grade C fertilizer is represented as z.

Content in bag of grade A fertilizer can be written as.

18x+20y+24z=26,400

Content in bag of grade B fertilizer can be written as.

4x+4y+3z=4900

Content in bag of grade C fertilizer can be written as.

5x+4y+6z=6200

Solve the above linear equation using Gauss-Jordan elimination method.

For the given system of linear equation, write the augmented matrix.

[182024443546|26,40049006200]=[xyz]

Using elementary row operation, obtain the row echelon form matrix.

[ 18 20 24 4 4 3 5 4 6| 26,400 4900 6200 ]R1R14R2,R3R3R2[ 2 4 12 4 4 3 1 0 3| 6800 4900 1300 ][ 2 4 12 4 4 3 1 0 3| 6800 4900 1300 ]R1R3[ 1 0 3 4 4 3 2 4 12| 1300 4900 6800 ][ 1 0 3 4 4 3 2 4 12| 1300 4900 6800 ]R2R24R1,R3R32R1[

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