A farmer has 200 acres of land suitable for cultivating crops A, B and C. The cost per acre of cultivating crop A, crop B and crop C is $40, $60 and $80, respectively. The farmer has $12,600 available for land cultivation. Each acre of crop A requires 20 labor-hours, each acre of crop B requires 25 labor-hours and each acre of crop C requires 40 labor-hours. The farmer has a maximum of 5950 labor-hours available. If he wishes to use all his cultivatable land, the entire budget and all the labor available, how many acres of each crop should he plant? [Please can you show me how to Use the Gauss Jordan Elimination method to solve]

To solve this problem using the Gauss Jordan Elimination method, we need to set up a system of linear equations based on the given information.

Let's denote the number of acres of crop A, crop B, and crop C as x, y, and z, respectively. The objective is to find the values of x, y, and z that satisfy the constraints.

First, let's set up the constraints:

1. The total number of acres of land is 200:
x + y + z = 200

2. The total cost of cultivation is $12,600:
40x + 60y + 80z = 12,600

3. The total labor-hours available is 5950:
20x + 25y + 40z = 5950

Now, we can solve this system of equations using the Gauss Jordan Elimination method. Let's write the augmented matrix:

[1 1 1 | 200]
[40 60 80 | 12600]
[20 25 40 | 5950]

Next, we'll perform row operations to transform the augmented matrix into row-echelon form.

Step 1: Get a leading 1 in the first row, first column (divide row 1 by 1):

[1 1 1 | 200]
[40 60 80 | 12600]
[20 25 40 | 5950]

Step 2: Use row operations to create zeros below the leading 1 in the first column (subtract 40 times row 1 from row 2, and subtract 20 times row 1 from row 3):

[1 1 1 | 200 ]
[0 20 40 | 5600]
[0 5 20 | 3950]

Step 3: Get a leading 1 in the second row, second column (divide row 2 by 20):

[1 1 1 | 200]
[0 1 2 | 280 ]
[0 5 20 | 3950]

Step 4: Use row operations to create zeros above and below the leading 1 in the second column (subtract 5 times row 2 from row 3, and subtract row 2 from row 1):

[1 0 -1 | -80 ]
[0 1 2 | 280 ]
[0 0 10 | 1650]

Step 5: Get a leading 1 in the third row, third column (divide row 3 by 10):

[1 0 -1 | -80 ]
[0 1 2 | 280 ]
[0 0 1 | 165 ]

Step 6: Use row operations to create zeros above the leading 1 in the third column (add row 3 to 1 times row 1, and subtract 2 times row 3 from row 2):

[1 0 0 | 5 ]
[0 1 0 | 150 ]
[0 0 1 | 165 ]

The row-echelon form of the augmented matrix is obtained. We can now interpret the solutions:

x = 5
y = 150
z = 165

Therefore, the farmer should plant 5 acres of crop A, 150 acres of crop B, and 165 acres of crop C to utilize all his cultivatable land, the entire budget, and all the labor available.