Simplex Method and Gauss-Jordan elimination method.
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1) Using the simplex method, solve the following linear programming equation:
Maximize: P= 5x + 2y
Subject to: 4x + 3y <= 30
2x ? 3y <= 6
x>=0 , y>=0
2) Solve the system of linear equation using the Gaus- Jordan elimination method.
a)
3x ? 3y + 4z = 16
2x + y ? 2z = -1
X + 4y ? 8z = -18
b)
3x ? 2y + 4z = 11
2x ? 4y + 5z = 4
X + 2y ? z = 10
c)
x ? 2y + 3z + 4w = 17
2x + y - 2z - 3w = -9
3x ? y ? 2z ? 4w = 0
4x + 2y -3z + w = -2
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Solution Summary
A linear programming question is solved using the simplex method and the solution is presented in full detail, step by step and annotated. In addition, three examples of a system of linear equation are solved using the Gauss-Jordan elimination method indicating every step in the procedure.
Solution Preview
We first convert the linear system into a matrix tableau adding the variables S1 and S2
4x + 3y + S1 = 30
2x - 3y + S2 = 6
The row Ci is the product of the multiplication of Xi column and each of the variable columns i.e. (0*4)+(0*2) etc.
X Y S1 S2
Xi 5
2 0 0
S1 0 4 3 1 0 30 7.5
S2 0
2 -3 0 1 6 3 We divide the answer column by the pivot column smallest positive one gives pivot row
Ci 0 0 0 0 0
Xi - Ci 5
2 0 0
Largest So this is pivot column
The number on both pivot column and pivot row must become 1 and the rest of the pivot column must become 0 so we divide the second row by 2 and then we subtract 2 times the second row from the first row
5 2
0 0
S1 0
0 9 1 -2 18 2 We divide the answer column by the pivot column smallest positive one gives pivot row
x 5 1 -3/2 0 1/2 3 -2
5 -15/2 0 5/2 15
0 19/2
0 -5/2
The number on both pivot column and pivot row must become 1 and the rest of the pivot column must become 0 so we divide the first row by 9 and then we add 1.5 times the first row from the second row
5 2 0 0
y 2 0 1 1/9 -2/9 2
X 5 1 0 1/6 1/6 6
5 2 19/18 7/18 34
0 0 -19/18 -7/18
We now stop as the last row Xi - Ci is all negative. The ...
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