IGNOU MTE 12 2018 - Determine all the basic feasible solutions to the equationsx1 + x2 +

Question

Determine all the basic feasible solutions to the equations
x₁ + x₂ + 2x₃ = 4

2x₁ - x₂ + x₃ = 2
Identify the degenerate basic feasible solutions.

14 Feb 2018

Answer :

Word Count : 620

To find the basic feasible solutions (BFS) of the system of equations: \[ x_1 + x_2 + 2x_3 = 4 \] \[ 2x_1 - x_2 + x_3 = 2 \] We first need to solve the system using the concept of linear programming, considering the basic variables and non-basic variables. ### Step 1: Express the system in augmented matrix form: \[ \begin{pmatrix} 1 & 1 & 2 & | & 4 \\ 2 & -1 & 1 & | & 2 \end{pmatrix} \] ### Step 2: Select sets of variables to be basic (pivot variables) and non-basic (variables set to 0). For a system with three variables, we need to consider combinations of two variables as basic, while the other will be set to 0. We have the variables \(x_1, x_2, x_3\), so the possible pairs of basic variables are: - \(x_1, x_2\) - \(x_1, x_3\) - \(x_2, x_3\) We will solve the system for each case where the non-basic variable(s) are set to 0. --- ### Case 1: \(x_3 = 0\), Basic Variables:

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