Question

Find all the basic solutions of the following system of equations:

x1 + 2x2 + x3 = 14

3x1 + x2 + x3 = 12

Which of these solutions are basic feasible solutions?

Answer 1

Answer on Question #65544, Math / Statistics and probability

Find all the basic solutions of the following system of equations:

$$\left\{ \begin{array}{l} x _ {1} + 2 x _ {2} + x _ {3} = 1 4 \\ 3 x _ {1} + x _ {2} + x _ {3} = 1 2 \end{array} \right.$$

Which of these solutions are basic feasible solutions?

Solution

Definition Basic Solution: A solution obtained by setting exactly $n - m$ variables to zero provided the determinant formed by the columns associated to the remaining $m$ variables is non-zero is called Basic Solution.

$$\left\{ \begin{array}{l} x _ {1} + 2 x _ {2} + x _ {3} = 1 4 \\ 3 x _ {1} + x _ {2} + x _ {3} = 1 2 \end{array} \right.$$ $$[ A, b ] = \left[ \begin{array}{c c c c} 1 & 2 & 1 & 1 4 \\ 3 & 1 & 1 & 1 2 \end{array} \right]$$ $$R _ {2} \rightarrow R _ {2} - (3) R _ {1}$$ $$\left[ \begin{array}{c c c c} 1 & 2 & 1 & 1 4 \\ 0 & - 5 & - 2 & - 3 0 \end{array} \right]$$ $$R _ {2} \rightarrow R _ {2} / (- 5)$$ $$\left[ \begin{array}{c c c c} 1 & 2 & 1 & 1 4 \\ 0 & 1 & 2 / 5 & 6 \end{array} \right]$$ $$R _ {1} \rightarrow R _ {1} - (2) R _ {2}$$ $$\left[ \begin{array}{c c c c} 1 & 0 & 1 / 5 & 2 \\ 0 & 1 & 2 / 5 & 6 \end{array} \right]$$

Basic solutions

$$x _ {1} = 2, \quad x _ {2} = 6, \quad x _ {3} = 0$$ $$x _ {1} = - 1, \quad x _ {2} = 0, \quad x _ {3} = 1 5$$ $$x _ {1} = 0, \quad x _ {2} = 2, \quad x _ {3} = 1 0$$

$x$ is a feasible basic solution if $x$ is basic and $x \geq 0$ .

Therefore

$x = (2,6,0)^T$ is a basic feasible solution

$x = (-1,0,15)^T$ is a basic feasible solution but not feasible

$x = (0,2,10)^T$ is a basic feasible solution

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