In L.P.P., `x_(j)` for all basic variable is equal to

Options

A. 1
B. `-1`
C. 0
D. none of these

Answer

To solve the question "In L.P.P., \( x_j \) for all basic variables is equal to...", we need to understand the concept of basic variables in the context of Linear Programming Problems (LPP).

### Step-by-step Solution:

1. **Understanding Basic Variables**:
In a Linear Programming Problem, we have a set of variables that are used to represent the decision variables in the objective function. The basic variables are those that correspond to the basic feasible solution of the linear programming model.

**Hint**: Remember that basic variables are those variables that are included in the solution of the system of equations formed by the constraints.

2. **Formulating the Problem**:
A standard LPP can be expressed in the form:
\[ \text{Maximize or Minimize } Z = c_1x_1 + c_2x_2 + ... + c_nx_n \]
subject to constraints like:
\[ a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n \leq b_1 \]
and the non-negativity constraints:
\[ x_j \geq 0 \text{ for all } j \]

**Hint**: Identify the objective function and the constraints to understand the context of the problem.

3. **Identifying Basic and Non-Basic Variables**:
In the context of the simplex method, basic variables are those that are part of the solution (i.e., they have non-zero values), while non-basic variables are set to zero.

**Hint**: Basic variables are typically those that correspond to the pivot columns in the tableau during the simplex method iterations.

4. **Values of Basic Variables**:
The basic variables can take values that are either zero or positive. This means that for any basic variable \( x_j \), it can be expressed as:
\[ x_j \geq 0 \]
However, it is not restricted to being exactly 0 or 1; it can take any non-negative value.

**Hint**: Recall that basic variables can be at their lower bound (0) or can take on any positive value depending on the constraints.

5. **Conclusion**:
Therefore, the statement "In L.P.P., \( x_j \) for all basic variables is equal to..." can be concluded as:
- \( x_j \) for basic variables can be either 0 or greater than 0.

The correct answer is that \( x_j \) for all basic variables is equal to either 0 or a positive value.

### Final Answer:
In L.P.P., \( x_j \) for all basic variables is equal to either 0 or a positive value.