Express the objective function of the following linear programming problem in terms of non-basic variables and hence evaluate the initial basic feasible solution.


Minimize.Z=4x_1+x_2
Subject to:
3x_1+x_2=3
4x_1+3x_2≥6
x_1+2x_2=≤4
x_1,x_2≥0

To express the objective function in terms of non-basic variables, we need to eliminate the basic variables.

Given the constraints:
1. 3x_1 + x_2 = 3
2. 4x_1 + 3x_2 ≥ 6
3. x_1 + 2x_2 ≤ 4

Let's solve the first constraint for x_2:
x_2 = 3 - 3x_1

Now, we can substitute this value of x_2 into the objective function to express it in terms of non-basic variables:
Z = 4x_1 + (3 - 3x_1)
Z = 4x_1 + 3 - 3x_1
Z = x_1 + 3

To evaluate the initial basic feasible solution, we substitute the values of the non-basic variables as 0 and solve for the basic variables.

For the initial basic feasible solution, we assume:
x_1 = 0
x_2 = 0

Using these values in the constraints, we have:
1. 3(0) + (0) = 3 (satisfied)
2. 4(0) + 3(0) ≥ 6 (not satisfied)
3. (0) + 2(0) ≤ 4 (satisfied)

Constraint number 2 is not satisfied, so this initial solution is not feasible. We need to find a different initial solution that satisfies all the constraints.

To express the objective function in terms of non-basic variables, we need to first determine the basic variables in the given linear programming problem.

In this case, the basic variables are the variables associated with the equations or inequalities that have equality constraints. Looking at the constraints, we can identify the following basic variables:

1. Equation 1: 3x_1 + x_2 = 3
2. Inequality 2: 4x_1 + 3x_2 ≥ 6
3. Inequality 3: x_1 + 2x_2 ≤ 4

Now, let's solve these equations/inequalities to find the values of the basic variables:

1. Solving Equation 1 for x_2, we get:
x_2 = 3 - 3x_1

2. Solving Inequality 2 for x_2, we get:
x_2 ≥ (6 - 4x_1) / 3

3. Solving Inequality 3 for x_2, we get:
x_2 ≤ (4 - x_1) / 2

To evaluate the initial basic feasible solution, we need to assign values to the non-basic variables (variables not associated with equality constraints). Since there are only 2 variables, x_1 and x_2, there are no non-basic variables in this case.

Now, substitute the expressions of x_2 obtained from the inequalities into the objective function:

Z = 4x_1 + x_2
= 4x_1 + (6 - 4x_1) / 3

Simplifying, we have:

Z = 4x_1 + (6 - 4x_1) / 3
= (12x_1 + 6 - 4x_1) / 3
= (8x_1 + 6) / 3

Now, we can evaluate the initial basic feasible solution by substituting the values of the basic variables (x_1 and x_2) into the expression for Z.

Since no specific values are provided for x_1 and x_2, we cannot evaluate the initial basic feasible solution. However, once values are assigned to x_1 and x_2, we can substitute them back into the equations/inequalities to check if they satisfy all the constraints and determine if they form a feasible solution.