Question

In the Linear Programming Problem for objective function \( Z = 18x + 10y \) subject to constraints \[ 4x + y \geq 20 \] \[ 2x + 3y \geq 30 \] \[ x, y \geq 0 \] Find the minimum value of \( Z \).

Hint:

Quick Tip: In linear programming problems, solving the system of equations obtained by converting inequalities into equalities helps find the corner points. The objective function value at these points will give the minimum or maximum value depending on the problem.

Answer 1

We are given the objective function and constraints: \[ Z = 18x + 10y \] subject to: \[ 4x + y \geq 20 \quad \text{(1)} \] \[ 2x + 3y \geq 30 \quad \text{(2)} \] \[ x, y \geq 0 \quad \text{(3)} \] First, we convert the inequalities into equalities to find the corner points:
- From equation (1): \( 4x + y = 20 \)
- From equation (2): \( 2x + 3y = 30 \)
Solve this system of equations to find the values of \( x \) and \( y \). \[ \text{Multiplying (1) by 3: } 12x + 3y = 60 \] Subtract equation (2) from this: \[ (12x + 3y) - (2x + 3y) = 60 - 30 \] \[ 10x = 30 \quad \Rightarrow \quad x = 3 \] Substitute \( x = 3 \) into equation (1): \[ 4(3) + y = 20 \quad \Rightarrow \quad 12 + y = 20 \quad \Rightarrow \quad y = 8 \] Thus, the corner point is \( (3, 8) \). Now, evaluate the objective function \( Z = 18x + 10y \) at this corner point: \[ Z = 18(3) + 10(8) = 54 + 80 = 134 \] Thus, the minimum value of \( Z \) is \( \boxed{134} \).