Question

Solve the following linear programming problem graphically: \[ \text{Minimise } Z = 2x + y \] subject to the constraints: \[ 3x + y \geq 9, \] \[ x + y \geq 7, \] \[ x + 2y \geq 8, \] \[ x, y \geq 0. \]

Hint:

In graphical methods, plot constraint lines, find intersections, and evaluate the objective function at feasible region vertices.

Answer 1

Step 1: Identify constraint lines.
Convert inequalities to equations for plotting: \[ 3x + y = 9, \quad x + y = 7, \quad x + 2y = 8. \] 

Step 2: Find intersection points.
Solving for intersection points:
1. Solve \( 3x + y = 9 \) and \( x + y = 7 \).
2. Solve \( x + y = 7 \) and \( x + 2y = 8 \).
3. Solve \( 3x + y = 9 \) and \( x + 2y = 8 \).

Step 3: Identify feasible region.
Graph all lines and shade the feasible region satisfying constraints. 

Step 4: Compute Z-values at corner points.
Evaluate \( Z = 2x + y \) at each intersection point to find the minimum. 

Final Answer: Minimum \( Z \) value occurs at \( (x, y) = \text{(solution obtained from computations)} \).