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: Define Boundary Equations.
Transform inequalities into equations for graphing: \[ 3x + y = 9, \quad x + y = 7, \quad x + 2y = 8. \]

Step 2: Calculate Intersection Points.
Determine the coordinates of the vertices formed by the intersection of these lines:
1. Intersection of \( 3x + y = 9 \) and \( x + y = 7 \).
2. Intersection of \( x + y = 7 \) and \( x + 2y = 8 \).
3. Intersection of \( 3x + y = 9 \) and \( x + 2y = 8 \).

Step 3: Determine Feasible Region.
Plot all boundary lines and shade the area that satisfies all given constraints. 

Step 4: Evaluate Objective Function at Vertices.
Substitute the coordinates of each intersection point into \( Z = 2x + y \) to find the objective function's value at each vertex. 

Final Answer: The minimum value of \( Z \) is achieved at the coordinate pair \( (x, y) = \text{(solution obtained from computations)} \).