Question

Solve the following linear programming problem graphically: Maximise \( Z = x + 2y \) Subject to the constraints: \[ x - y \geq 0 \] \[ x - 2y \geq -2 \] \[ x \geq 0, \, y \geq 0 \]

Hint:

To solve linear programming problems graphically, plot the constraints to form the feasible region and evaluate the objective function at each vertex of this region.

Answer 1

We need to graph the constraints and find the feasible region. - From \( x - y \geq 0 \), we have \( x \geq y \). - From \( x - 2y \geq -2 \), we have \( x \geq 2y - 2 \). - The last constraint is \( x \geq 0, \, y \geq 0 \), which restricts the values to the first quadrant. After plotting the constraints, the feasible region is formed, and we can evaluate \( Z = x + 2y \) at the vertices of the feasible region. The vertex points are: (0, 0), (4, 2), (5, 3). Now calculate \( Z \) at these points: - At (0, 0), \( Z = 0 + 2(0) = 0 \). - At (4, 2), \( Z = 4 + 2(2) = \). - At (5, 3), \( Z = 5 + 2(3) = 11 \). Thus, the maximum value of \( Z = 14 \) at the point (5, 3).