Solve the following linear programming problem graphically:
Minimise \( z = 5x - 2y \)
Subject to the constraints:
\[
x + 2y \leq 120, \\
x + y \geq 60, \\
x - 2y \geq 0, \\
x \geq 0, \\
y \geq 0.
\]
Show Hint
In graphical linear programming, identify the feasible region, locate the corner points, and evaluate the objective function at each corner point to find the optimal solution.
To solve this linear programming problem graphically, follow these steps:
Step 1: Plot the constraints. \( x + 2y \leq 120 \): Rewrite as \( y \leq \frac{120 - x}{2} \). This is a line passing through \( (0, 60) \) and \( (120, 0) \). \( x + y \geq 60 \): Rewrite as \( y \geq 60 - x \). This is a line passing through \( (0, 60) \) and \( (60, 0) \). \( x - 2y \geq 0 \): Rewrite as \( y \leq \frac{x}{2} \). This is a line passing through \( (0, 0) \) and \( (120, 60) \). \( x \geq 0, y \geq 0 \): These are the \( x \)- and \( y \)-axes, restricting the feasible region to the first quadrant.
Step 2: Identify the feasible region.
The feasible region is the area that satisfies all the constraints. Plot the lines and shade the overlapping region that satisfies the inequalities.
Step 3: Determine the corner points of the feasible region.
The corner points are the intersections of the constraint lines:
\[ \text{Solve } x + 2y = 120 \text{ and } x + y = 60. \]
\[ y = 60, \quad x = 0 \Rightarrow (0, 60). \]
\[ \text{Solve } x + y = 60 \text{ and } x - 2y = 0. \]
\[ 2y + y = 60 \Rightarrow y = 20, \quad x = 40 \Rightarrow (40, 20). \]
\[ \text{Solve } x - 2y = 0 \text{ and } x + 2y = 120. \]
\[ 2y + 2y = 120 \Rightarrow y = 30, \quad x = 60 \Rightarrow (60, 30). \]
Step 4: Evaluate the objective function at each corner point.
