Solve the following linear programming problem graphically:
\[
\text{Minimise } Z = 2x + y
\]
subject to the constraints:
\[
3x + y \geq 9,
\]
\[
x + y \geq 7,
\]
\[
x + 2y \geq 8,
\]
\[
x, y \geq 0.
\]
Show Hint
Solution and Explanation
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 \( (x, y) = \text{(solution obtained from computations)} \).
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