Question

Solve the L.P.P graphically
Maximize \( Z = 7x + 8y \) subject to constraints \[ x + y \geq 50, \quad x + 2y \leq 100, \quad x - y \geq 10, \quad x \geq 0, \quad y \geq 0 \]

Hint:

For graphical L.P.P problems, always graph the constraints, find the feasible region, and evaluate the objective function at the vertices to find the optimal solution.

Answer 1

Step 1: Graph the constraints.
We begin by graphing the given constraints on the coordinate plane.
- \( x + y \geq 50 \): This represents a line with slope \( -1 \) and y-intercept \( 50 \). The region satisfying this inequality is to the right of the line.
- \( x + 2y \leq 100 \): This represents a line with slope \( -\frac{1}{2} \) and y-intercept \( 50 \). The region satisfying this inequality is below the line.
- \( x - y \geq 10 \): This represents a line with slope \( 1 \) and y-intercept \( -10 \). The region satisfying this inequality is to the right of the line.
- \( x \geq 0 \): This is the right half of the coordinate plane.
- \( y \geq 0 \): This is the upper half of the coordinate plane.
Step 2: Find the feasible region.
The feasible region is the region where all the inequalities overlap. We find the points of intersection of the boundary lines of these inequalities.
Step 3: Evaluate the objective function at the vertices.
After plotting the feasible region, we evaluate the objective function \( Z = 7x + 8y \) at each vertex of the feasible region to find the maximum value.
Step 4: Conclusion.
The solution to the problem is the vertex that gives the maximum value of \( Z \).