Question

Find the maximum value of \( Z = 4x + y \) under the given constraints by graphical method: \[ x + y \leq 50, \quad 3x + y \leq 90, \quad x \geq 0, \quad y \geq 0. \]

Hint:

To maximize a function under constraints, use the graphical method by plotting constraints and evaluating the objective function at corner points.

Answer 1

We solve the system graphically by plotting the lines: \[ x + y = 50 \quad \text{(boundary line for first constraint)} \] \[ 3x + y = 90 \quad \text{(boundary line for second constraint)} \] Finding intersection points, we evaluate \( Z = 4x + y \) at feasible points and determine the maximum value.