Problem Statement
Solve the following Linear Programming Problem graphically:
$$\text{Maximize } Z = 4x + y$$
Subject to constraints:
$$\begin{aligned} x + y &\le 50 \\ 3x + y &\le 90 \\ x \ge 0, &\quad y \ge 0 \end{aligned}$$
Verified Solution & Marking Scheme
Find intersection points of boundary lines
Line 1: $x + y = 50$. Passes through $(50, 0)$ and $(0, 50)$.
Line 2: $3x + y = 90$. Passes through $(30, 0)$ and $(0, 90)$.
Solving simultaneously: subtracting Line 1 from Line 2 gives $2x = 40 \implies x = 20$.
Then $y = 50 - 20 = 30$. Point of intersection is $B(20, 30)$.
Identify Corner Points of Feasible Region
The feasible region bounded by constraints and non-negative axes $x \ge 0, y \ge 0$ has corner points:
$O(0, 0)$, $A(30, 0)$, $B(20, 30)$, and $C(0, 50)$.
Evaluate Objective Function Z = 4x + y at Corner Points
$$\begin{aligned} \text{At } O(0, 0): &\quad Z = 4(0) + 0 = 0 \\ \text{At } A(30, 0): &\quad Z = 4(30) + 0 = 120 \\ \text{At } B(20, 30): &\quad Z = 4(20) + 30 = 80 + 30 = 110 \\ \text{At } C(0, 50): &\quad Z = 4(0) + 50 = 50 \end{aligned}$$
The maximum value is $120$ occurring at point $A(30, 0)$.