Problem Statement
Maximize $Z = x + 2y$ subject to $2x + y \le 10$, $x + 2y \le 8$, and $x, y \ge 0$.
Verified Solution & Marking Scheme
Determine Boundary Line Intercepts
Line 1: $2x + y = 10 \implies (5, 0)$ and $(0, 10)$.
Line 2: $x + 2y = 8 \implies (8, 0)$ and $(0, 4)$.
Find Intersection of Constraint Boundaries
$2x + y = 10 \implies y = 10 - 2x$
Substitute into line 2: $x + 2(10 - 2x) = 8 \implies x + 20 - 4x = 8 \implies -3x = -12 \implies x = 4$.
$y = 10 - 2(4) = 2$.
Intersection point is $(4, 2)$.
Evaluate Objective Function at All Vertices
- At $(0, 0)$: $Z = 0$
- At $(5, 0)$: $Z = 5 + 0 = 5$
- At $(0, 4)$: $Z = 0 + 2(4) = 8$
- At $(4, 2)$: $Z = 4 + 2(2) = 4 + 4 = 8$
Maximum value is $8$, occurring at all points on the line segment joining $(0, 4)$ and $(4, 2)$.