Question

Minimize \( Z=5 x+8 y \) Subject to the constraints : \[ \begin{array}{r} 2 x+y \geq 8 \\ 6 x+y \geq 12 \\ x+3 y \geq 9 \\ x \geq 0 \\ y \geq 0 \end{array} \]

Answer 1

\(z = 5x + 8y\)

s.t. 

\(2x +y \ge 8\\6x+y\ge 12\\x +3y\ge 9\\x\ge 0\\y\ge 0\)

Corner points are A(0, 12), B(1, 6), C(3, 2) & D(9, 0).

Corner points	z = 5x + 8y
A(0, 12)	96 (Maximum)
B(1, 6)	53
C(3, 2)	31 (Minimum)
D(9, 0)	45

∵ Region is unbounded

∴ We have to draw 5x + 8y = 31 with given graph.

Since, line does not crossing from the shaded area.

∴ We obtained minima at x = 3 & y = 2 and minimum value is 31.