Question

Consider the following linear programming problem:

Maximize	$5X+6Y$
Subject to:	$4X+2Y\le 420$
	$1X+2Y\le 120$
	all variables $\ge 0$

Which of the following points $(X,Y)$ is feasible?

• A. $(50,40)$
• B. $(30,50)$
• C. $(60,30)$
• D. $(90,20)$
• E. None of these

Answer 1

Answer: (C)

$(60,30)$

Feasible points are the points that satisfy the constraints.

Therefore, substitute the options in the constraint equations and verify.

$A.$ substituting $(50,40)$ in $4x+2y\le 420$

we get $4\times 50+2\times 40\le 420$

$⟹200+80\le 420⟹280\le 420$ True

substituting $(50,40)$ in $1x+2y\le 120$

we get $1\times 50+2\times 40\le 120$

$⟹50+80\le 120⟹130\le 120$ False

$B.$ substituting $(30,50)$ in $4x+2y\le 420$

we get $4\times 30+2\times 50\le 420$

$⟹120+100\le 420⟹220\le 420$ True

substituting $(30,50)$ in $1x+2y\le 120$

we get $1\times 30+2\times 50\le 120$

$⟹30+100\le 120⟹130\le 120$ False

$C.$ substituting $(60,30)$ in $4x+2y\le 420$

we get $4\times 60+2\times 30\le 420$

$⟹240+60\le 420⟹300\le 420$ True

substituting $(60,30)$ in $1x+2y\le 120$

we get $1\times 60+2\times 30\le 120$

$⟹60+60\le 120⟹120\le 120$ True

$A.$ substituting $(90,20)$ in $4x+2y\le 420$

we get $4\times 90+2\times 20\le 420$

$⟹360+40\le 420⟹400\le 420$ True

substituting $(90,20)$ in $1x+2y\le 120$

we get $1\times 90+2\times 20\le 120$

$⟹90+40\le 120⟹130\le 120$ False

Therefore option C (60,30) is the feasible point