Use graphical methods to solve the linear programming problem.
Maximize: $z= 4x+2y$
subject to : $x-y\le 7$
$19x+12y\le 228$
$18x+18y \le 324$
$x\ge 0,y\ge 0$
the max is ?? when x= ?? and y=???
my effort
$x-y\le 7$
when $x= 0 , y=0$,we get,$(0, -7) ,(7, 0)$
$19x+12y\le 228$
when $x= 0 , y=0$,we get $(12, 0) ,(0, 19)$
$18x+18y \le 324$
when$ x= 0 , y=0$, we get, $ (18, 0),(0, 18)$
We evaluate the objective function to maximize
at each corner point:
$z= 4x+2y$
$(0,0)\Rightarrow 0$
$(7, 0)\Rightarrow 28$
$(12, 0)\Rightarrow 48$
$(18, 0)\Rightarrow 72$
The maximum value of $z$ is $72$, and will occur at $(18,0)$, when $x=18$ and $y=0$
Please review my answer.
Best Answer
Try plotting the region using WolframAlpha
(The code looks like this:
RegionPlot[x <= 7 + y && 19 x + 12 y <= 228 && x + y <= 18, {x, -10, 10}, {y, -20, 30}])It has 4 vertices, figure out where is the extreme value of the objective function you are looking for...