Question

graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.
{6<= x + y <= 13
{x >=3
{y >=1
Maximum for
P=4x + 3y

Answer 1

Do you know how to graph these inequalities?

Answer 2

No I have no clue :-(

Answer 3

well the first one is a bit strange, but it means that the line y = -x is between 6 and 13 on the y axis (basically have the line y = -x+6 and y = -x+13 and shade everything between them)

Answer 4

x >= 3 means you have a vertical line at x = 3 and you shade to the right

Answer 5

y >= 1 means you have a horizontal line at y = 1 and you shade above this line

Answer 6

ok That makes sense so far.

Answer 7

all 4 regions will overlap to form a new region

Answer 8

the corners of this region are potential max/min points

so you must find each corner point and you must test each one in P=4x + 3y to see which point is the max

Answer 9

how do I find each corner point?

Answer 10

each corner is created by 2 lines intersecting

Answer 11

so you must solve a system of equations (2 equations 2 unknowns) to find each corner point

Answer 12

for instance, solving the system

-x - y = -6
y = 1

gives you the corner point (5,1)

Answer 13

oh ok

Answer 14

you should get 4 corner (or vertex) points

Answer 15

tell me which vertex points you get

Answer 16

ok still trying to figure it out

Answer 17

that's ok

Answer 18

would one of the corners be (3,1)

Answer 19

no

Answer 20

that's one intersection, but it's not in the shaded region

Answer 21

(3, -9)

Answer 22

no

Answer 23

here's what the shaded region looks like

Answer 24

I dont know why I have such a problem grasping this. I have never been good at graphs and such

Answer 25

you'll get better with more practice

Answer 26

So from this graph, we see that the vertex points are


(3, 3), (5, 1), (3, 10), (12, 1)

Answer 27

ok

Answer 28

From here, plug each vertex point into the objective function P = 4x+3y to find the min and max


Plug in (3,3)


P = 4x+3y


P = 4(3)+3(3)


P = 21


---------------------------------------------

Plug in (5, 1)


P = 4x+3y


P = 4(5)+3(1)


P = 23


---------------------------------------------

Plug in (3,10)


P = 4x+3y


P = 4(3)+3(10)


P = 42


---------------------------------------------

Plug in (12, 1)


P = 4x+3y


P = 4(12)+3(1)


P = 51


---------------------------------------------

Summary


Min is P = 21 which occurs at (3,3)


Max is P = 51 which occurs at (12,1)

Answer 29

Ohh okay. That part makes sense! Thank you so much! Now I just hope I can get that graphing part down for good, :-)

Answer 30

you will, just spend more time doing it and you'll get better

Answer 31

I would recommend practicing graphing linear equations. The inequality part is just the shading (and how the line is drawn -- either solid or dashed).

Answer 32

Okay. I will definitely do that. Thanks for all your help.

Answer 33

yw