Question

1. Rewrite the constraints in slope-intercept form.


2. List the values of the objective function for each vertex

Answer 1

Slope intercept form is \(y = mx + b\)
So you just have to solve for y.

\[-3x + 2y \leq 8\]\[2y \leq 8 + 3x\]\[y \leq 4 + \frac{3}{2}x\]
You can probably do the other one.

Answer 2

ok thanks @SACAPUNTAS but when it says list the values of the objective function for each vertex what does that mean?

Answer 3

It means after you've found the coordinates of each vertex you plug its values into the objective function to get a value of P.

Answer 4

For example if there is a vertex at (1, 2)
(I'm not saying there is, just, as an example)
You'd plug it in with x = 1 and y = 2 so \(P = 13(1) + 2(2)\)

Answer 5

WHAT? im so lost

Answer 6

Sorry. Do you get how linear programming works? I can start at the beginning!

Answer 7

i do not understand linear prog. we are studying it but idk wats going on

Answer 8

Ok. Do you have the two inequalities in slope intercept form?

Answer 9

no just 1

Answer 10

Ok. Well, first we get the other one. That's just solving for y. :D

Answer 11

y≤4+32x is one and hold on let me find the other one

Answer 12

3/2, not 32. But yes.

Answer 13

sorry i copied n pasted that

Answer 14

\[y \le 4+\frac{ 3 }{ 2 }x\] is the other one

Answer 15

That's the same one.
The other one is
\[-8x + y \geq -48\]So it's actually really simple to solve it for y. :P

Answer 16

ok −8x+y≥−48 changes to y≥8x-48 right?

Answer 17

Right.

Answer 18

so now what?

Answer 19

So, we can graph these inequalities. They want you to do it on a graphing calculator I guess but on here I'll let Wolfram Alpha do it.
http://www.wolframalpha.com/input/?i=graph+y+%3C%3D+4+%2B+%283%2F2%29x+and+y+%3E%3D+-48+%2B+8x+and+x+%3E%3D+0+and+y+%3E%3D+0

Do you see how since they are inequalities (and not equations) each one has a region associated with it, like, it's not just a line. And by graphing them all at the same time, we constrain a specific area?

Answer 20

yes i see that.

Answer 21

Ok. So basically the idea of linear programming is that once we've constrained the specific area, we have to test whatever function we're trying to optimize at each of the vertices of that area.

Answer 22

For example, (0,0) is one of our vertices.
So we can plug x = 0, y = 0 into our objective function.
\[P = 13(0) + 2(0) = 0\](That one's simple)

Answer 23

Essentially, you're making a shape with the inequalities and then test your objective function at the coordinates of each of its corners.

Answer 24

ohhhhhh so we just plug in the vertices to P=13x+2y?

Answer 25

Right! And then you pick the vertex with the highest value for P, since you are trying to maximize.

Answer 26

ahhhhhhh thanks sooooooooooo much i understand it now

Answer 27

Well, that's what you'd do in real linear programming.
They just want you to list the P for every vertex.

Answer 28

Good. :D