Question

Given the system of constraints, name all vertices of the feasible region. Then find the maximum value of the given objective function.

Answer 1

https://www.desmos.com/calculator/iakkh60qtm

Answer 2

By the way, what is the given objective function?

Answer 3

C=6x-4y
I had the desmo calc open but I couldn't figure out how to use it haha.

Answer 4

What do the bottom 4 mean?

Answer 5

Those are just line segments I graphed that correspond with specific lengths along the x or y axis.

Answer 6

I see, so I just use the points to find the maximum. I have another question I am almost finished with, can you check it?

Answer 7

Yes, you can use the points to find the maximum. I can check your other work.

Answer 8

k, give me a minute to finish it.

Answer 9

This is the question:
Use the elimination method.
-2x+2y+3z=0
-2x-y+z=-3
2x+3y+3z=5

I added:
-2x-y+z=-3
+ 2x+3y+3z=5

and got 2y+4z=2

I got stuck here. I think I am supposed to multiply one of the equations so I can eliminate another variable, but Im not sure how

Answer 10

Using equations 2 and 3 you eliminated x.
Use equations 1 and 2 and eliminate x again.

Answer 11

I usually eliminate z variable.

Answer 12

Don't I have to multiply one or both the equations by a number so I get additive inverses?

Answer 13

That way, I'm left with a two equation system in terms of x and y.

Answer 14

From there, it is easy.

Answer 15

But in this case, it is probably wise to eliminate x variable.

Answer 16

So, to do what ranga said, I have to multiply the 1st and 2nd equation by a number to get inverses? and then I can add them to eliminate

Answer 17

You are allowed to subtract equations too. You can subtract equation 2 from equation 1 to get rid of x.

Answer 18

Or think of it as multiplying equation 2 by -1 and adding it to equation 1.

Answer 19

alright, let me do it and I'll tell you what I got.

Answer 20

okay, btw, the first one you got 2y+4z=2 can be simplified further by dividing by 2.

Answer 21

I haven't learned to do that yet. I think I am just supposed to do it this way.
I got 2x+1-z=3

Answer 22

and now I am going to add the equations together

Answer 23

it should be 2x + y - z = 3

Answer 24

That's what I meant haha sorry
I got:
3y+2z=3

Answer 25

Yes. You got an earlier equation with just y and z and now you have a second one. You can eliminate z from both.

Answer 26

So, do I add the two new equations and solve for z?

Answer 27

Write down your two equations here first.

Answer 28

2y+4z=2
3y+2z=3

Answer 29

Yes. Multiply the second by -2 and add it to the first.

Answer 30

When I multiplied the 2nd eq by -2 I got:
-6y-4z=-6

Answer 31

Yes. Add it to first to get rid of z.

Answer 32

I got -4y=-4 and simplified it to y=1

Answer 33

Yes. Put y = 1 in any one of the two equations and find z.

Answer 34

into one of the original equations, right?

Answer 35

No the two we were just using that had only y and z variables.

Answer 36

2y+4z=2 or
3y+2z=3

Answer 37

I got 4z=0 and if I use the other I get 2z=0
idk what I should do

Answer 38

Either one means z is 0. Zero multiplied by 2 or 4 is still 0.

Answer 39

oh, ok. and then I plug in z into an original eq?

Answer 40

So you have y = 1, z = 0. Pick any one of the original 3 equations and substitute the values for y and z and then solve for x.

Answer 41

I got x=1

Answer 42

so the solution is (1,1,0)

Answer 43

Correct. I always put the values in all three original equations to see if it satisfies all three equations. This is one way to double check that we did not make any calculation mistake anywhere.

Answer 44

k, let me check

Answer 45

they are all correct. thank you so much.

Answer 46

you are welcome