Help with Algebra II? (Systems of Inequalities/Linear Programming)

Okay I made a second attempt that yielded a completely different graph with vertices at (0,0), (0, 150), (6, 75) and (8, 12.5)
 
Here's my graph (without the dotted lines for x=0 and y=0).

SaucyGraph.jpg


Try to figure out what you're doing wrongly.
 
Okay I think I see what I did wrong. I just had to readjust how I numbered the axes. So the 4 vertices are (0,0), (0, 150) and the other two are a bit harder to measure. On my graph they seem to be around (150,70) and (210, 0) but that may be a bit off
 
The intersection point of the two lines may be found by solving an equation for x, followed by substituting the solution into either of the linear equations.

-5/4x + 262.5 = -1/2x + 150

The vertex on the positive x-axis may be found by finding that line's x-intercept.
 
Okay so I solved that equation so one of the points is (150, 75) and (0, 210). Which completes the 4 vertices at (0,0), (0, 150) and the two listed. What's the next step if this is correct?
 
The next step is to answer the question posed in activity 1:

Select one solution of the system and determine how many peppers you will
have left over.

In other words, all of the points in that overlapping region represent solutions to the system of four inequalities.

Pick any one point, and use those x- and y- values as the number of bottles produced. How many peppers of each type are used, with that particular solution? The leftovers are the difference.
 
Okay so if I use (150, 75) and then plug it into the equation 5x + 4y <= 1050 then there will be 0 peppers left over. Is that all I have to do for that step?
 
There are two types of peppers involved, in this exercise. I think that the question in activity 1 is not asking specifically for the number of green peppers left over, but rather how many peppers of each type are left over.
 
Hmmm, hmmm, hmmm. Fresh Dover Sole, organic asparagus lightly steamed, and risotto with a quick, homemade mushroom and garlic sauce. So good!

Your exercise made me very hungry. In honor of you, I just sprinkled Try-Me brand Cajun Sunshine hot pepper sauce over the whole plate! Yum. :D
 
Okay so does that mean I have to apply the (150, 75) to both equations? In that case, there wouldnt be any peppers remaining in either equation. If i'm wrong, how do you answer the question in activity 1? And that sounds delicious :D. Lucky you!
 
prestico23 said:
Okay so does that mean I have to apply the (150, 75) to both equations?
Click to expand...

I would say "both expressions".

Remember? The expression 5x+4y gives the number of green peppers used. The expression 4x+8y gives the number of chili peppers used.

Yes, you just happened to pick a point where all of the peppers get used up.

If you had picked -- for example -- the solution point at (10,10), then there would be:

50+40 green peppers used, leaving 1050-90 left over

40+80 chili peppers used, leaving 1200-120 left over

There are many, many number of possibilities, for the answer to that question.


Next, activity 2 wants to talk about profit.

The objective is to maximize profit, yes? Hence, we need an objective function -- that is, we need a function that inputs the numbers of bottles produced (x and y) and outputs the profit realized when that many bottles are produced and sold.

Can you write a function of two variables that outputs the profit?
 
Honestly I don't think I can. We haven't really covered functions yet this year so my knowledge is limited to what I remember from Algebra I two years ago and I don't think we ever got to functions with multiple variables.
 
Oh, you have worked with such functions (at least once). You just don't realize it. :cool:

G(x,y) = 5x + 4y

C(x,y) = 4x + 8y

Function G inputs the numbers of bottles produced, and outputs the number of Green peppers used

Function C inputs the numbers of bottles produced, and outputs the number of Chili peppers used

In other words, you just used these functions unwittingly to answer the question in activity 1, even though we did not discuss those expressions in terms of function notation.


P(x,y) = ?

Function P inputs the numbers of bottles produced, and outputs the profit realized.


Re-read the info under activity 2, and take a guess at defining P(x,y).
 
P (x,y) = $1.20x + $1.00y? Lol kinda taking a shot in the dark but that's what I came up with
 
Sweat is running from my forehead down my left cheek, and my lips are burning.

LOL!


Okay, that's a very good guess.

We multiply the profit-per-bottle of Red Sauce by the number of bottles of Red Sauce; that gives the profit realized on that sauce.

Likewise, we multiply Scorchin' Sauce profit-per-bottle by the number of those bottles produced, too.

The two profits added together form the total profit from producing and selling both types.

Leave units off of numbers, when writing expressions/equations.

P(x,y) = 1.2x + y


As I mentioned earlier, with this type of linear-programming exercise, the ultimate solution always comes from coordinates located at a vertex of the region formed by solving the system of inequalities arising from the given constraints of the scenario.


Therefore, you need to substitute the (x,y) values at each vertex -- one pair at a time -- into the objective function P.

Then compare the four values of P(x,y) that you get, and the answer to both questions in activity 2 should be obvious.

What do you get?
 
Okay I plugged the 4 vertices of the equation into the function and the results were:

P (x,y) = 1.2 (0) + (o)

P (x,y) = 1.2 (0) + 150

P(x,y)= 1.2 (150) + 75

P(x,y)= 1.2 (0) + 210.

The third one leads to the highest profit so for activity 2, you should make 150 pints of the Scorching and 75 of the Red to bring a maximum profit of 255? Is that correct? Lol
 
prestico23 said:
Okay I plugged the 4 vertices of the equation into the function and the results were:

P (x,y) = 1.2 (0) + (o)

P (x,y) = 1.2 (0) + 150

P(x,y)= 1.2 (150) + 75

P(x,y)= 1.2 (0) + 210.

The third one leads to the highest profit ← Correct

you should make 150 pints of the Scorching and 75 of the Red ← Incorrect

to bring a maximum profit of 255? ← Correct
Click to expand...

Your mistake in misstating the answer above is an excellent reason for why you should always write down your definitions for symbols that you choose to use in word problems.

It's very easy to get symbols mixed up, when an exercise has so many steps.

If you always write down the definitions, then you have something to look at, when you need to remind yourself of the meaning for x and y.
 
Yeah I think I've completely lost track of what x and y are... I knew it the second I typed that answer too lol.
 
Learning math is a process of making mistakes, recognizing them, understanding and fixing them, and movin' on.

In summary, this type of maximization exercise is known as "linear programming".

It involves setting up a system of inequalities, and determining the solution region.

We finish by testing vertex coordinates of the system's solution region in some objective function.


Technically speaking, only the vertex (150,75) needed to be tested because the other three are not part of the solution region.


If you don't remember graphing inequalities, then you may not understand the reason why, but it has to do with the fact that x>0 and y>0 means only points in Quadrant I are possible solutions to the system of inequalities.

The axes are not a part of Quadrant I (or any other quadrant).

The origin, and points on the x-axis have y-value zero, and that's not greater than zero.

Likewise, the origin, and points on the y-axis have x-value zero, and that's not greater than zero.

Hence, the axes are excluded from the solution of the inequalities.


I wanted you to test all four vertices, anyway, because often these exercises involve x>=0 and y>=0 instead.


If you don't mind answering, I would like to know a few things.

How long has it been, since you did math, prior to enrolling in this course?

Is this an on-line course? If so, from where?


Cheers
 
Yeah it's an online course. Was it that obvious? Lol. It's my first year doing it online (at Laurel Springs). Last year I went to a regular high school and took a geometry course. The year before, I took Algebra I but I never did very well in it which might be why I'm having such a problem in Algebra II. Math has just never really been my thing.
 
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