Need help with Pre-AP Algebra II Linear Programming

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Drtwl

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Given the following problem:

Daniel, Kendall and Caitlin are going to three different water parks. Splash Mountain costs $15 per person, typhoon lagoon costs $25 per person and Blizzard beach costs $22 per person. With all their money put together they have $490. They go to the parks 23 times in all. If they go to splash mountain two more times than they go to blizzard beach, how many times do they go to each park?


Equasions I developed


45x + 75y + 66z = 490


x - z = 2


x + y + z = 23
 
Drtwl said:
Daniel, Kendall and Caitlin are going to three different water parks. Splash Mountain costs $15 per person, typhoon lagoon costs $25 per person and Blizzard beach costs $22 per person. With all their money put together they have $490. They go to the parks 23 times in all. If they go to splash mountain two more times than they go to blizzard beach, how many times do they go to each park?


Equasions I developed


45x + 75y + 66z = 490

x - z = 2

x + y + z = 23
Click to expand...

It's always good form to explicitly define symbols that you pick. In other words, don't forget to state the meaning of your symbols x, y, and z at the very beginning.

I don't view this exercise as linear programming. Linear programming is a method to minimize or maximize a quantity, based on a system of inequalities.

This exercise seems to require solving a system of equalities, instead.

Your equation involving only symbols x and z may be solved for z. That will give you an expression that may be substituted for z in the other two equations. Doing this creates a new system of two equations in x and y.

What methods have you learned for solving a system of two equations for two variables?

Cheers :cool:
 
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I just solved your system, and I get a negative value for y.

Please double-check your post to ensure that you have accurately given us the complete exercise as worded.

Cheers :cool:
 
mmm4444bot said:
I just solved your system, and I get a negative value for y.

Please double-check your post to ensure that you have accurately given us the complete exercise as worded.

Cheers :cool:
Click to expand...

Linear equations and substitution for both.
That is the exact problem. I can't find an answer, so are you saying that this is impossible. I can't get 23 total visits for $490. I found x=2 y=14 and z=4 which is way more than $490.
 
Yes -- the exercise seems goofed up. It is not properly worded for a linear-programming setup.

Their cash limit is $490, so:

45x + 75y + 66z ≤ 490

Substituting z=x-2 gives:

y ≤ -(37/25)x + 622/75

In other words, (x,y) solutions need to be x<5 and y<8.

The exercise not make sense, to me. :cool:
 
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Hello, Drtwl!

Your first equation is incorrect.

Daniel, Kendall and Caitlin are going to three different water parks.
Splash Mountain costs $15 per person, Typhoon Lagoon costs $25 per person, Blizzard Beach costs $22 per person.
They go to the parks 23 times in all and spend $490.
If they go to Splash Mountain two more times than Blizzard Beach, how many times do they go to each park?
Click to expand...

Name your variables:

\(\displaystyle S\) = number of visits to Splash Mountain.
\(\displaystyle T\) = number of visits to Typhoon Lagoon.
\(\displaystyle B\) = number of visits to Blizzard Beach.


23 visits: .\(\displaystyle S + T + B \:=\:23\)

Cost $490: .\(\displaystyle 15S + 25T + 22B \:=\:490\)

. . . . \(\displaystyle S \:=\:B+2 \quad\Rightarrow\quad S - B \:=\:2\)


You can solve the system of equations by Gaussian elimination:

We are given: .\(\displaystyle \left|\begin{array}{ccc|c} 1 & 1 & 1 & 23 \\ 15 & 25 & 22 & 490 \\ 1 & 0 & \text{-}1 & 2 \end{array}\right|\)


If you are careful, you will get: .\(\displaystyle \left|\begin{array}{ccc|c}1&0&0&7 \\ 0&1&0&11 \\ 0&0&1&5 \end{array}\right|\)

Therefore: .\(\displaystyle \begin{Bmatrix}S &=& 7 \\ T &=& 11 \\ B &=& 5\end{Bmatrix}\)
 
Soroban, how do you interpret the givens? Are you treating "they" as a group? The question asks, "How many times do they go to each park?"
 
Drtwl said:
Given the following problem:

Daniel, Kendall and Caitlin are going to three different water parks. Splash Mountain costs $15 per person, typhoon lagoon costs $25 per person and Blizzard beach costs $22 per person. With all their money put together they have $490. They go to the parks 23 times in all. If they go to splash mountain two more times than they go to blizzard beach, how many times do they go to each park?


Equasions I developed


45x + 75y + 66z = 490


x - z = 2


x + y + z = 23
Click to expand...

If all three of them go to park together - then they made (23*3 =) 69 person.park visit. That will at least cost (69 * 15= ) $1035 - no way!!!!
 
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Soroban's set-up suggests a scenario where the question would be "how many total trips were made to each park, regardless of who went where and with whom (per trip)?"
 
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I would to tell my daughter's teacher that however she is only a sophomore and will have this teacher again for advanced algebra.
 
This is why - whenever I make up a "question", I ask my colleagues to solve it and critique it
 
Drtwl said:
I would [like] to tell my daughter's teacher that however she is only a sophomore and will have this teacher again for advanced algebra.
Click to expand...

That's okay; we're kidding, of course. I'm not surprised that your daughter did not understand the exercise.

I think something like the following wording would have been better.

Daniel visited three different water parks, making at least one trip to each park. Splash Mountain costs $15 per person, Typhoon Lagoon costs $25 per person, and Blizzard Beach costs $22 per person. Daniel made 23 trips, and he spent a total of $490. If Daniel went to Splash Mountain two more times than he went to Blizzard Beach, how many times did he visit each park?
Click to expand...
 
I totally agree with the rewording. Ten weeks into the year and this is the first time we have had a question. It's been 30 years since I had advanced algebra, but I do enjoy the challenges this brings. Thanks to all of you and I now know where to turn to if I/we have any questions in the next years.
 
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