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Math Help - General and Particular Solutions

  1. #1
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    General and Particular Solutions

    Hi all, I think I'm in the correct section to post this. I'm here for some help on a specific problem. The problem is asking me to find "a general solution and 3 particular solutions" to the problem. However, I have no idea what a general or particular solution is. And my professor nor my math book have made any mention of either So if anyone could explain that it would be much appreciated.

    Here's the actual problem:

    "California coffee roasters sells Kona coffee for $22.95 per pound and Columbian coffee for $6.75 per pound. Suppose they offer a blend of the two coffees for $10.80 per pound. What amounts of Kona and Colombian coffee's should be blended to obtain the desired mixture?"

    The problem to go along with it is:

    "Is there sufficient data to determine a unique solution?"

    I'm pretty sure there is not.

    "If no, find a general solution to the problem and write 3 particular solutions from it."

    And this is where I'm stuck. I'm not looking for someone to come and answer the problem for me, I just don't know what it is I'm supposed to find.
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  2. #2
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    Quote Originally Posted by qckdrw777 View Post
    Hi all, I think I'm in the correct section to post this. I'm here for some help on a specific problem. The problem is asking me to find "a general solution and 3 particular solutions" to the problem. However, I have no idea what a general or particular solution is. And my professor nor my math book have made any mention of either So if anyone could explain that it would be much appreciated.

    Here's the actual problem:

    "California coffee roasters sells Kona coffee for $22.95 per pound and Columbian coffee for $6.75 per pound. Suppose they offer a blend of the two coffees for $10.80 per pound. What amounts of Kona and Colombian coffee's should be blended to obtain the desired mixture?"

    The problem to go along with it is:

    "Is there sufficient data to determine a unique solution?"

    I'm pretty sure there is not.

    "If no, find a general solution to the problem and write 3 particular solutions from it."

    And this is where I'm stuck. I'm not looking for someone to come and answer the problem for me, I just don't know what it is I'm supposed to find.
    Answer by analogy.

    Suppose we're asked to solve the equation x/y = 1/2 where x and y are integers. Then the set of solutions is infinite. General solution could be given as: (x,y) = (k, 2k) where k ranges over all the nonzero integers. Some particular solutions are (1,2), (2,4), and (3,6).
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  3. #3
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    So In my situation, would

    22.95x + 6.75y = 10.80 be the general solution?

    And for my particulars I'd have to find values for x and y to satisfy the equation?

    Or am I looking at this the wrong way?
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  4. #4
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    Quote Originally Posted by qckdrw777 View Post
    So In my situation, would

    22.95x + 6.75y = 10.80 be the general solution?

    And for my particulars I'd have to find values for x and y to satisfy the equation?

    Or am I looking at this the wrong way?
    I would look at it this way.

    Suppose we want to make 1 pound of the blend using some amount x of Kona coffee and some amount y of Colombian. If you think about it you should realize that this uniquely determines how much of each type you must use. (But see notes below.) So you are missing an equation in your setup, should be the system of equations

    x + y = 1
    22.95x + 6.75y = 10.80

    Solve for x and y, and then notice that it's not necessary to make 1 pound of blend; we could just as easily make 2 pounds of blend, etc. These give us particular solutions. The general solution will specify the ratio between x and y.
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  5. #5
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    I think I get it now. Those equations work out to make

    x = .25, y = .75

    So my general equation would be something like

    (x,y) = (k, 3k)

    And for my particular solutions I can plug in any numbers like (1, 3), (3,9), (2,6), etc?

    And those would be particular solutions for how much coffee to mix with how much coffee.
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  6. #6
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    Quote Originally Posted by qckdrw777 View Post
    I think I get it now. Those equations work out to make

    x = .25, y = .75

    So my general equation would be something like

    (x,y) = (k, 3k)

    And for my particular solutions I can plug in any numbers like (1, 3), (3,9), (2,6), etc?

    And those would be particular solutions for how much coffee to mix with how much coffee.
    Since x and y aren't required to be integers, you can just say that for any arbitrary (nonzero) amount x of Kona coffee, mix it with 3x of Colombian (save some letters!). But yes, your solution is good.

    By the way, even though there is a system of linear equations here, I wouldn't consider this university level linear algebra. I'm marking this for mods to move to the Pre-Algebra and Algebra subforum.
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  7. #7
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    Quote Originally Posted by undefined View Post
    Since x and y aren't required to be integers, you can just say that for any arbitrary (nonzero) amount x of Kona coffee, mix it with 3x of Colombian (save some letters!). But yes, your solution is good.

    By the way, even though there is a system of linear equations here, I wouldn't consider this university level linear algebra. I'm marking this for mods to move to the Pre-Algebra and Algebra subforum.
    Oops, sorry. I didn't know quite which forum to post in. It's from a math 132 course at my university (aka Finite Mathmatics) so I thought it would go under the college level mathematics.

    My mistake!
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