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Math Help - finding linear equations for this word problem

  1. #1
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    Oct 2007
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    finding linear equations for this word problem

    I have to find a coefficient matrix to represent this word problem, but I can't for the life of me decide what to even make the variables...

    A paint store has to fill an order for 244 gallons of paint, .5 red paint, .25 blue and .25 green. They have large quantities of greenish paint( 12.5% red, 75% green, 12.5% blue), bluish( 20% red, 20% 60% blue), and redish (2/3 red, 1/6 green, and 1/6 blue).
    How many gallons of each must be mixed to fill the order.


    So I tried to set variables r + b + g = 244 for r = redish etc.
    but then I can't figure out the other equations.
    I don't think I can make the variables r, b, and g represent the colors themselves...

    can anyone figure this out? thanks a ton for any help
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  2. #2
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    Mar 2008
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    A Little Late

    Well, I'll respond to the question hoping it will help someone else in the future. The reason why you're having difficulty setting up the matrix is because you're too focused on the actual colors rather than the three mixes.

    We want a total of 244 gallons of paint with:

    50% red = 122 gallons of the initial 244
    25% green = 61 gallons of the initial 244
    25% blue = 61 gallons of the initial 244.

    Don't give the variable names to the colors (we know how much we want) but rather to mix 1, mix 2, and mix 3:

    MIXES RED GREEN BLUE
    x = mix 1 .125x .75x .125x
    y = mix 2 .2y .2y .6y
    z = mix 3 (2/3)z (1/6)z (1/6)z
    TOTAL 122 61 61

    Thus your matrix should be:
    Red: .125x + .2y + (2/3)z = 122
    Green: .75x + .2y + (1/6)z = 61
    Blue: .125x + .6y + (1/6) = 61

    .125 .2 2/3 | 122
    .75 .2 1/6 | 61
    .125 .6 1/6 | 61

    Solving for this gives: 32 gallons of mix 1; 50 gallons of mix 2, and 162 gallons of mix 3.
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  3. #3
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    Oct 2007
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    definitely on time enough to help me. the lab isn't due till midnight, thanks
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