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Math Help - Simultaneous equations - 3 variables

  1. #1
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    Question Simultaneous equations - 3 variables

    Hi!

    I've got myself into this rather unhelpful question. I have tried many different ways to solve it, but all of them have turned out to fail miserably .The question is:

    Solve,
    xy+5(x+y)=47
    yz+5(y+z)=65
    zx+5(z+x)=55

    I thought it would be reasonably simple to solve, however, (for me at least!) it is harder than it looks!

    Could anybody help me a little here?
    Thanks very much!
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  2. #2
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    Re: Simultaneous equations - 3 variables

    2 integer solutions:
    a = 3, b = 4, c = 5
    a = -13, b = -14, c = -15
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  3. #3
    MHF Contributor Reckoner's Avatar
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    Re: Simultaneous equations - 3 variables

    xy+5(x+y)=47 (1)
    yz+5(y+z)=65 (2)
    zx+5(z+x)=55 (3)

    Solving (1) for x yields

    x=\frac{47-5y}{y+5} (4)

    And solving (2) for z,

    z=\frac{65-5y}{y+5} (5)

    Then, substituting (4) and (5) into (3):

    \left(\frac{65-5y}{y+5}\right)\left(\frac{47-5y}{y+5}\right)+5\left(\frac{65-5y}{y+5}+\frac{47-5y}{y+5}\right)=55

    \Rightarrow\frac{5\left(5y^2-112y+611\right)}{(y+5)^2}+\frac{10(56-5y)}{y+5}=55

    \Rightarrow5\left(5y^2-112y+611\right)+10(56-5y)(y+5)=55(y+5)^2

    \Rightarrow80y^2+800y-4480=0

    \Rightarrow80(y+14)(y-4)=0

    \Rightarrow y=-14\mathrm{\quad or\quad}y=4

    And substituting both of these into (4) and (5) gives us x and z.
    Thanks from BobtheBob
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  4. #4
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    Re: Simultaneous equations - 3 variables

    Oh right thanks! I must have made a mistake when I did a similar thing.

    Thanks very much!
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