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The point (x,y) lies on both conics x2+xy+x=81 and y2+xy+y=51. Given that x+y is positive, determine the value of x+y.

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    mmmh, been messing around this for a while

    from 1st:
    x(x + y + 1) = 81
    from 2nd:
    y(y + x + 1) = 51
    divide them:
    x/y = 81/51
    81y = 51x -----> y = 51x/81

    sub that into x^2 + xy + x = 81
    81x^2 + x(51x/81) + x = 81
    81x^2 + 51x^2 + 81x = 6561
    132x^2 + 81x - 6561 = 0
    x = (-81 ± √3470769)/ 264
    = (-81 ± 1863)/264 , which reduces to
    = 27/4 or -81/11

    ahhh, I guess we could have factored it, lol
    132x^2 + 81x - 6561 = 0
    44x^2 + 27x - 2187 = 0
    (4x - 27)(11x + 81) = 0

    anyway, too late,
    if x = 27/4 , from y = 51x/81 --> y = 17/4
    if x = -81/11, ------> y = -51/11

    for first point:
    x + y = 27/4 + 17/4 = 44
    for 2nd point:
    x+y = negative

    so for your condition:
    x+y = 44

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