Math grade 9 and up

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let s and w represent positive integers where x,y satisfy x/s+y/w=1 and s/x+w/y=4, find x+y in terms of s and w

It's really hard even for the teachers it took them a while. So I I was wondering how to solve it

  • Math grade 9 and up -

    x/s + y/w = 1
    s/x + w/y = 4
    You can use substitution.
    From the first equation,
    wx + sy = sw
    x = (sw - sy)/w
    Substitute to the second equation:
    s/x + w/y = 4
    sy + wx = 4xy
    sy + (w)((sw - sy)/w) = 4((sw - sy)/w)y
    sy + sw - sy = 4y(sw - sy)/w
    sw^2 = 4swy - 4sy^2
    4sy^2 - 4swy + sw^2 = 0
    s(4y^2 - 4wy + w^2) = 0
    Factoring,
    s(2y - w)(2y - w) = 0
    y = w/2
    Substituting this to x:
    y = w/2:
    x = (sw - sy)/w
    x = (sw - s(w/2))/w
    x = (sw - sw/2)/w
    x = (1/2)s
    x = s/2

    Thus, x + y = w/2 + s/2

    Hope this helps :)

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