# 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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