Math grade 9 and up
posted by Xein on .
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

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 :)