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[@SanePerson1]

Square equation 2 and subtract equation 1 from the result, then divide that by 2 to obtain xy + (x + y)√(xy) = 80. Now do a substitution u = √(xy) and v = x + y ⇒ u² + vu = u(u + v) = 80 and equation 1 is just u + v = 16. Therefore, u = √(xy) = 5 or xy = 25. Putting that into equation 1, we get x² + y² = 71 (= 96 - 25). Now the problem can be seen as finding the intersection points between a circle of radius √71 and the hyperbola xy = 25. I find it often helps to set problems up graphically as well as algebraically, You can also see the symmetry in the result both ways, you can permute x and y in the solutions because both the circle and the hyperbola have mirror symmetry across the x = y line and, of course, you can also permute x and y in the original equations without altering them.

[@timofej8507]

x+y = 16 - sqrt(xy) => x*x + xy + y*y = 256 - 32*sqrt(xy) => sqrt(xy) = 160/32 = 5 and so on

[@giuseppemalaguti435]

1)..(x+y)^2=96+xy..2)..(x+y)^2=(16-√xy)^2=256+xy-32√xy...32√xy=256-96=160...xy=25. dalla1) (x+y)^2=96+25=121...(x+y)=11..-11...poi è semplice..x1=(11+√21)/2..y1=(11-√21)/2..X2,y2.viceversa..poi dovrei verificare anche il -11

[@hakanerci4372]

x+y=11 y=11-x, x2+y2=71 x2+ (11-x)2=71, x2-x2+22x+121=71 22x=-50 x=-50/22 , x+y= 11 -50/22+y=11 y= 292/22 (x=-25/11, y= 146/11)
x*y not equal with 25 but where is my fault?

[@davidmalerich8469]

You've got a couple signs reversed. Should be x2 + x2 - 22x + 121 = 71.

[@hakanerci4372]

@@davidmalerich8469 👍

[@cdy4901]

Can you write ²